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"COORD_OP_METHOD_CODE","COORD_OP_METHOD_NAME","REVERSE_OP","FORMULA","EXAMPLE","REMARKS","INFORMATION_SOURCE","DATA_SOURCE","REVISION_DATE","CHANGE_ID","DEPRECATED"
1024,"Popular Visualisation Pseudo Mercator",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
This method is utilised by some popular web mapping and visualisation applications. It applies standard Mercator (Spherical) formulas (method code 1026) to ellipsoidal coordinates and the sphere radius is taken to be the semi-major axis of the ellipsoid. This approach only approximates to the more rigorous application of ellipsoidal formulas to ellipsoidal coordinates (as given in EPSG dataset coordinate operation method codes 9804 and 9805). Unlike either the spherical or ellipsoidal Mercator projection methods, this method is not conformal: scale factor varies as a function of azimuth, which creates angular distortion. Despite angular distortion there is no convergence in the meridian.
The formulas to derive projected Easting and Northing coordinates from ellipsoidal latitude (lat) and longitude (lon) first derive the radius of the sphere (R) from:
R = a
Then applying spherical Mercator formulae:
E = FE + R(lon - lonO)
N = FN + R ln[tan(pi/4 + lat/2)]
where FE and FN are false easting and false nothing at the projection origin, other symbols are as listed above and logarithms are natural.
If latitude lat = 90?, N is infinite. The above formula for N will fail near to the pole, and should not be used poleward of 88?.
The reverse formulas to derive latitude and longitude on the sphere from E and N values are:
D = -(N-FN)/R = (FN-N)/R
lat = pi/2 - 2 atan(e^D) where e=base of natural logarithms, 2.7182818...
lon = [(E - FE)/R] + lonO
If q_alpha is the scale factor at a given azimuth alpha, it is a function of R', the radius of curvature at that azimuth derived from:
R' = rho nu / (nu cos^2alpha + rho sin^2alpha)
q_alpha = R / (R' cos lat)
where rho and nu are the radii of curvature of the ellipsoid at latitude lat in the plane of the meridian and perpendicular to the meridian respectively;
rho = a(1 - e^2)/(1 - e^2 sin^2(lat))^3/2
nu = a /(1 - e^2 sin^2(lat))^1/2
Then when the azimuth is 0?, 180?, 90? or 270? the scale factors in the meridian (h) and on the parallel (k) are:
q_0 = q_180 = h = R / (rho cos(lat))
q_90 = q_270 = k = R / (nu cos(lat))
which demonstrates the non-conformallity of the Pseudo Mercator method.
Maximum angular distortion omega is a function of latitude and is found from:
omega = 2 asin{[ABS(h - k)] / (h + k)}","For Projected Coordinate Reference System: WGS 84 / Pseudo-Mercator
Parameters:
Ellipsoid: WGS 84 a = 6378137.0 metres 1/f = 298.2572236
Latitude of natural origin (latO) = 0?00'00.000""N = 0.0 rad
Longitude of natural origin (lonO) = 0?00'00.000""E = 0.0 rad
False easting (FE) = 0.00 metres
False northing (FN) = 0.00 metres
Forward calculation for the same coordinate values as used for the Mercator (1SP) (Spherical) example (method code 9841):
Latitude (lat) = 24?22'54.433""N = 0.425542460 rad
Longitude (lon) = 100?20'00.000""W = -1.751147016 rad
R = 6378137.0
whence
E = -11 169 055.58 m
N = 2 800 000.00 m
and
h = 1.1034264
k = 1.0972914
omega = 0?19'10.01""
Reverse calculation for a point 10km north on the grid (-11 169 055.58 m E, 2 810 000.00m N) first gives:
D = -0.44056752
Then Latitude (lat) = 0.426970023 rad = 24?27'48.889""N
Longitude (lon) = -1.751147016 rad = 100?20'00.000""W","Applies spherical formulas to the ellipsoid. As such does not have the properties of a true Mercator projection.","OGP Guidance Note 7-2","IOGP",9/22/2017 0:00:00,"2008.114 2009.023 2017.030",0
1025,"Geographic3D to GravityRelatedHeight (EGM2008)",0,"This transformation involves the application of a geoid-ellipsoid separation value interpolated from a geoid model. The model provides separation values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the separation is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the geoid-ellipsoid separation (N) above the ellipsoid of the source Geographic 3D CRS.
Then:
H = h - N
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","For earlier EGM84 and EGM96 models see Geographic3D to GravityRelatedHeight (EGM), method code 9661.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",12/15/2016 0:00:00,"2008.097 2016.050",0
1026,"Mercator (Spherical)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The formulas to derive projected Easting and Northing coordinates from spherical latitude lat and longitude lon are:
E = FE R (lon - lonO)
N = FN R ln[tan(pi/4 lat/2)]
where lonO is the longitude of natural origin and FE and FN are false easting and false nothing.
R is the radius of the sphere and will normally be one of the CRS parameters. If the figure of the earth used is an ellipsoid rather than a sphere then R should be calculated as the radius of the conformal sphere at the projection origin at latitude latO using the formula for Rc given in EPSG Guidance Note 7-2, section 1.2, table 3. Note however that if applying spherical formula to ellipsoidal coordinates, the projection properties are not preserved.
If latitude lat = 90?, N is infinite. The above formula for N will fail near to the pole, and should not be used poleward of 88?.
The reverse formulas to derive latitude and longitude on the sphere from E and N values are:
D = -(N - FN)/R = (FN - N)/R
lat = pi/2 - 2 atan(e^D) where e=base of natural logarithms, 2.7182818...
lon = [(E - FE)/R] + lonO
Note that in these formulas, the parameter latitude of natural origin (*O) is not used. However for the Merctor (Spherical) method, for completeness in CRS labelling the EPSG dataset includes this parameter, which must have a value of zero.","For Projected Coordinate Reference System: World Spherical Mercator (Note: CRS not in EPSG dataset)
Parameters:
Sphere: R = 6371007.0 metres
Latitude of natural origin (latO) = 0?00'00.000""N = 0.0 rad
Longitude of natural origin (lonO) = 0?00'00.000""E = 0.0 rad
False easting (FE) = 0.00 metres
False northing (FN) = 0.00 metres
Forward calculation for:
Latitude (lat) = 24?22'54.433""N = 0.425542460 rad
Longitude (lon) = 100?20'00.000""W = -1.751147016 rad
whence
E = -11 156 569.90 m
N = 2 796 869.94 m
Reverse calculation for the same point (-11 156 569.90 m E, 2 796 869.94m N) first gives:
D = -0.438999665
Then Latitude (lat) = 0.425542460 rad = 24?22'54.433""N
Longitude (lon) = -1.751147016 rad = 100?20'00.000""W",,"EPSG guidance note #7-2, http://www.epsg.org","OGP",6/7/2009 0:00:00,"2008.114 2009.023",0
1027,"Lambert Azimuthal Equal Area (Spherical)",1,"See information source.
R is the radius of the sphere and will normally be one of the CRS parameters. If the figure of the earth used is an ellipsoid rather than a sphere then R should be calculated as the radius of the authalic sphere using the formula for RA given in EPSG Guidance Note 7-2, section 1.2, table 3. Note however that if applying spherical formula to ellipsoidal coordinates, the authalic projection properties are not preserved.","See information source.","This is the spherical form of the projection. See coordinate operation method Lambert Azimuthal Equal Area (code 9820) for ellipsoidal form. Differences of several tens of metres result from comparison of the two methods.","USGS Professional Paper 1395, ""Map Projections - A Working Manual"" by John P. Snyder.","OGP",6/7/2009 0:00:00,"2009.007 2009.023",0
1028,"Equidistant Cylindrical",1,"See EPSG Guidance Note #7-2.","See EPSG Guidance Note #7-2.","See method code 1029 for spherical development. See also Pseudo Plate Carree, method code 9825.","ESRI","OGP",6/7/2009 0:00:00,"2009.023",0
1029,"Equidistant Cylindrical (Spherical)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
This method has one of the simplest formulas available. If the latitude of natural origin (lat1) is at the equator the method is also known as Plate Carr?e. It is not used for rigorous topographic mapping because its distortion characteristics are unsuitable. Formulas are included to distinguish this map projection method from an approach sometimes mistakenly called by the same name and used for simple computer display of geographic coordinates - see Pseudo Plate Carr?e (coordinate operation method code 9825).
For the forward calculation:
E = FE + R . (lon - lonO) . cos(lat1)
N = FN + R . lat
where lat1, lonO, lat and lon are expressed in radians.
R is the radius of the sphere and will normally be one of the CRS parameters. If the figure of the earth used is an ellipsoid rather than a sphere then R should be calculated as the radius of the conformal sphere at the projection origin at latitude lat1 using the formula for RC given in EPSG Guidance Note 7-2, section 1.2, table 3. Note however that if applying spherical formula to ellipsoidal coordinates, the equidistant projection properties are not preserved.
For the reverse calculation:
lat = (N - FN)/ R
lon = lonO + ([E - FE] / R cos(lat1))
where R is as for the forward method.","See information source.","See method code 1028 for ellipsoidal development. If the latitude of natural origin is at the equator, also known as Plate Carr?e. See also Pseudo Plate Carree, method code 9825.","US Geological Survey Professional Paper 1395; ""Map Projections - A Working Manual""; J. Snyder.","OGP",6/7/2009 0:00:00,"2009.023",0
1030,"Geographic3D to GravityRelatedHeight (NZgeoid)",0,"This transformation involves the application of a geoid-ellipsoid separation value interpolated from a quasigeoid model. The model provides separation values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the separation is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the geoid-ellipsoid separation (N) above the ellipsoid of the source Geographic 3D CRS.
Then:
H = h - N
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","EPSG initially gave this method the name ""Geographic3D to GravityRelatedHeight (NZgeoid2009)"". As the same file format was retained for the 2016 geoid, date removed from the method name.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",12/15/2016 0:00:00,"2009.081 2016.031",0
1031,"Geocentric translations (geocentric domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Xt = Xs + dX; Yt = Ys + dY; Zt = Zs + dZ","Given a three parameter datum shift from WGS 84 to ED50 for this North Sea area is given as
dX = +84.87m, dY = +96.49m, dZ = +116.95m.
The given WGS 84 geocentric values are:
XA = 3771 793.97m
YA = 140 253.34m
ZA = 5124 304.35m
Applying the three geocentric translations to these source CRS coordinates we obtain new geocentric values now related to the target CRS ED50:
XB = 3771 878.84m
YB = 140 349.83m
ZB = 5124 421.30m","This method allows calculation of geocentric coords in the target system by adding the parameter values to the corresponding coordinates of the point in the source system. See methods 1035 and 9603 for similar tfms operating between other CRSs types.","EPSG guidance note #7-2, http://www.epsg.org","OGP",11/17/2009 0:00:00,"2009.083",0
1032,"Coordinate Frame rotation (geocentric domain)",1,"<<<<>>>>
Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Although being common practice particularly in the European E&P industry, the Position Vector transformation sign convention is not universally accepted. A variation on this formula is also used, particularly in the USA E&P industry. That formula is based on the same definition of translation and scale parameters, but a different definition of the rotation parameters. The associated convention is known as the ""Coordinate Frame rotation"" convention.
The formula is:
(Xt) ( 1 +rZ -rY) (Xs) (tX)
(Yt) = M * ( -rZ 1 +rX) * (Ys) + (tY)
(Zt) ( +rY -rX 1 ) (Zs) (tZ)
and the parameters are defined as:
(tX, tY, tZ): Translation vector, to be added to the point's position vector in the source coordinate reference system in order to transform from source coordinate reference system to target coordinate reference system; also: the coordinates of the origin of source coordinate reference system in the target frame.
(rX, rY, rZ): Rotations to be applied to the coordinate reference frame. The sign convention is such that a positive rotation of the frame about an axis is defined as a clockwise rotation of the coordinate reference frame when viewed from the origin of the Cartesian coordinate reference system in the positive direction of that axis, that is a positive rotation about the Z-axis only from source coordinate reference system to target coordinate reference system will result in a smaller longitude value for the point in the target coordinate reference system. Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.
M: Multiplication factor to be applied to the position vector in the source coordinate reference system in order to obtain the correct scale of the target coordinate reference system. M = (1+dS) where dS is the scale difference. When dS is expressed in parts per million, M = (1+dS*10^-6). When dS is the scale difference expressed in parts per billion, M = (1+dS*10^-9).
In the absence of rotations the two formulas are identical; the difference is solely in the rotations. The name of the second method reflects this.
Note that the same rotation that is defined as positive in the first method is consequently negative in the second and vice versa. It is therefore crucial that the convention underlying the definition of the rotation parameters is clearly understood and is communicated when exchanging datum transformation parameters, so that the parameters may be associated with the correct coordinate transformation method (algorithm).","The same example as for the Position Vector transformation (coordinate operation method 1033) can be calculated, however the following transformation parameters have to be applied to achieve the same input and output in terms of coordinate values:
Transformation parameters for the Coordinate Frame rotation convention:
tX (m) = 0.000
tY (m) = 0.000
tZ (m) = +4.5
rX ("") = 0.000
rY ("") = 0.000
rZ ("") = -0.554 = -0.000002685868 radians
dS (ppm) = +0.219
M = 1 + dS = 1.000000219
Please note that only the rotation has changed sign as compared to the Position Vector transformation.","This method is a specific case of the Molodensky-Badekas (CF) method (code 1034) in which the evaluation point is at the geocentre with coordinate values of zero. Note the analogy with the Position Vector method (code 1033) but beware of the differences!","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2009.083 2013.021 2014.039 2018.001 2019.006",0
1033,"Position Vector transformation (geocentric domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The transformation between source and target CRS geocentric coordinates is usually described as a simplified 7-parameter Helmert transformation, expressed in matrix form with 7 parameters, in what is known as the ""Bursa-Wolf"" formula:
(Xt) ( 1 -rZ +rY) (Xs) (tX)
(Yt) = M * ( +rZ 1 -rX) * (Ys) + (tY)
(Zt) ( -rY +rX 1 ) (Zs) (tZ)
The parameters are commonly referred to defining the transformation ""from source coordinate reference system to target coordinate reference system"" in which (Xs, Ys, Zs) are the coordinates of the point in the source geocentric coordinate reference system and (Xt, Yt, Zt) are the coordinates of the point in the target geocentric coordinate reference system. But that does not define the parameters uniquely; neither is the definition of the parameters implied in the formula, as is often believed. However, the following definition, which is consistent with the 'Position Vector Transformation' convention is common E&P survey practice,
(tX, tY, tZ): Translation vector, to be added to the point's position vector in the source coordinate reference system in order to transform from source system to target system; also: the coordinates of the origin of the source coordinate reference system in the target coordinate reference system.
(rX, rY, rZ): Rotations to be applied to the point's vector. The sign convention is such that a positive rotation about an axis is defined as a clockwise rotation of the position vector when viewed from the origin of the Cartesian coordinate reference system in the positive direction of that axis; e.g. a positive rotation about the Z-axis only from source system to target system will result in a larger longitude value for the point in the target system. Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.
M: Multiplication factor to be applied to the position vector in the source coordinate reference system in order to obtain the correct scale of the target coordinate reference system. M = (1+dS) where dS is the scale difference. When dS is expressed in parts per million, M = (1+dS*10^-6). When dS is the scale difference expressed in parts per billion, M = (1+dS*10^-9).
<<<<>>>>","Input point:
Coordinate reference system: WGS 72
Cartesian geocentric coords:
X = 3 657 660.66 (m)
Y = 255 768.55 (m)
Z = 5 201 382.11 (m)
Transformation parameters WGS 72 to WGS 84:
tX (m) = 0.000
tY (m) = 0.000
tZ (m) = +4.5
rX ("") = 0.000 = 0.0 radians
rY ("") = 0.000 = 0.0 radians
rZ ("") = +0.554 = 0.000002685868 radians
dS (ppm) = +0.219
First M = 1 + dS = 1.000000219
Then application of this 7 parameter Position Vector transformation results in WGS 84 geocentric coordinates of:
X = 3 657 660.78 (m)
Y = 255 778.43 (m)
Z = 5 201 387.75 (m)","This method is a specific case of the Molodensky-Badekas (PV) method (code 1061) in which the evaluation point is the geocentre with coordinate values of zero. Note the analogy with the Coordinate Frame method (code 1032) but beware of the differences!","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2009.083 2013.021 2014.039 2018.001 2019.006",0
1034,"Molodensky-Badekas (CF geocentric domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
To eliminate high correlation between the translations and rotations in the derivation of parameter values for the Helmert transformation methods (coordinate operation method codes 1032 and 1033), instead of the rotations being derived about the geocentric coordinate reference system origin they may be derived at a location within the points used in the determination. Three additional parameters, the coordinates of the rotation point, are then required. The formula is:
(Xt) ( 1 +rZ -rY) (Xs - Xp) (Xp) (tX)
(Yt) = M * ( -rZ 1 +rX) * (Ys - Yp) + (Yp) + (tY)
(Zt) ( +rY -rX 1 ) (Zs - Zp) (Zp) (tZ)
and the parameters are defined as:
(tX, tY, tZ): Translation vector, to be added to the point's position vector in the source coordinate system in order to transform from source coordinate reference system to target coordinate reference system; also: the coordinates of the origin of source coordinate reference system in the target frame.
(rX, rY, rZ): Rotations to be applied to the coordinate reference frame. The sign convention is such that a positive rotation of the frame about an axis is defined as a clockwise rotation of the coordinate reference frame when viewed from the origin of the Cartesian coordinate system in the positive direction of that axis, that is a positive rotation about the Z-axis only from source coordinate reference system to target coordinate reference system will result in a smaller longitude value for the point in the target coordinate reference system. Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.
(Xp, Yp, Zp): Coordinates of the point about which the coordinate reference frame is rotated, given in the source Cartesian coordinate reference system.
M: Multiplication factor to be applied to the position vector in the source coordinate reference system in order to obtain the correct scale of the target coordinate reference system. M = (1+dS) where dS is the scale difference. When dS is expressed in parts per million, M = (1+dS*10^-6). When dS is the scale difference expressed in parts per billion, M = (1+dS*10^-9).
Reversibility.
The Molodensky-Badekas transformation in a strict mathematical sense is not reversible, i.e. in principle the same parameter values cannot be used to execute the reverse transformation. This is because the evaluation point coordinates are in the forward direction source coordinate reference system and the rotations have been derived about this point. They should not be applied about the point having the same coordinate values in the target coordinate reference system, as is required for the reverse transformation. However, in practical application there are exceptions when applied to the approximation of small differences between the geometry of a set of points in two different coordinate reference systems. The typical vector difference in coordinate values is in the order of 6*10^1 to 6*10^2 metres, whereas the evaluation point on or near the surface of the earth is 6.3*10^6 metres from the origin of the coordinate systems at the Earth's centre. This difference of four or five orders of magnitude allows the transformation in practice to be considered reversible. Note that in the reverse transformation, only the signs of the translations and rotation parameter values are reversed; the coordinates of the evaluation point remain unchanged.","Input point:
Coordinate reference system: La Canoa
Cartesian geocentric coords:
Xs = 2 550 408.965 m
Ys = -5 749 912.266 m
Zs = 1 054 891.114 m
Transformation parameters La Canoa to REGVEN:
tX = -270.933 m
tY = +115.599 m
tZ = -360.226 m
rX = -5.266 sec = -0.000025530288 radians
rY = -1.238 sec = -0.000006001993 radians
rZ = +2.381 sec = 0.000011543414 radians
dS = -5.109 ppm
Ordinate 1 of evaluation point = 2464351.59 m
Ordinate 2 of evaluation point = -5783466.61 m
Ordinate 3 of evaluation point = 974809.81 m
First, M = 1 + dS = 0.999994891
Then application of the 10 parameter Molodenski-Badekas Transformation results in REGVEN geocentric coordinates of:
Xt = 2 550 138.467 m
Yt = -5 749 799.862 m
Zt = 1 054 530.826 m","See method codes 1039 and 9636 for this operation in other coordinate domains and method code 1061 for opposite rotation convention in geocentric domain.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2009.083 2013.021 2014.039 2014.065 2019.006",0
1035,"Geocentric translations (geog3D domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Coordinate Frame Rotation (geog2D domain) transformation has 3 steps:
(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Geocentric translations(geocentric domain) method, EPSG method code 1031;
(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","(For the geocentric step only). Given a three parameter datum shift from WGS 84 to ED50 for this North Sea area is given as
dX = +84.87m, dY = +96.49m, dZ = +116.95m.
The WGS84 geographical coordinates convert to the following WGS 84 geocentric values using
the above formulas for X, Y, Z:
XA = 3771 793.97m
YA = 140 253.34m
ZA = 5124 304.35m
Applying the given datum shifts to these, we obtain new geocentric values now related
to ED50:
XB = 3771 878.84m
YB = 140 349.83m
ZB = 5124 421.30m","See methods 1031 and 9603 for similar tfms operating between other CRSs types.","EPSG guidance note #7-2, http://www.epsg.org","OGP",11/17/2009 0:00:00,"2009.083",0
1036,"Cartesian Grid Offsets from Form Function",1,"See information source. The Cartesian offset to be applied is determined through interpolation within an irregular grid of points at which coordinates in both source and target coordinate reference systems are given. The interpolation uses a finite element method form function procedure. The interpolation uses a finite element method form function procedure described in the Information Source.",,"Used in German state of Schleswig-Holstein.","Zeitschrift fur Vermessungswesen (ZfV, the Journal of the German Association of Surveying) volume 128 of April 2003 pages 244-250 and volume 129 of April 2004 pages 258-260.","OGP",11/14/2009 0:00:00,"2009.071",0
1037,"Position Vector transformation (geog3D domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Position Vector (geog3D domain) transformation has 3 steps:
(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Position Vector (geocentric domain) method, EPSG method code 1033;
(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","Input point:
Coordinate reference system: WGS 72 (geographic 3D)
Latitude = 55 deg 00 min 00 sec N
Longitude = 4 deg 00 min 00 sec E
Ellipsoidal height hS = 0 m
This transforms to Cartesian geocentric coords:
Xs = 3 657 660.66 (m)
Ys = 255 768.55 (m)
Zs = 5 201 382.11 (m)
Transformation parameters WGS 72 to WGS 84:
tX (m) = 0.000
tY (m) = 0.000
tZ (m) = 4.5
rX ("") = 0.000 = 0.0 radians
rY ("") = 0.000 = 0.0 radians
rZ ("") = 0.554 = 0.000002685868 radians
dS (ppm) = 0.219
from which M = 1.000000219
Application of the 7 parameter Position Vector transformation results in WGS 84 coordinates of:
Xt = 3 657 660.78 (m)
Yt = 255 778.43 (m)
Zt = 5 201 387.75 (m)
This converts into:
Latitude = 55 deg 00 min 00.090 sec N
Longitude = 4 deg 00 min 00.554 sec E
Ellipsoidal height = 3.22 m
on the WGS 84 geographic 3D coordinate reference system.","Note the analogy with the Coordinate Frame rotation (code 1038) but beware of the differences! The Position Vector convention is used by IAG and recommended by ISO 19111. See methods 1033 and 9606 for similar tfms operating between other CRS types.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2009.083 2013.021 2018.001 2019.006",0
1038,"Coordinate Frame rotation (geog3D domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Coordinate Frame rotation (geog3D domain) transformation has 3 steps:
(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Coordinate Frame rotation (geocentric domain) method, EPSG method code 1032;
(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","The same example as for the Position Vector transformation (coordinate operation method 1037) can be calculated, however the following transformation parameters have to be applied to achieve the same input and output in terms of coordinate values:
Transformation parameters Coordinate Frame rotation convention:
tX (m) = 0.000
tY (m) = 0.000
tZ (m) = 4.5
rX ("") = 0.000
rY ("") = 0.000
rZ ("") = -0.554 = -0.000002685868 radians
dS (ppm) = 0.219
from which M = 1.000000219
Please note that only the rotation has changed sign as compared to the Position Vector transformation.","Note the analogy with the Position Vector tfm (code 1037) but beware of the differences! The Position Vector convention is used by IAG and recommended by ISO 19111. See methods 1032 and 9607 for similar tfms operating between other CRS types.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2009.083 2013.021 2018.001 2019.006",0
1039,"Molodensky-Badekas (CF geog3D domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Molodensky-Badekas (geog3D domain) transformation has 3 steps:
(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Molodensky-Badekas (geocentric domain) method, EPSG method code 1034;
(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","Input point:
Coordinate reference system: La Canoa (geographic 3D)
Latitude = 9 deg 35 min 00.386 sec N
Longitude = 66 deg 04 min 48.091 sec W
Ellipsoidal height hS = 201.465 m
This transforms to Cartesian geocentric coords:
Xs = 2 550 408.965 m
Ys = -5 749 912.266 m
Zs = 1 054 891.114 m
Transformation parameters La Canoa to REGVEN:
tX = -270.933 m
tY = +115.599 m
tZ = -360.226 m
rX = -5.266 sec = -0.000025530288 radians
rY = -1.238 sec = -0.000006001993 radians
rZ = +2.381 sec = 0.000011543414 radians
dS = -5.109 ppm
Coordinate 1 of evaluation point = 2464351.59 m
Coordinate 2 of evaluation point = -5783466.61 m
Coordinate 3 of evaluation point = 974809.81 m
from which M = 0.999994891
Application of the Molodenski-Badekas (geocentric domain) Transformation results in REGVEN geocentric coordinates of:
Xt = 2 550 138.467 m
Yt = -5 749 799.862 m
Zt = 1 054 530.826 m
This converts into:
Latitude = 9 deg 34 min 49.001 sec N
Longitude = 66 deg 04 min 54.705 sec W
Ellipsoidal height = -18.10 m
on the REGVEN geographic 3D coordinate reference system.","See method codes 1034 and 9636 for this operation in other coordinate domains and method code 1062 for opposite rotation convention in geographic 3D domain.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2009.083 2013.021 2014.039 2014.065 2019.006",0
1040,"GNTRANS",1,"GNTRANS is a multistage three-dimensional transformation approach. Starting with a conventional seven-parameter transformation, the next stage functionally models the residuals at identical points using a unique, continuous spline surface. The final stage is a stochastic prediction to describe remaining discrepancies. For application and to improve performance, topocentric coordinate offsets are interpolated from a regular grid derived from the above transformation to determine coordinates of an arbitrary input point in the target system and vice-versa.",,,"Geo++.","OGP",3/30/2010 0:00:00,"2010.037",0
1041,"Krovak (North Orientated)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
From the defining parameters the following constants for the projection may be calculated :
A = a(1 - e^2)^0.5 / [1 - e^2 sin^2.(latC)]
B = {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5
gammao = asin[sin(latC) / B]
to = tan(pi/4 + gammao/2).[(1 + e sin(latC)) / (1 - e sin(latC))]^(e.B/2) / [tan(pi/4 + latC/2)]^B
n = sin(latp)
ro = kp.A / tan(latp)
To derive the projected Southing and Westing coordinates of a point with geographical coordinates (lat, lon) the formulas for the Krovak are:
U = 2(atan{to.tan^B(lat/2 + pi/4) / [(1 + e sin(lat)) / (1 - e sin(lat))]^[e.B/2]} - pi/4)
V = B(lonO - lon) where lonO and lon must both be referenced to the same prime meridian.
T = asin[cos(alphaC).sin(U) + sin(alphaC).cos(U). cos(V)]
D = asin[cos(U).sin(V)/cos(T)]
theta = n.D
r = ro.tan^n(pi/4 + latp/2) / tan^n(T/2 + pi/4)
Xp = r.cos(theta)
Yp = r.sin(theta)
Southing = Xp + FN
Westing = Yp + FE
Then Easting = -Westing
Northing = -Southing
Note also that the formula for D is satisfactory for the normal use of the projection within the pseudo-longitude range on the conformal sphere of ?90 degrees from the central line of the projection. Should there be a need to exceed this range (which is not necessary for application in Czechia and Slovakia) then for the calculation of D use:
sin(D1) = cos(U) * sin(V) / cos(T)
cos(D1) = {[cos(alphaC)*sin(T) ? sin(U)] / [sin(alphaC)*cos(T)]}
D = atan2(sin(D1) , cos(D1)) (see implementation notes in preface for atan2 convention)
The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:
Southing = -Northing
Westing = -Easting
Xp' = Southing ? FN
Yp' = Westing ? FE
r' = [(Yp')^2 + (Xp')^2]^(1/2)
theta' = atan2[Yp' , Xp'] (see GN7-2 implementation notes in preface for atan2 convention)
D' = theta' / sin(latp)
T' = 2{atan[((ro / r')^(1/n)).tan(pi/4 + latp/2)] - pi/4}
U' = asin[cos(alphaC).sin(T') - sin(alphaC).cos(T').cos(D')]
V' = asin(cos(T').sin(D') / cos(U'))
Then latitude lat is found by iteration using U' as the value for lat(j-1) in the first iteration:
lat(j) = 2*(atan{to^(-1/B) tan^(1/B).(U'/2 + pi/4).[(1 + e sin(lat(j-1)) / (1 - e sin(lat(j-1))]^(e/2)} - pi/4)
Then
lon = lonO - V' / B where lon is referenced to the same prime meridian as lonO.","For Projected Coordinate Reference System: S-JTSK (Ferro) / Krovak East North
Parameters:
Ellipsoid Bessel 1841 a = 6377397.155m 1/f = 299.15281
then e = 0.081696831 e^2 = 0.006674372
Latitude of projection centre = 49?30'00""N = 0.863937979 rad
Longitude of Origin = 42?30'00""E of Ferro = 0.741764932 rad
Co-latitude of cone axis = 30?17'17.30311"" = 0.528627763 rad
Latitude of pseudo standard parallel = 78?30'00""N = 1.370083463 rad
Scale factor on pseudo Standard Parallel (ko) = 0.9999
False Easting = 0.00 m
False Northing = 0.00 m
Calculated projection constants:
A = 6380703.611
B = 1.000597498
gammao = 0.863239103
to = 1.003419164
n = 0.979924705
ro = 1298039.005
Forward calculation for:
Latitude = 50?12'32.442""N = 0.876312568 rad
Longitude = 34?30'59.179""E of Ferro = 0.602425500 rad
Then the forward calculation first gives
U = 0.875596949
V = 0.139422687
T = 1.386275049
D = 0.506554623
theta = 0.496385389
r = 1194731.014
Xp = 1050538.643
Yp = 568990.997
Southing = 1050538.643
Westing = 568990.997
Then
Easting (X) = -568991.00 m
Northing (Y) = -1050538.64 m
Reverse calculation for the same Easting and Northing:
Southing = 1050538.643
Westing = 568990.997
Xp' = 1050538.643
Yp' = 568990.997
r' = 1194731.014
theta' = 0.496385389
D' = 0.506554623
T' = 1.386275049
U' = 0.875596949
V' = 0.139422687
lat(iteration 1) = 0.876310601
lat(iteration 2) = 0.876312560
lat(iteration 3) = 0.876312566
Latitude = 0.876312566 rad = 50?12'32.442""N
Longitude = 0.60242500 rad = 34?30'59.179""E of Ferro.",,"Land Survey Office (ZU), Prague, www.cuzk.cz/zu, and Geodesy, Cartography and Geodesy, Cartography and Cadastre Authority of Slovak Republic (UGKK SR).","IOGP",5/17/2019 0:00:00,"2010.071 2010.100 2017.037 2017.024 2019.030",0
1042,"Krovak Modified",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
From the defining parameters the following constants for the projection may be calculated :
A = a(1 - e^2)^0.5 / [1 - e^2 sin^2.(latC)]
B = {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5
gammao = asin[sin(latC) / B]
to = tan(pi/4 + gammao/2).[(1 + e sin(latC)) / (1 - e sin(latC))]^(e.B/2) / [tan(pi/4 + latC/2)]^B
n = sin(latp)
ro = kp.A / tan(latp)
To derive the projected Southing and Westing coordinates of a point with geographical coordinates (lat, lon) the formulas for the Krovak are:
U = 2(atan{to.tan^B(lat/2 + pi/4) / [(1 + e sin(lat)) / (1 - e sin(lat))]^[e.B/2]} - pi/4)
V = B(lonO - lon) where lonO and lon must both be referenced to the same prime meridian.
T = asin[cos(alphaC).sin(U) + sin(alphaC).cos(U). cos(V)]
D = asin[cos(U).sin(V)/cos(T)]
theta = n.D
r = ro.tan^n(pi/4 + latp/2) / tan^n(T/2 + pi/4)
Xp = r.cos(theta)
Yp = r.sin(theta)
Xr = Xp ? Xo
Yr = Yp ? Yo
dX = C1 + C3.Xr ? C4.Yr ? 2.C6.Xr.Yr + C5.(Xr^2 ? Yr^2) + C7.Xr.(Xr^2 ? 3.Yr^2) ? C8.Yr.(3.Xr^2 ? Yr^2) + 4.C9.Xr.Yr.(Xr^2 ? Yr^2) + C10.(Xr^4 + Yr^4 ? 6.Xr^2.Yr^2)
dY = C2 + C3.Yr + C4.Xr + 2.C5.Xr.Yr + C6.(Xr^2 ? Yr^2) + C8.Xr.(Xr^2 ? 3.Yr^2)+ C7.Yr.(3.Xr^2 ? Yr^2) ? 4.C10.Xr.Yr.(Xr^2 ? Yr^2) + C9.(Xr^4 + Yr^4 ? 6.Xr^2.Yr^2)
Southing X = FN + Xp ? dX
Westing Y = FE + Yp ? dY
The reverse formulas to derive the latitude and longitude of a point from its Southing and Westing values are:
Xr' = (Southing ? FN) ? Xo
Yr' = (Westing ? FE) ? Yo
dX' = C1 + C3.Xr' ? C4.Yr' ? 2.C6.Xr'.Yr' + C5.(Xr'^2 ? Yr'^2) + C7.Xr'.(Xr'^2 ? 3.Yr'^2) ? C8.Yr'.(3.Xr'^2 ? Yr'^2) + 4.C9.Xr'.Yr'.(Xr'^2 ? Yr'^2) + C10.(Xr'^4 + Yr'^4 ? 6.Xr'^2.Yr'^2)
dY' = C2 + C3.Yr' + C4.Xr' + 2.C5.Xr'.Yr' + C6.(Xr'^2 ? Yr'^2) + C8.Xr'.(Xr'^2 ? 3.Yr'^2)
+ C7.Yr'.(3.Xr'^2 ? Yr'^2) - 4.C10.Xr'.Yr'.(Xr'^2 ? Yr'^2) + C9.(Xr'^4 + Yr'^4 ? 6.Xr'^2.Yr'^2)
Xp' = (Southing ? FN) + dX'
Yp' = (Westing ? FE) + dY'
r' = [(Yp')^2 + (Xp')^2]^(1/2)
theta' = atan2[Yp' , Xp'] (see GN7-2 implementation notes in preface for atan2 convention)
D' = theta' / sin(latp)
T' = 2{atan[((ro / r')^(1/n)).tan(pi/4 + latp/2)] - pi/4}
U' = asin[cos(alphaC).sin(T') - sin(alphaC).cos(T').cos(D')]
V' = asin(cos(T').sin(D') / cos(U'))
Then latitude lat is found by iteration using U' as the value for lat(j-1) in the first iteration:
lat(j) = 2*(atan{tO^(-1/B) tan^(1/B).(U'/2 + pi/4).[(1 + e sin(lat(j-1)) / (1 - e sin(lat(j-1))]^(e/2)} - pi/4)
Then
lon = lonO - V' / B where lon is referenced to the same prime meridian as lonO.","For Projected Coordinate Reference System: S-JTSK/05 (Ferro) / Modified Krovak
Parameters:
Ellipsoid Bessel 1841 a = 6377397.155m 1/f = 299.15281
then e = 0.081696831 e^2 = 0.006674372
Latitude of projection centre = 49?30'00""N = 0.863937979 rad
Longitude of Origin = 42?30'00""E of Ferro = 0.741764932 rad
Co-latitude of cone axis = 30?17'17.30311"" = 0.528627763 rad
Latitude of pseudo standard parallel = 78?30'00""N 1.370083463 rad
Scale factor on pseudo Standard Parallel (ko) = 0.9999
False Easting = 5000000.00 m
False Northing = 5000000.00 m
Ordinate 1 of evaluation point Xo = 1089000.00 m
Ordinate 2 of evaluation point Yo = 654000.00 m
C1 = 2.946529277E-02
C2 = 2.515965696E-02
C3 = 1.193845912E-07
C4 = -4.668270147E-07
C5 = 9.233980362E-12
C6 = 1.523735715E-12
C7 = 1.696780024E-18
C8 = 4.408314235E-18
C9 = -8.331083518E-24
C10 = -3.689471323E-24
Calculated projection constants:
A = 6380703.611
B = 1.000597498
gammao = 0.863239103
to = 1.003419164
n = 0.979924705
ro = 1298039.005
Forward calculation for:
Latitude = 50?12'32.442""N = 0.876312568 rad
Longitude = 34?30'59.179""E of Ferro = 0.602425500 rad
U = 0.875596951
V = 0.139422687
S = 1.386275051
D = 0.506554627
theta = 0.496385393
r = 1194731.002
Xp = 1050538.631
Yp = 568990.995
Xr = -38461.369
Yr = -85009.005
dX = -0.077
dY = 0.088
and
Southing X = 6050538.71 m
Westing Y = 5568990.91 m
Reverse calculation for the same Southing and Westing:
Xr' = -38461.292
Yr' = -85009.093
dX' = -0.077
dY' = 0.088
Xp' = 1050538.631
Yp' = 568990.995
r' = 1194731.002
theta' = 0.496385393
D' = 0.506554627
T' = 1.386275051
U' = 0.875596951
V' = 0.139422687
lat(iteration 1) = 0.876310603
lat(iteration 2) = 0.876312562
lat(iteration 3) = 0.876312568
Latitude = 0.876312568 rad = 50?12'32.442""N
Longitude = 0.294083997 rad = 16?50'59.179""E","Incorporates a polynomial transformation which is defined to be exact and for practical purposes is considered to be a map projection.","Land Survey Office (ZU), Prague. www.cuzk.cz/zu","IOGP",8/29/2018 0:00:00,"2010.071 2010.100 2012.011 2017.037 2017.024",0
1043,"Krovak Modified (North Orientated)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
From the defining parameters the following constants for the projection may be calculated :
A = a(1 - e^2)^0.5 / [1 - e^2 sin^2.(latC)]
B = {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5
gammao = asin[sin(latC) / B]
to = tan(pi/4 + gammao/2).[(1 + e sin(latC)) / (1 - e sin(latC))]^(e.B/2) / [tan(pi/4 + latC/2)]^B
n = sin(latp)
ro = kp.A / tan(latp)
To derive the projected Southing and Westing coordinates of a point with geographical coordinates (lat, lon) the formulas for the Krovak are:
U = 2(atan{to.tan^B(lat/2 + pi/4) / [(1 + e sin(lat)) / (1 - e sin(lat))]^[e.B/2]} - pi/4)
V = B(lonO - lon) where lonO and lon must both be referenced to the same prime meridian.
T = asin[cos(alphaC).sin(U) + sin(alphaC).cos(U). cos(V)]
D = asin[cos(U).sin(V)/cos(T)]
theta = n.D
r = ro.tan^n(pi/4 + latp/2) / tan^n(T/2 + pi/4)
Xp = r.cos(theta)
Yp = r.sin(theta)
Xr = Xp ? Xo
Yr = Yp ? Yo
dX = C1 + C3.Xr ? C4.Yr ? 2.C6.Xr.Yr + C5.(Xr^2 ? Yr^2) + C7.Xr.(Xr^2 ? 3.Yr^2) ? C8.Yr.(3.Xr^2 ? Yr^2) + 4.C9.Xr.Yr.(Xr^2 ? Yr^2) + C10.(Xr^4 + Yr^4 ? 6.Xr^2.Yr^2)
dY = C2 + C3.Yr + C4.Xr + 2.C5.Xr.Yr + C6.(Xr^2 ? Yr^2) + C8.Xr.(Xr^2 ? 3.Yr^2)+ C7.Yr.(3.Xr^2 ? Yr^2) - 4.C10.Xr.Yr.(Xr^2 ? Yr^2) + C9.(Xr^4 + Yr^4 ? 6.Xr^2.Yr^2)
Southing = FN + Xp ? dX
Westing = FE + Yp ? dY
Easting = -(Westing)
Northing = -(Southing)
The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:
Southing = -(Northing)
Westing = -(Easting)
Xr' = (Southing - FN) ? Xo
Yr' = (Westing - FE) ? Yo
dX' = C1 + C3.Xr' ? C4.Yr' ? 2.C6.Xr'.Yr' + C5.(Xr'^2 ? Yr'^2) + C7.Xr'.(Xr'^2 ? 3.Yr'^2) ? C8.Yr'.(3.Xr'^2 ? Yr'^2) + 4.C9.Xr'.Yr'.(Xr'^2 ? Yr'^2) + C10.(Xr'^4 + Yr'^4 ? 6.Xr'^2.Yr'^2)
dY' = C2 + C3.Yr' + C4.Xr' + 2.C5.Xr'.Yr' + C6.(Xr'^2 ? Yr'^2) + C8.Xr'.(Xr'^2 ? 3.Yr'^2)
+ C7.Yr'.(3.Xr'^2 ? Yr'^2) - 4.C10.Xr'.Yr'.(Xr'^2 ? Yr'^2) + C9.(Xr'^4 + Yr'^4 ? 6.Xr'^2.Yr'^2)
Xp' = (Southing - FN) + dX'
Yp' = (Westing - FE) + dY'
r' = [(Yp')^2 + (Xp')^2]^(1/2)
theta' = atan2[Yp' , Xp'] (see GN7-2 implementation notes in preface for atan2 convention)
D' = theta' / sin(latp)
T' = 2{atan[((ro / r')^(1/n)).tan(pi/4 + latp/2)] - pi/4}
U' = asin[cos(alphaC).sin(T') - sin(alphaC).cos(T').cos(D')]
V' = asin(cos(T').sin(D') / cos(U'))
Then latitude lat is found by iteration using U' as the value for lat(j-1) in the first iteration:
lat(j) = 2*(atan{tO^(-1/B) tan^(1/B).(U'/2 + pi/4).[(1 + e sin(lat(j-1)) / (1 - e sin(lat(j-1))]^(e/2)} - pi/4)
Then
lon = lonO - V' / B where lon is referenced to the same prime meridian as lonO.","For Projected Coordinate Reference System: S-JTSK (Ferro) / Modified Krovak
Parameters:
Ellipsoid Bessel 1841 a = 6377397.155m 1/f = 299.15281
then e = 0.081696831 e^2 = 0.006674372
Latitude of projection centre = 49?30'00""N = 0.863937979 rad
Longitude of Origin = 42?30'00""E of Ferro = 0.741764932 rad
Co-latitude of cone axis = 30?17'17.30311"" = 0.528627763 rad
Latitude of pseudo standard parallel = 78?30'00""N = 1.370083463 rad
Scale factor on pseudo Standard Parallel (ko) = 0.9999
False Easting = 0.00 m
False Northing = 0.00 m
Ordinate 1 of evaluation point Xo = 1089000.00 m
Ordinate 2 of evaluation point Yo = 654000.00 m
C1 = 2.946529277E-02
C2 = 2.515965696E-02
C3 = 1.193845912E-07
C4 = -4.668270147E-07
C5 = 9.233980362E-12
C6 = 1.523735715E-12
C7 = 1.696780024E-18
C8 = 4.408314235E-18
C9 = -8.331083518E-24
C10 = -3.689471323E-24
Calculated projection constants:
A = 6380703.611
B = 1.000597498
gammao = 0.863239103
to = 1.003419164
n = 0.979924705
ro = 1298039.005
Forward calculation for:
Latitude = 50?12'32.442""N = 0.876312568 rad
Longitude = 34?30'59.1790""E of Ferro = 0.602425500 rad
Then the forward calculation first gives
U = 0.875596951
V = 0.139422687
T = 1.386275051
D = 0.506554627
theta = 0.496385393
r = 1194731.002
Xp = 1050538.631
Yp = 568990.995
Xr = -38461.369
Yr = -85009.005
dX = -0.077
dY = 0.088
Southing = 6050538.71 m
Westing = 5568990.91 m
and then
Easting X = -5568990.91 m
Northing Y = -6050538.71 m
Reverse calculation for the same Easting and Northing:
Southing = 6050538.71 m
Westing = 5568990.91 m
Xr' = -38461.292
Yr' = -85009.093
dX' = -0.077
dY' = 0.088
Xp' = 1050538.631
Yp' = 568990.995
r' = 1194731.002
theta' = 0.496385393
D' = 0.506554627
T' = 1.386275051
U' = 0.875596951
V' = 0.139422687
lat(iteration 1) = 0.876310603
lat(iteration 2) = 0.876312562
lat(iteration 3) = 0.876312568
Latitude = 0.876312568 rad = 50?12'32.442""N
Longitude = 0.602425500 rad = 34?30'59.179""E of Ferro.","Incorporates a polynomial transformation which is defined to be exact and for practical purposes is considered to be a map projection.","Land Survey Office (ZU), Prague. www.cuzk.cz/zu","IOGP",8/29/2018 0:00:00,"2010.071 2010.100 2012.011 2017.037 2017.024",0
1044,"Mercator (variant C)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The formulas to derive projected Easting and Northing coordinates are:
ko = cos(latSP1)/(1 - e^2*sin^2(latSP1))^0.5
where latSP1 is the absolute value of the first standard parallel (i.e. positive).
E = FE + a*ko(lon - lonF)
M = a ko ln{tan(pi/4 + latF/2)[(1 ? e sin(latF))/(1 + e sin(latF))]^(e/2)}
N = (NF ? M) + a ko ln{tan(pi/4 + lat/2)[(1 ? e sin(lat))/(1 + e sin(lat))]^(e/2)}
where logarithms are natural.
The reverse formulas to derive latitude and longitude from E and N values are:
lat = chi + (esq/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2chi)
+ (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4chi)
+ (7e^6/120 + 81e^8/1120) sin(6chi) + (4279e^8/161280) sin(8chi)
where chi = pi/2 - 2 atan(t)
t = B^((NF-M-N)/(a*ko))
B = base of the natural logarithm, 2.7182818...
and ko is calculated as for the forward transformation above.
lon = ((E - EF)/(a*ko)) + lonF","Parameters:
Ellipsoid Krassowsky 1940 a = 6378245.00m 1/f = 298.300
then e = 0.08181333 and e^2 = 0.00669342
Latitude first SP = 42?00'00""N = 0.73303829 rad
Longitude of natural origin = 51?00'00""E = 0.89011792 rad
Latitude of false origin = 42?00'00""N = 0.73303829 rad
Eastings at false origin EF = 0.00 m
Northing at false origin NF = 0.00 m
Forward calculation for:
Latitude = 53?00'00.00""N = 0.9250245 rad
Longitude = 53?00'00.00""E = 0.9250245 rad
ko = 0.744260894
M = 3819897.85
Easting E = 165704.29 m
Northing N = 1351950.22 m
Reverse calculation for same easting and northing first gives :
t = 0.336391288
chi = 0.921795958
Latitude = 53?00'00.000""N
Longitude = 53?00'00.000""E",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",5/3/2019 0:00:00,"2010.058 2010.103 2017.024 2019.018",0
1045,"Geographic3D to GravityRelatedHeight (OSGM02-Ire)",0,"This transformation involves the application of a height difference interpolated from a height correction model. The model provides height difference values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the height difference is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the height correction (C) of the target datum above the ellipsoid of the source Geographic 3D CRS. C differs from the geoid-ellipsoid separation N because a vertical datum is a realisation of the geoid surface, not the geoid itself.
Then:
H = h - C
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","Transformation of the vertical component of a Geographic 3D CRS to a Vertical CRS.","EPSG guidance note #7-2, http://www.epsg.org","OGP",1/14/2011 0:00:00,"2010.104",0
1046,"Vertical Offset and Slope",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
In Europe, national height systems are related to the pan-European height system through three transformation parameters and the formula:
Ht = Hs + A + [IncLat * rhoO * (Lat ? LatO)] + [IncLong * nuO * (Long
? LongO) * cos(Lat)]
where
Ht = gravity-related height value in the target vertical coordinate reference system.
Hs = gravity-related height value in the source vertical coordinate reference system.
A is the value of the vertical offset to be applied.
IncLat is the value in radians of the inclination parameter in the latitude domain, i.e. in the plane of the meridian, derived at an evaluation point with coordinates of LatO, LongO.
IncLon is the value of the inclination parameter in the longitude domain, i.e. perpendicular to the plane of the meridian.
rhoO is the radius of curvature of the meridian at latitude LatO, where rhoO = a(1 ? e^2)/(1 ? e^2 * sin^2(LatO))^1.5
nuO is the radius of curvature on the prime vertical (i.e. perpendicular to the meridian) at latitude LatO, where nuO = a /(1 ? e^2 * sin^2(LatO))^0.5
LatO , LongO are the coordinates of the evaluation point in the 'Horizontal CRS' coordinate reference system, in radians. The ID of the 'Horizontal CRS' is a defining parameter.
Lat , Long are the horizontal coordinates of the point in the same coordinate reference system as the evaluation point (the 'Horizontal CRS'), in radians.
The horizontal location of the point must always be given in 'Horizontal CRS' terms. If given in another CRS the coordinates must first be transformed to 'Horizontal CRS' values. The method is reversible.","For coordinate transformation LN02 to EVRF2000 (1)
Ordinate 1 of evaluation point: 46deg 55min N = 0.818850307 radians
Ordinate 2 of evaluation point: 8deg 11min E of Greenwich = 0.142826110 radians
Transformation Parameters:
A = -0.245m
IncLat = -0.210"" = -0.000001018 radians
IncLong = -0.032"" = -0.000000155 radians
Consider a point having a gravity-related height in the LN02 system (Hs) of 473.0m and with horizontal coordinates in the ETRS89 geographical coordinate reference system (CRS code 4258) of:
ETRS89 latitude: 47deg 20 min N = 0.826122513 radians
ETRS89 longitude: 9 deg 40min E of Greenwich = 0.168715161 radians
Then rhoO = 6369526.88 m
IncLat term = -0.047 m
nuO = 6389555.64 m
incLong term = -0.017 m
whence EVRF2000 height (Ht) = 473.0 +(-0.245) + (-0.047) + (-0.017) = 472.690 m.","This transformation allows calculation of height in the target system by applying the parameter values to the height value of the point in the source system.","OGP","IOGP",10/16/2017 0:00:00,"2011.108 2014.027",0
1047,"Geographic3D to GravityRelatedHeight (Gravsoft)",0,"This transformation involves the application of a geoid-ellipsoid separation value interpolated from a quasigeoid model. The model provides separation values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the separation is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the geoid-ellipsoid separation (N) above the ellipsoid of the source Geographic 3D CRS.
Then:
H = h - N
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",12/15/2016 0:00:00,"2011.083 2016.050",0
1048,"Geographic3D to GravityRelatedHeight (Ausgeoid v2)",0,"This transformation involves the application of a geoid-ellipsoid separation value interpolated from a geoid model. The model provides separation values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the separation is achieved through a bi-cubic interpolation of the grid, using the latitude and longitude of the point. This step provides the geoid-ellipsoid separation (N) above the ellipsoid of the source Geographic 3D CRS.
Then:
H = h - N
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","The Information Source references software which offers both bi-cubic and bi-linear interpolation methods. Unlike earlier Ausgeoid98 method which used bi-linear interpolation, Ausgeoid v2 uses bi-cubic. See Info Source for file format documentation.","Geoscience Australia; http://www.ga.gov.au, geodesy/ausgeoid pages.","IOGP",4/20/2018 0:00:00,"2012.009 2016.050 2018.009",0
1049,"P6 (I = J-90?) seismic bin grid transformation",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The P6 (I = J-90?) seismic bin grid transformation is a special case of the Affine Geometric Transformation (method code 9623). In practice bin grid definitions are based on an identified map grid. As such the example bin grids in the EPSG Dataset are described with the map grid (projected CRS) as source CRS. However to retain consistency with the syntax used in the description of the affine geometric transformation, in the formulas below the bin grid is the affine source CRS and the map grid is the affine target CRS.
Map grid Easting = XT = XT0 - XS . k . dSX . cos q + YS . k . dSY . sin q
Map grid Northing = YT = YT0 + XS . k . dSX . sin q + YS . k . dSY . cos q
Bin grid I = XS = XSO - {[( XT ? XTO) * cos q ? (YT ? YTO) * sin q ] * [IncSX / (k * MX)]}
Bin grid J = YS = YSO + {[(XT ? XTO) * sin q + (YT ? YTO) * cos q] * [IncSY / (k * MY)]}
where:
XT0 ,YT0 = the map grid coordinates (easting, northing) of the origin point of the bin grid;
dSX , dSY = the length of one unit of the bin, expressed in units of the map grid, for the bin grid I and J axes respectively;
k = point scale factor in the map grid at a chosen reference point;
q = the angle through which the bin grid axes must be rotated to coincide with the map grid axes (counter-clockwise is positive). Alternatively, the bearing (clockwise positive) of the bin grid system J-axis measured relative to map grid north.","This example is given in the OGP P6/11 user guide.
The bin grid is based on projected CRS NAD27/ BLM 16N (ftUS) in which the origin of the bin grid is defined at east = 871200.0 ftUS, north = 10280160.0 ftUS. As the survey was acquired on the map grid, the bin grid scale factor at the bin grid origin is defined to be 1.
The bin width on the I-axis (XS axis) is 82.5 US survey feet, whilst the bin width on the J-axis (YS axis) is 41.25 US survey feet. The origin of the bin grid has bin values of I=5000, J=0.
In the map grid, the bearing of the bin grid I and J axes are 250? and 340? respectively. The latter is one of the transformation parameters.
The transformation parameter values are:
Bin grid origin I: 5000 bins
Bin grid origin J: 0 bins
Bin grid origin Easting: 871200 ftUS
Bin grid origin Northing: 10280160 ftUS
Scale factor of bin grid: 1.0
Bin Width on I-axis: 82.5 ftUS
Bin Width on J-axis: 41.25 ftUS
Map grid bearing of bin grid J-axis: 340?
Bin node increment on I-axis: 1
Bin node increment on J-axis: 1
Calculation of map grid coordinates for centre of bin with coordinates: I = 4700, J = 247:
Easting = XT = XTO - [(XS ? XSO) * cos q * k * MX / IncSX] + [(YS ? YSO) * sin q * k * MY / IncSY]
= 890972.63 ftUS.
Northing = YT = YTO + [(XS ? XSO) * sin q * k * MX / IncSX] + [(YS ? YSO) * cos q * k * MY / IncSY]
= 10298199.29 ftUS.
Calculation of bin grid coordinates for this point, map grid location east = 890972.63ftUS, north = 10298199.29ftUS:
Bin grid I = XS = XSO - {[( XT ? XTO) * cos q ? (YT ? YTO) * sin q ] * [IncSX / (k * MX)]}
= 4700 bins,
Bin grid J = YS = YSO + {[(XT ? XTO) * sin q + (YT ? YTO) * cos q] * [IncSY / (k * MY)]}
= 247 bins","Use only when the bin grid I-axis is rotated 90? counter-clockwise from the J-axis when viewed from above the plane containing the two axes (CS code 1034). If the I-axis is rotated 90? clockwise from the J-axis, method 9666 should be used.","OGP P6/11 Seismic Bin Grid Data Exchange Format - User Guidelines, version 1.0, July 2012.","OGP",7/28/2014 0:00:00,"2011.106 2014.044",0
1050,"Geographic3D to GravityRelatedHeight (CI)",0,"This transformation involves the application of a geoid-ellipsoid separation value interpolated from a quasigeoid model. The model provides separation values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the separation is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the geoid-ellipsoid separation (N) above the ellipsoid of the source Geographic 3D CRS. In this method the separation is given in feet.
Then:
H = (h/0.3048) - N
where h = the height above the ellipsoid in the source geographic 3D CRS (in metres)
and H = the gravity-related height in the target vertical CRS.","(none)",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",10/16/2017 0:00:00,"2012.095 2016.050 2014.027",0
1051,"Lambert Conic Conformal (2SP Michigan)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
To derive the projected Easting and Northing coordinates of a point with geographical coordinates (lat,lon) the formulas for the one standard parallel case are:
E = EF + r sin(theta)
N = NF + rF - r cos(theta)
where
m = cos(lat)/(1 - e^2 sin^2(lat))^0.5 for m1, lat1, and m2, lat2 where lat1 and lat2 are the latitudes of the two standard parallels.
t = tan(pi/4 - lat/2)/[(1 - e sin(lat))/(1 + e sin(lat))]^(e/2) for t1, t2, tF and t using lat1, lat2, latF and lat respectively.
n = (loge(m1) - loge(m2))/(loge(t1) - loge(t2))
F = m1/(n t1^n)
r = a K F t^n for rF and r, where rF is the radius of the parallel of latitude of the false origin and K is the ellipsoid scaling factor.
theta = n(lon - lon0)
The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:
lat = pi/2 - 2atan{t'[(1 - esin(lat))/(1 + esin(lat))]^(e/2)}
lon = theta'/n +lon0
where
r' = +/-[(E - EF)^2 + {rF - (N - NF)}^2]^0.5 , taking the sign of n
t' = (r'/(aKF))^(1/n)
theta' = atan2 [(E- EF),(rF - (N- NF))] (see GN7-2 implementation notes in preface for atan2 convention)
and n, F, and rF are derived as for the forward calculation.
Note that the formula for lat requires iteration. First calculate t' and then a trial value for lat* using
lat = p/2-2atan(t'). Then use the full equation for lat substituting the trial value into the right hand side of the equation. Thus derive a new value for lat. Iterate the process until lat does not change significantly. The solution should quickly converge, in 3 or 4 iterations.","For Projected Coordinate System NAD27 / Michigan Central
Parameters:
Ellipsoid Clarke 1866, a = 6378206.400 metres = 20925832.16 US survey feet
1/f = 294.97870
then e = 0.08227185 and e^2 = 0.00676866
First Standard Parallel = 44?11'00""N = 0.771144641 rad
Second Standard Parallel = 45?42'00""N = 0.797615468 rad
Latitude False Origin = 43?19'00""N = 0.756018454 rad
Longitude False Origin = 84?20'00""W = -1.471894336 rad
Easting at false origin = 2000000.00 US survey feet
Northing at false origin = 0.00 US survey feet
Ellipsoid scaling factor = 1.0000382
Forward calculation for:
Latitude = 43?45'00.00""N = 0.763581548 rad
Longitude = 83?10'00.00""W = -1.451532161 rad
first gives :
m1 = 0.718295175 m2 = 0.699629151
t = 0.429057680 tF = 0.433541026
t1 = 0.424588396 t2 = 0.409053868
n = 0.706407410 F = 1.862317735
r = 21436775.51 rF = 21594768.40
theta = 0.014383991
Then Easting X = 2308335.75 US survey feet
Northing Y = 160210.48 US survey feet
Reverse calculation for same easting and northing first gives:
theta' = 0.014383991 r' = 21436775.51
t' = 0.429057680
Then Latitude = 43?45'00.000""N
Longitude = 83?10'00.000""W",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"2013.020 2017.024",0
1052,"Colombia Urban",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The forward conversion from latitude and longitude to easting and northing is:
E = FE + A * nu * cos(lat) * (lon ? lonO)
N = FN + G * rhoO * [(lat ? latO) + {B *(lon ? lonO)^2 * nu^2 * cos^2(lat)}]
where
latO, lonO and hO are the latitude, longitude and height of the projection origin,
rho is the radius of curvature of the ellipsoid in the plane of the meridian at latitude lat, rhoO at latitude latO, and rhoM at mid-latitude (latO + lat)/2; rho = a(1 ? e^2)/(1 ? e^2 * sin^2(lat))^1.5 for lat, latO and latM
nu is the radius of curvature of the ellipsoid perpendicular to the meridian at latitude lat with nuO at latitude latO; nu = a /(1 ? e^2 * sin^2(lat))^0.5 for lat and latO
e is the eccentricity of the ellipsoid and e^2 = (a^2 ? b^2)/a^2 = 2f ? f^2
a is the ellipsoidal semi-major and 1/f is the inverse flattening
A = 1 + hO/nuO
B = tan(latO) / 2 * rhoO * nuO
G = 1 + hO/latM
The reverse conversion from easting and northing to latitude and longitude is:
lat = latO + [(N ? FN) / D] ? B [(E?FE) / C]^2
lon = lonO + (E ? FE) / C *nu * cos(lat)
where
B is as in the forward conversion,
C = 1 + hO/a
D = rhoO * (1 + hO/a(1-e^2))","For Projected Coordinate Reference System: MAGNA-SIRGAS / Bogota urban grid
Parameters:
Ellipsoid: GRS 1980 a = 6378137.0 metres 1/f = 298.2572221
then e^2 = 0.006694380
Latitude of natural origin = 4?40'49.75""N = 0.081689893 rad
Longitude of natural origin = 74?08'47.73""W = -1.294102154 rad
False easting = 92334.879 metres
False northing = 109320.965 metres
Height of projection origin = 2550.000 metres
Forward calculation for:
Latitude = 4?48'00.000""N = 0.083775804 rad
Longitude = 74?15'00.000""W = -1.295906970 rad
First calculate constants for the projection:
rho_O = 6335862.944
nu_O = 6378279.154
A = 1.000399794
B = 1.012970E-15
C = 1.000399803
D = 6338413.114
Then
nu = 6378286.489
G = 1.000402470
and then
E = 80859.033m
N = 122543.174m
Reverse calculation for the same Easting and Northing (80859.033E, 122543.174N) first gives:
Latitude = 4?48'00.000""N
then nu = 6378286.489
and Longitude = 74?15'00.000""W",,"EPSG guidance note #7-2, http://www.epsg.org","OGP",5/20/2013 0:00:00,"2012.044",0
1053,"Time-dependent Position Vector tfm (geocentric)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The transformation between source and target CRS geocentric coordinates is usually described as a simplified 7-parameter Helmert transformation, expressed in matrix form with 7 parameters:
(Xt) ( 1 -rZ' +rY') (Xs) (tX')
(Yt) = M' * ( +rZ' 1 -rX') * (Ys) + (tY')
(Zt) ( -rY' +rX' 1 ) (Zs) (tZ')
where
tX' = tX + dtX (t ? t0)
tY' = tY + dtY (t ? t0)
tZ' = tZ + dtZ (t ? t0)
rX' = rX + drX (t ? t0)
rY' = rY + drY (t ? t0)
rZ' = rZ + drZ (t ? t0)
dS' = dS + ddS (t ? t0)
M' = 1 + dS'
and the parameters are defined as:
(tX', tY', tZ'): Translation vector, to be added to the point's position vector in the source coordinate reference system in order to transform from source system to target system; also: the coordinates of the origin of the source coordinate reference system in the target coordinate reference system.
(rX', rY', rZ'): Rotations to be applied to the point's vector. The sign convention is such that a positive rotation about an axis is defined as a clockwise rotation of the position vector when viewed from the origin of the Cartesian coordinate reference system in the positive direction of that axis; e.g. a positive rotation about the Z-axis only from source system to target system will result in a larger longitude value for the point in the target system. Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.
M': The time-adjusted scale correction to be made to the position vector in the source coordinate reference system in order to obtain the correct scale in the target coordinate reference system. M' = (1 + dS'). When the time-adjusted scale difference dS' is expressed in parts per million, M' = (1 + dS'*10^-6). When the time-adjusted scale difference dS' is expressed in parts per billion, M' = (1 + dS'*10^-9).","Transformation from ITRF2008 to GDA94 at epoch 2013.90.
Input point coordinate reference system: ITRF2008 at epoch 2013.90 (geocentric Cartesian coordinates):
Xs = -3789470.710 m
Ys = 4841770.404 m
Zs = -1690893.952 m
t = 2013.90
Transformation parameter values:
tX = ?84.68 mm
dtX = +1.42 mm/yr
tY = ?19.42 mm
dtY = +1.34 mm/yr
tZ = +32.01 mm
dtZ = +0.90 mm/yr
rX = +0.4254 msec
drX = ?1.5461 msec/yr
rY = ?2.2578 msec
drY = ?1.1820 msec/yr
rZ = ?2.4015 msec
drZ = ?1.1551 msec/yr
dS = +0.00971 ppm
ddS = +0.000109 ppm/yr
t0 = 1994.00
First calculate the correction due to rate of change to each of the 7 transformation parameters for the period (t-t0), taking care to convert the translations to the same units as the source CRS (in this case metres) and the rotations to radians:
tX' = ?84.68 + [+1.42 * (2013.90-1994.00)] = ?56.42 mm = ?0.056 m
tY' = ?19.42 + [+1.34 * (2013.90-1994.00)] = +7.25 mm = +0.007 m
tZ' = +32.01 + [+0.90 * (2013.90-1994.00)] = +49.92 mm = +0.050 m
rX' = 0.4254 + [?1.5461 * (2013.90-1994.00)] = ?30.34 msec = ?1.471021E-07 rad
rY' = ?2.2578 + [?1.1820 * (2013.90-1994.00)] = ?25.78 msec = ?1.249830E-07 rad
rZ' = ?2.4015 + [?1.1551 * (2013.90-1994.00)] = ?25.39 msec = ?1.230844E-07 rad
dS' = +0.00971 + [+0.000109 * (2013.90-1994.00)] = +0.01188 ppm
Then M' = 1.00000001188
Using these parameter values, application of the 7 parameter Position Vector transformation to the given (source) ITRF2008 coordinates results in:
Xt = -3789470.004m
Yt = 4841770.686 m
Zt = -1690895.108 m
on the GDA94 (target) geocentric coordinate reference system.
For the reverse transformation from GDA94 coordinates to ITRF08 coordinates at epoch 2013.90, the signs of all parameters need to be reversed except for the reference epoch. Then:
tX' = +84.68 + [(?1.42 * (2013.90 ?1994.00)] = +56.42 mm = +0.056 m
and similarly for the other six parameters. Hence tY' = ?0.007 m, tZ' = ?0.050 m, rX' = 1.471021E-07 rad, rY' = 1.249830E-07 rad, rZ' = 1.230844E-07 rad and dS' = ?0.01188 ppm.
Using these time-adjusted parameters values, M' = 0.99999998812 and then applying the 7 parameter Position Vector transformation fomula to the GDA94 coordinates of
Xs = ?3789470.004 m
Ys = 4841770.686 m
Zs = ?1690895.108 m
results in ITRF08 coordinates at epoch 2013.90 of:
Xt = ?3789470.710 m
Yt = 4841770.404 m
Zt = ?1690893.952 m","Note the analogy with the Time-dependent Coordinate Frame rotation (code 1056) but beware of the differences! The Position Vector convention is used by IAG. See method codes 1054 and 1055 for similar methods operating between other CRS types.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2013.021 2013.065 2014.068 2018.001 2019.006",0
1054,"Time-dependent Position Vector tfm (geog2D)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Time-dependent Position Vector transformation (geog2D domain) has 5 steps:
(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;
(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Time-dependent Position Vector (geocentric domain) method, EPSG method code 1053;
(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","Input point:
Coordinate reference system: WGS 84 (G1674) = ITRF08 (geographic 2D)
Latitude = 15 deg 28 min 32.368 sec S
Longitude = 128 deg 02 min 56.198 sec E
at epoch 2013.90
This is taken to be geographic 3D with an assumed ellipsoidal height hs = 46.244 m
This transforms to WGS 84 (G1674) = ITRF2008 Cartesian geocentric coordinates:
Xs = -3789470.702 m
Ys = 4841770.411 m
Zs = -1690893.950 m
(at epoch 2013.90)
Transformation parameters ITRF08 to GDA94:
tX = ?84.68 mm
dtX = +1.42 mm/yr
tY = ?19.42 mm
dtY = +1.34 mm/yr
tZ = +32.01 mm
dtZ = +0.90 mm/yr
rX = 0.4254 msec
drX = -1.5461 msec/yr
rY = -2.2578 msec
drY = -1.1820 msec/yr
rZ = -2.4015 msec
drZ = -1.1551 msec/yr
dS = +0.00971 ppm
ddS = +0.000109 ppm/yr
t0 = 1994.00
Corrections due to rate of change to each of the 7 transformation parameters for the period (t-t0), taking care to convert the translations to the same units as the source CRS (in this case metres) and the rotations to radians:
tX' = ?84.68 + [(1.42 * (2013.90-1994.00)] = ?56.42 mm = ?0.056m
tY' = +0.007 m
tZ' = +0.050 m
rX' = ?0.4254 + [(1.5461 * (2013.90-1994.00)] = +30.3420 msec = 1.471021E-07rad
rY' = 1.249830E-07 rad
rZ' = 1.230844E-07 rad
dS' = 0.00971 + [0.000109 * (2013.90-1994.00)] = 0.01188 ppm
from which M' = 1.00000001188
Using these parameter values, application of the 7 parameter Coordinate Frame transformation results in:
Xt = -3789470.008 m
Yt = 4841770.685 m
Zt = -1690895.103 m
on the GDA94 geocentric coordinate reference system.
This converts into:
Latitude = 15 deg 28 min 32.406 sec S
Longitude = 128 deg 02 min 56.174 sec E
Ellipsoidal height = +46.244m
on the GDA94 geographic 3D coordinate reference system. For the 2D equivalent the height is ignored.","Note the analogy with the Time-dependent Coordinate Frame rotation (code 1057) but beware of the differences! The Position Vector convention is used by IAG. See method codes 1053 and 1055 for similar methods operating between other CRS types.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2013.021 2018.001 2019.006",0
1055,"Time-dependent Position Vector tfm (geog3D)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Time-dependent Position Vector (geog3D domain) transformation has 3 steps:
(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Time-dependent Position Vector (geocentric domain) method, EPSG method code 1053;
(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","Input point:
Coordinate reference system: WGS 84 (G1674) = ITRF08 (geographic 2D)
Latitude = 15 deg 28 min 32.368 sec S
Longitude = 128 deg 02 min 56.198 sec E
Ellipsoidal height hs = 46.244 m
at epoch 2013.90
This transforms to WGS 84 (G1674) = ITRF2008 Cartesian geocentric coordinates:
Xs = -3789470.702 m
Ys = 4841770.411 m
Zs = -1690893.950 m
(at epoch 2013.9)
Transformation parameters ITRF08 to GDA94:
tX = ?84.68 mm
dtX = +1.42 mm/yr
tY = ?19.42 mm
dtY = +1.34 mm/yr
tZ = +32.01 mm
dtZ = +0.90 mm/yr
rX = 0.4254 msec
drX = -1.5461 msec/yr
rY = -2.2578 msec
drY = -1.1820 msec/yr
rZ = -2.4015 msec
drZ = -1.1551 msec/yr
dS = +0.00971 ppm
ddS = +0.000109 ppm/yr
t0 = 1994.00
Corrections due to rate of change to each of the 7 transformation parameters for the period (t-t0), taking care to convert the translations to the same units as the source CRS (in this case metres) and the rotations to radians:
tX' = ?84.68 + [(1.42 * (2013.90-1994.00)] = ?56.42 mm = ?0.056m
tY' = +0.007 m
tZ' = +0.050 m
rX' = ?0.4254 + [(1.5461 * (2013.90-1994.00)] = +30.3420 msec = 1.471021E-07rad
rY' = 1.249830E-07 rad
rZ' = 1.230844E-07 rad
dS' = 0.00971 + [0.000109 * (2013.90-1994.00)] = 0.01188 ppm
from which M' = 1.00000001188
Using these parameter values, application of the 7 parameter Position Vector transformation results in:
Xt = -3789470.008 m
Yt = 4841770.685 m
Zt = -1690895.103 m
on the GDA94 geocentric coordinate reference system.
This converts into:
Latitude = 15 deg 28 min 32.406 sec S
Longitude = 128 deg 02 min 56.174 sec E
Ellipsoidal height = +46.244m
on the GDA94 geographic 3D coordinate reference system.","Note the analogy with the Coordinate Frame rotation (code 1058) but beware of the differences! The Position Vector convention is used by IAG. See method codes 1053 and 1054 for similar methods operating between other CRS types.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2013.021 2018.001 2019.006",0
1056,"Time-dependent Coordinate Frame rotation (geocen)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The transformation between source and target CRS geocentric coordinates is a two-step process. The first step is to compute the values of seven Helmert transformation parameters at the required epoch. The second step is to apply a seven-parameter Helmert transformation to the source CRS coordinates. This Coordinate Frame rotation is expressed in matrix form as:
(Xt) ( 1 +rZ' -rY') (Xs) (tX')
(Yt) = M' * ( -rZ' 1 +rX') * (Ys) + (tY')
(Zt) ( +rY' -rX' 1 ) (Zs) (tZ')
where
tX' = tX + dtX (t ? t0)
tY' = tY + dtY (t ? t0)
tZ' = tZ + dtZ (t ? t0)
rX' = rX + drX (t ? t0)
rY' = rY + drY (t ? t0)
rZ' = rZ + drZ (t ? t0)
dS' = dS + ddS (t ? t0)
M' = 1 + dS'
and the parameters are defined as:
(tX', tY', tZ'): Translation vector, to be added to the point's position vector
in the source coordinate reference system in order to transform from source
coordinate reference system to target coordinate reference system; also: the
coordinates of the origin of source coordinate reference system in the target
frame.
(rX', rY', rZ'): Rotations to be applied to the coordinate reference frame.
The sign convention is such that a positive rotation of the frame about an
axis is defined as a clockwise rotation of the coordinate reference frame when
viewed from the origin of the Cartesian coordinate reference system in the positive
direction of that axis, that is a positive rotation about the Z-axis only from
source coordinate reference system to target coordinate reference system will
result in a smaller longitude value for the point in the target coordinate reference
system. Although rotation angles may be quoted in any angular unit of measure,
the formula as given here requires the angles to be provided in radians.
M': The time-adjusted scale correction to be made to the position vector in the source coordinate reference system in order to obtain the correct scale in the target coordinate reference system. M' = (1 + dS'). When the time-adjusted scale difference dS' is expressed in parts per million, M' = (1 + dS'*10^-6). When the time-adjusted scale difference dS' is expressed in parts per billion, M' = (1 + dS'*10^-9).","Transformation from ITRF2008 to GDA94 at epoch 2013.90.
Source CRS ITRF2008 coordinates at epoch 2013.9
(geocentric Cartesian coordinates):
Xs = ?3789470.710 m
Ys = 4841770.404 m
Zs = ?1690893.952 m
t = 2013.90
Transformation parameter values:
tX = ?84.68 mm
dtX = +1.42 mm/yr
tY = ?19.42 mm
dtY = +1.34 mm/yr
tZ = +32.01 mm
dtZ = +0.90 mm/yr
rX = ?0.4254 msec
drX = +1.5461 msec/yr
rY = +2.2578 msec
drY = +1.1820 msec/yr
rZ = +2.4015 msec
drZ = +1.1551 msec/yr
dS = +0.00971 ppm
ddS = +0.000109 ppm/yr
t0 = 1994.00
First apply the correction due to rate of change to each of the 7 transformation parameters for the period (t-t0), taking care to convert the translations to the same units as the source CRS (in this case metres) and the rotations to radians:
tX' = ?84.68 + [+1.42 * (2013.90-1994.00)] = ?56.42 mm = ?0.056m
tY' = ?19.42 + [+1.34 * (2013.90-1994.00)] = +7.25 mm = +0.007 m
tZ' = +32.01 + [+0.90 * (2013.90-1994.00)] = 49.92 mm = +0.050 m
rX' = ?0.4254 + [(+1.5461 * (2013.90-1994.00)] = +30.3420 msec = +1.471021E-07rad
rY' = +2.2578 + [(1.1820 * (2013.90-1994.00)] = +25.7796 msec = +1.249830E-07 rad
rZ' = +2.4015 + [(1.1551 * (2013.90-1994.00)] = +25.3880 msec = +1.230844E-07 rad
dS' = +0.00971 + [+0.000109 * (2013.90-1994.00)] = +0.01188 ppm
Then M' = 1.00000001188
Using these time-adjusted parameter values, application of the 7 parameter Coordinate Frame rotation formula to the given (source) ITRF2008 coordinates results in:
Xt = ?3789470.004 m
Yt = 4841770.686 m
Zt = ?1690895.108 m
on the GDA94 (target) geocentric coordinate reference system.
For the reverse transformation from GDA94 coordinates to ITRF08 coordinates at epoch 2013.9, the signs of all parameters need to be reversed except for the reference epoch. Then:
tX' = +84.68 + [(?1.42 * (2013.90 ?1994.00)] = +56.42 mm = +0.056 m
and similarly for the other six parameters. Hence tY' = ?0.007 m, tZ' = ?0.050 m, rX' = ?1.471021E-07 rad, rY' = ?1.249830E-07 rad, rZ' = ?1.230844E-07 rad and dS' = ?0.01188 ppm.
Using these time-adjusted parameters values, M' = 0.99999998812 and then applying the 7 parameter Coordinate Frame rotation fomula to the GDA94 coordinates of
Xs = ?3789470.004 m
Ys = 4841770.686 m
Zs = ?1690895.108 m
results in ITRF08 coordinates at epoch 2013.9 of:
Xt = ?3789470.710 m
Yt = 4841770.404 m
Zt = ?1690893.952 m","Note the analogy with the Time-dependent Position Vector transformation (code 1053) but beware of the differences! The Position Vector convention is used by IAG. See method codes 1057 and 1058 for similar methods operating between other CRS types.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2013.021 2014.068 2019.006",0
1057,"Time-dependent Coordinate Frame rotation (geog2D)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Time-dependent Coordinate Frame rotation (geog2D) has 5 steps:
(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;
(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Time-dependent Coordinate Frame rotation (geocentric domain) method, EPSG method code 1056;
(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","Input point:
Coordinate reference system: WGS 84 (G1674) = ITRF08 (geographic 2D)
Latitude = 15 deg 28 min 32.368 sec S
Longitude = 128 deg 02 min 56.198 sec E
at epoch 2013.90
This is taken to be geographic 3D with an assumed ellipsoidal height hs = 46.244 m
This transforms to WGS 84 (G1674) = ITRF2008 Cartesian geocentric coordinates:
Xs = -3789470.702 m
Ys = 4841770.411 m
Zs = -1690893.950 m
(at epoch 2013.90)
Transformation parameters ITRF08 to GDA94:
tX = ?84.68 mm
dtX = +1.42 mm/yr
tY = ?19.42 mm
dtY = +1.34 mm/yr
tZ = +32.01 mm
dtZ = +0.90 mm/yr
rX = -0.4254 msec
drX = +1.5461 msec/yr
rY = +2.2578 msec
drY = +1.1820 msec/yr
rZ = +2.4015 msec
drZ = +1.1551 msec/yr
dS = +0.00971 ppm
ddS = +0.000109 ppm/yr
t0 = 1994.00
Then corrections due to rate of change to each of the 7 transformation parameters for the period (t-t0), taking care to convert the translations to the same units as the source CRS (in this case metres) and the rotations to radians, are:
tX' = ?84.68 + [(1.42 * (2013.90-1994.00)] = ?56.42 mm = ?0.056m
Similarly, tY' = +0.007 m, tZ' = +0.050 m, rX' = 1.471021E-07 rad,
rY' = 1.249830E-07 rad, rZ' = 1.230844E-07 rad and dS' = 0.01188 ppm from which M' = 1.00000001188
Using these parameter values, application of the 7 parameter Coordinate Frame transformation results in:
Xt = -3789470.008 m
Yt = 4841770.685 m
Zt = -1690895.103 m
on the GDA94 geocentric coordinate reference system.
This converts into:
Latitude = 15 deg 28 min 32.406 sec S
Longitude = 128 deg 02 min 56.174 sec E
Ellipsoidal height = 46.244m
on the GDA94 geographic 3D coordinate reference system. For the 2D equivalent the height is ignored.","Note the analogy with the Time-dependent Position Vector transformation (code 1054) but beware of the differences! The Position Vector convention is used by IAG. See methods 1056 and 1058 for similar tfms operating between other CRS types.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2013.021 2018.001 2019.006",0
1058,"Time-dependent Coordinate Frame rotation (geog3D)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Time-dependent Coordinate Frame rotation (geog3D domain) transformation has 3 steps:
(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Time-dependent Coordinate Frame rotation(geocentric domain) method, EPSG method code 1056;
(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","Input point:
Coordinate reference system: WGS 84 (G1674) = ITRF08 (geographic 2D)
Latitude = 15 deg 28 min 32.368 sec S
Longitude = 128 deg 02 min 56.198 sec E
Ellipsoidal height hs = 46.244 m
at epoch 2013.90
This transforms to WGS 84 (G1674) = ITRF2008 Cartesian geocentric coordinates:
Xs = -3789470.702 m
Ys = 4841770.411 m
Zs = -1690893.950 m
(at epoch 2013.9)
Transformation parameters ITRF08 to GDA94:
tX = ?84.68 mm
dtX = +1.42 mm/yr
tY = ?19.42 mm
dtY = +1.34 mm/yr
tZ = +32.01 mm
dtZ = +0.90 mm/yr
rX = -0.4254 msec
drX = +1.5461 msec/yr
rY = +2.2578 msec
drY = +1.1820 msec/yr
rZ = +2.4015 msec
drZ = +1.1551 msec/yr
dS = +0.00971 ppm
ddS = +0.000109 ppm/yr
t0 = 1994.00
Corrections due to rate of change to each of the 7 transformation parameters for the period (t-t0), taking care to convert the translations to the same units as the source CRS (in this case metres) and the rotations to radians, are:
tX' = ?84.68 + [(1.42 * (2013.90-1994.00)] = ?56.42 mm = ?0.056m
tY' = +0.007 m
tZ' = +0.050 m
rX' = +1.471021E-07 rad
rY' = +1.249830E-07 rad
rZ' = +1.230844E-07 rad
dS' = +0.01188 ppm
from which M = 1.00000001188
Using these parameter values, application of the 7 parameter Coordinate Frame rotation formula results in:
Xt = ?3789470.008 m
Yt = 4841770.685 m
Zt = ?1690895.103 m
on the GDA94 geocentric coordinate reference system.
This converts into:
Latitude = 15 deg 28 min 32.406 sec S
Longitude = 128 deg 02 min 56.174 sec E
Ellipsoidal height = +46.244m
on the GDA94 geographic 3D coordinate reference system.","Note the analogy with the Time-dependent Position Vector transformation (code 1055) but beware of the differences! The Position Vector convention is used by IAG. See method codes 1056 and 1057 for similar methods operating between other CRS types.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2013.021 2019.006",0
1059,"Geographic3D to GravityRelatedHeight (PNG)",0,"This transformation involves the application of a height difference interpolated from a height correction model. The model provides height difference values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the height difference is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the height correction (C) of the target datum above the ellipsoid of the source Geographic 3D CRS. C differs from the geoid-ellipsoid separation N because a vertical datum is a realisation of the geoid surface, not the geoid itself.
Then:
H = h - C
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","Transformation of the vertical component of a Geographic 3D CRS to a Vertical CRS.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",12/15/2016 0:00:00,"2015.052 2016.050",0
1060,"Geographic3D to GravityRelatedHeight (CGG2013)",0,"This transformation involves the application of a geoid-ellipsoid separation value interpolated from a geoid model. The model provides separation values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the separation is achieved through a bi-quadratic interpolation of the grid, using the latitude and longitude of the point. This step provides the geoid-ellipsoid separation (N) above the ellipsoid of the source Geographic 3D CRS.
Then:
H = h - N
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","For consistency with earlier models the Information Source references software which uses bi-quadratic interpolation. Because of denser node spacing in CGG2013 bi-linear interpolation will be sufficient for most uses. See Info Source for file format doc.","Geodetic Survey Division, Natural Resources Canada.","IOGP",12/15/2016 0:00:00,"2013.064 2016.050",0
1061,"Molodensky-Badekas (PV geocentric domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
To eliminate high correlation between the translations and rotations in the derivation of parameter values for the Helmert transformation methods (coordinate operation method codes 1032 and 1033), instead of the rotations being derived about the geocentric coordinate reference system origin they may be derived at a location within the points used in the determination. Three additional parameters, the coordinates of the rotation point, are then required. The formula is:
(Xt) ( 1 -rZ +rY) (Xs - Xp) (Xp) (tX)
(Yt) = M * ( +rZ 1 -rX) * (Ys - Yp) + (Yp) + (tY)
(Zt) ( -rY +rX 1 ) (Zs - Zp) (Zp) (tZ)
and the parameters are defined as:
(tX, tY, tZ): Translation vector, to be added to the point's position vector in the source coordinate system in order to transform from source coordinate reference system to target coordinate reference system; also: the coordinates of the origin of source coordinate reference system in the target frame.
(rX, rY, rZ): Rotations to be applied to the point's vector. The sign convention is such that a positive rotation about an axis is defined as a clockwise rotation of the position vector when viewed from the origin of the Cartesian coordinate reference system in the positive direction of that axis; e.g. a positive rotation about the Z-axis only from source system to target system will result in a larger longitude value for the point in the target system. Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.
(Xp, Yp, Zp): Coordinates of the point about which the coordinate reference frame is rotated, given in the source Cartesian coordinate reference system.
M: Multiplication factor to be applied to the position vector in the source coordinate reference system in order to obtain the correct scale of the target coordinate reference system. M = (1+dS) where dS is the scale difference. When dS is expressed in parts per million, M = (1+dS*10^-6). When dS is the scale difference expressed in parts per billion, M = (1+dS*10^-9).
Reversibility.
The Molodensky-Badekas (PV) transformation in a strict mathematical sense is not reversible, i.e. in principle the same parameter values cannot be used to execute the reverse transformation. This is because the evaluation point coordinates are in the forward direction source coordinate reference system and the rotations have been derived about this point. They should not be applied about the point having the same coordinate values in the target coordinate reference system, as is required for the reverse transformation. However, in practical application there are exceptions when applied to the approximation of small differences between the geometry of a set of points in two different coordinate reference systems. The typical vector difference in coordinate values is in the order of 6*10^1 to 6*10^2 metres, whereas the evaluation point on or near the surface of the earth is 6.3*10^6 metres from the origin of the coordinate systems at the Earth's centre. This difference of four or five orders of magnitude allows the transformation in practice to be considered reversible. Note that in the reverse transformation, only the signs of the translations and rotation parameter values are reversed; the coordinates of the evaluation point remain unchanged.","Input point:
Coordinate reference system: La Canoa
Cartesian geocentric coords:
Xs = 2 550 408.965 m
Ys = -5 749 912.266 m
Zs = 1 054 891.114 m
Transformation parameters La Canoa to REGVEN:
tX = -270.933 m
tY = +115.599 m
tZ = -360.226 m
rX = 5.266 sec = 0.000025530288 radians
rY = 1.238 sec = 0.000006001993 radians
rZ = -2.381 sec = -0.000011543414 radians
dS = -5.109 ppm
Ordinate 1 of evaluation point = 2464351.59 m
Ordinate 2 of evaluation point = -5783466.61 m
Ordinate 3 of evaluation point = 974809.81 m
First M = 1+dS = 0.999994891
Then application of the 10 parameter Molodenski-Badekas (PV) transformation results in REGVEN geocentric coordinates of:
Xt = 2 550 138.467 m
Yt = -5 749 799.862 m
Zt = 1 054 530.826 m","See method codes 1062 and 1063 for this operation in other coordinate domains and method code 1034 for opposite rotation convention in geocentric domain.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2014.039 2014.065 2019.006",0
1062,"Molodensky-Badekas (PV geog3D domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Molodensky-Badekas (geog3D domain) transformation has 3 steps:
(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Molodensky-Badekas (PV geocentric domain) method, EPSG method code 1061;
(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","Input point:
Coordinate reference system: La Canoa (geographic 3D)
Latitude = 9 deg 35 min 00.386 sec N
Longitude = 66 deg 04 min 48.091 sec W
Ellipsoidal height hS = 201.465 m
This transforms to Cartesian geocentric coords:
Xs = 2 550 408.965 m
Ys = -5 749 912.266 m
Zs = 1 054 891.114 m
Transformation parameters La Canoa to REGVEN:
tX = -270.933 m
tY = +115.599 m
tZ = -360.226 m
rX = 5.266 sec = 0.000025530288 radians
rY = 1.238 sec = 0.000006001993 radians
rZ = -2.381 sec = -0.000011543414 radians
dS = -5.109 ppm
Coordinate 1 of evaluation point = 2464351.59 m
Coordinate 2 of evaluation point = -5783466.61 m
Coordinate 3 of evaluation point = 974809.81 m
from which M = 0.999994891
Application of the Molodenski-Badekas (geocentric domain) Transformation results in REGVEN geocentric coordinates of:
Xt = 2 550 138.467 m
Yt = -5 749 799.862 m
Zt = 1 054 530.826 m
This converts into:
Latitude = 9 deg 34 min 49.001 sec N
Longitude = 66 deg 04 min 54.705 sec W
Ellipsoidal height = -18.10 m
on the REGVEN geographic 3D coordinate reference system.","See method codes 1061 and 1063 for this operation in other coordinate domains and method code 1039 for opposite rotation convention in geographic 3D domain.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2014.039 2014.065 2019.006",0
1063,"Molodensky-Badekas (PV geog2D domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Molodensky-Badekas (PV geog2D domain) transformation has 5 steps:
(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;
(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Molodensky-Badekas (PV geocentric domain) method, EPSG method code 1061;
(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","Input point:
Coordinate reference system: La Canoa (geographic 2D)
Latitude = 9 deg 35 min 00.386 sec N
Longitude = 66 deg 04 min 48.091 sec W
This is taken to be geographic 3D with an assumed Ellipsoidal height hs = 201.465 m
This transforms to Cartesian geocentric coords:
Xs = 2 550 408.965 m
Ys = -5 749 912.266 m
Zs = 1 054 891.114 m
Transformation parameters La Canoa to REGVEN:
tX = -270.933 m
tY = +115.599 m
tZ = -360.226 m
rX = 5.266 sec = 0.000025530288 radians
rY = 1.238 sec = 0.000006001993 radians
rZ = -2.381 sec = -0.000011543414 radians
dS = -5.109 ppm
Coordinate 1 of evaluation point = 2464351.59 m
Coordinate 2 of evaluation point = -5783466.61 m
Coordinate 3 of evaluation point = 974809.81 m
from which M = 0.999994891
Application of the 10 parameter Molodenski-Badekas Transformation results in REGVEN geocentric coordinates of:
Xt = 2 550 138.467 m
Yt = -5 749 799.862 m
Zt = 1 054 530.826 m
This converts into:
Latitude = 9 deg 34 min 49.001 sec N
Longitude = 66 deg 04 min 54.705 sec W
Ellipsoidal height = -18.10 m
on the REGVEN geographic 3D coordinate reference system.
Because the source coordinate reference system was 2D, the target system ellipsoidal height is ignored and the results treated as a geographic 2D coordinate reference system:
Latitude = 9 deg 34 min 49.001 sec N
Longitude = 66 deg 04 min 54.705 sec W","See method codes 1061 and 1062 for this operation in other coordinate domains and method code 9636 for opposite rotation in geographic 2D domain.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2014.039 2014.065 2018.001 2019.006",0
1064,"Point motion (geocentric Cartesian)",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The Cartesian coordinates of a point at coordinate epoch t1 may be calculated at any other coordinate epoch t2 from:
[X(t2)] = [X(t1)] + (t2 - t1) * [Vx]
[Y(t2)] = [Y(t1)] + (t2 - t1) * [Vy]
[Z(t2)] = [Z(t1)] + (t2 - t1) * [Vz]","Given a point with geocentric Cartesian 3D coordinates at coordinate epoch 2005.00 and having linear velocities:
X = 2845456.0813 m VX = ?0.0212 m/yr
Y = 2160954.2453 m VY = +0.0124 m/yr
Z = 5265993.2296 m VZ = +0.0072 m/yr
t = 2005.00 (years)
whose coordinates are required at coordinate epoch 2010.00.
X(t=2010.00) = 2845456.0813 + (?0.0212) * (2010.00 - 2005.00)
Y(t=2010.00) = 2160954.2453 + (+0.0124) * (2010.00 - 2005.00)
Z(t=2010.00) = 5265993.2296 + (+0.0072) * (2010.00 - 2005.00)
Then
X = 2845455.9753 m
Y = 2160954.3073 m
Z = 5265993.2656 m
t = 2010.00 (years)",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",9/23/2017 0:00:00,"2016.060",0
1065,"Time-specific Position Vector transform (geocen)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The time-specific ooordinate transformation is an alternative approach to the time-dependent coordinate transformation. In this method the transformation parameter values are declared for a specified transformation reference epoch (an additional parameter for the transformation) and coordinates are adjusted to this epoch before the transformation is applied. This requires that the velocities for points whose coordinates are to be transformed are available. The time-specific transformation may then be applied as part of a concatenated coordinate operation in conjunction with one or two point motion operations (see EPSG Guidance Note 7-3 section 3) in two or three steps:
i) change the source CRS Cartesian coordinates from their dataset coordinate epoch to the transformation reference epoch at which the time-specific transformation is valid;
ii) apply the Time-specific Position Vector transformation;
iii) change the target CRS Cartesian coordinates from the transformation reference epoch at which the time-specific transformation is valid to any other desired coordinate epoch.
The transformation reference epoch of the Time-specific Position Vector transformation is used as the target epoch (t2) in step (i) and as the source epoch (t1) in step (iii). It is not used in step (ii).","The same example as for the Time-specific Coordinate Frame rotation (coordinate operation method 1066) can be calculated, however in Step 2 the following transformation parameters have to be applied to achieve the same input and output in terms of coordinate values:
Transformation parameters for the Position Vector transformation convention:
tX = ?0.003 m
tY = ?0.001 m
tZ = 0.000 m
rX = -0.019 msec
rY = 0.042 msec
rZ = -0.002 msec
dS = 0.000 ppm
from which M = 1.0
t = 2010.00 years
Please note that only the rotation has changed sign as compared to the Coordinate Frame rotation.","Note the analogy with the Time-specifc Coordinate Frame method (code 1066) but beware of the differences!","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2016.061 2019.006",0
1066,"Time-specific Coordinate Frame rotation (geocen)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The time-specific ooordinate transformation is an alternative approach to the time-dependent coordinate transformation. In this method the transformation parameter values are declared for a specified transformation reference epoch (an additional parameter for the transformation) and coordinates are adjusted to this epoch before the transformation is applied. This requires that the velocities for points whose coordinates are to be transformed are available. The time-specific transformation may then be applied as part of a concatenated coordinate operation in conjunction with one or two point motion operations (see EPSG Guidance Note 7-3 section 3) in two or three steps:
i) change the source CRS Cartesian coordinates from their dataset coordinate epoch to the transformation reference epoch at which the time-specific transformation is valid;
ii) apply the Time-specific Coordinate Frame rotation transformation;
iii) change the target CRS Cartesian coordinates from the transformation reference epoch at which the time-specific transformation is valid to any other desired coordinate epoch.
The transformation reference epoch of the Time-specific Coordinate Frame rotation transformation is used as the target epoch (t2) in step (i) and as the source epoch (t1) in step (iii). It is not used in step (ii).","Initial coordinates are referenced to the ITRF2008 CRS (EPSG code 5332) at coordinate epoch 2005.00. (Note: this happens to be the frame reference epoch for ITRF2008 but in other circumstancs the coordinate set could be referenced to a coordinate epoch different from the CRS's frame reference epoch).
Coordinates are required to be referenced to the PZ-90.11 CRS (EPSG code 7679) at coordinate epoch 2013.90.
Input point geocentric Cartesian coordinates and linear velocities:
XS = 2845 456.0813 m VX = ?0.0212 m/yr
YS = 2160 954.2453 m VY = +0.0124 m/yr
ZS = 5265 993.2296 m VZ = +0.0072 m/yr
t = 2005.00 (years)
These are referenced to ITRF2008 at coordinate epoch 2005.00
Step 1: convert the coordinates in the source CRS from the dataset coordinate epoch (2005.00) to the epoch of validity of the time-specific transformation (2010.00) using the Point Motion (geocentric Cartesian) method described in EPSG Guidance Note 7-2.
XS(t=2010.00) = 2845456.0813 + (?0.0212) * (2010.0 - 2005.0) * 2845455.9753 m
YS(t=2010.00) = 2160954.2453 + (+0.0124) * (2010.0 - 2005.0) * 2160954.3073 m
ZS(t=2010.00) = 5265993.2296 + (+0.0072) * (2010.0 - 2005.0) * 5265993.2656 m
Step 2: transform the coordinates in the source CRS at coordinate epoch 2010.00 to the target CRS at coordinate epoch 2010.00 using the Coordinate Frame Rotation (geocentric domain) method described in EPSG Guidance Note 7-2. The time-specific transformation from PZ-90.11 to ITRF2008 (EPSG Dataset transformation code 7960) parameter values are:
tX = ?0.003 m
tY = ?0.001 m
tZ = 0.000 m
rX = 0.019 msec
rY = ?0.042 msec
rZ = 0.002 msec
dS = 0.000 ppm
from which M = 1.0
t = 2010.00 years
This uses the Coordinate Frame rotation convention.
Because the transformation is required in the reverse direction to that defined, the relevant parameters need to have their signs reversed. Transformation parameter values (from ITRF2008 to PZ-90.11) then are:
tX = 0.003 m
tY = 0.001 m
tZ = 0.000 m
rX = ?0.019 msec = ?9.21145994108117E-11 radians
rY = 0.042 msec = 2.03621746066005E-10 radians
rZ = ?0.002 msec = ?9.69627362219071E-12 radians
M = 1.0
t = 2010.00 (years)
Then:
XT(t=2010.00) = 2845455.9772m
YT(t=2010.00) = 2160954.3078m
ZT(t=2010.00) = 5265993.2652m
Step 3: convert the coordinates in the target CRS from epoch 2010.00 to the required coordinate epoch 2013.90 using the Point Motion (geocentric Cartesian) method:
XT(t=2013.90) = 2845455.9772 + [?0.0212 * (2013.90 - 2010.00)] = 2845455.8945m
YT(t=2013.90) = 2160954.3078 + [+0.0124 * (2013.90 - 2010.00)] = 2160954.3562m
ZT(t=2013.90) = 5265993.2652 + [+0.0072 * (2013.90 - 2010.00)] = 5265993.2945m","Note the analogy with the Time-specific Position Vector method (code 1065) but beware of the differences!","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2016.061 2018.001 2019.006",0
1067,"Point motion (ellipsoidal)",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The ellipsoidal coordinates of a point at coordinate epoch t1 may be calculated at any other coordinate epoch t2 from:
[Lat(t2)] = [Lat(t1)] + (t2-t1) * [V_lat]
[Lon(t2)] = [Lon(t1)] + (t2-t1) * [V_lon]
[h(t2)] = [h(t1)] + (t2-t1) * [V_up]
where V describes the velocities of the ellipsoidal coordinates. These are usually given as a rate in linear units (say millimetres per year) resolved into north, east and up components VN, VE and Vh. The north and east components are converted into latitude and longitude components by:
V_lat = VN / (rho+h)
V_lon = VE / [(nu+h) cos(lat)]
where
rho is radius of curvature of the CRS's ellipsoid in the plane of the meridian at latitude lat, where
rho = a(1 ? e^2)/(1 ? e^2sin^2(lat))^3/2
nu is radius of curvature of the ellipsoid perpendicular to the meridian at latitude lat, where
nu = a /(1 ? e^2sin^2(lat))^1/2","Given a point with ellipsoidal coordinates at coordinate epoch 2017.55 and having linear velocities:
lat = 51.0?N VN = +15.12 mm/yr
lon = 141.0?W VE = ?2.86 mm/yr
h = 1000 m V_up = +1.10 mm/yr
t = 2017.55 (years)
whose coordinates are required at epoch 1997.00. The point is referenced to a CRS which uses the GRS 1980 ellipsoid for which a = 6378137.0m and e^2 = 0.006694380.
Then (after conversion to appropriate units):
V_lat = 2.37174E-09 rad/yr
V_lon = -7.10972E-07 rad/yr
V_up = 0.0011 m / yr
and
lat = 0.890117919 + (1997.00-2017.55) * 2.37174E-09 = 0.890117870 radians = 50?59'59.990""N
lon = 0.890117919 + (1997.00-2017.55) * -7.10972E-07 = -2.460914231 radians = 140?59'59.997""W
h = 1000.0 + (1997.00-2017.55) * 0.0011 = 999.977m
t = 1997.00",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",9/23/2017 0:00:00,"2016.060",0
1068,"Height Depth Reversal",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
If there is a requirement to either (i) change a height in a source CRS to a depth in a target CRS referenced to the same vertical datum, or (ii) change a depth in the a source CRS to a height in a target CRS referenced to the same vertical datum, then the following generic formula can be used:
X2 = ?1 * X1","1. Source CRS = MSL height, in which a height = -300m. Then target CRS = MSL depth, and the depth = (-300 * -1) = 300m
2. Source CRS = NAVD88 depth (ftUS), in which a depth = 500 ftUS. Then target CRS = NAVD88 height (ftUS), and the height = (500 * -1) = -500 ftUS","This is a parameter-less conversion.","EPSG Guidance Note #7-2.","IOGP",2/21/2019 0:00:00,"2014.027 2019.002",0
1069,"Change of Vertical Unit",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
If there is a requirement to change the linear unit of a vertical axis, the following formulae can be used:
for the forward transformation: X2 = (X1 * U1) / U2
for the reverse transformation: X1 = (X2 * U2) / U1
where U1 and U2 are unit conversion ratios for CRS1 and CRS2 respectively. U = [(factor b) / (factor c)] from the EPSG Dataset Unit of Measure table, populated with respect to the SI base unit, for linear units the metre.
Then
for the forward transformation:
X2 = (X1 * (b1 / c1)) / (b2 / c2) = (X1 * b1 * c2) / (b2 * c1)
for the reverse transformation:
X1 = [X2 * (b2 / c2)]/ (b1 / c1) = (X2 * b2 * c1) / (b1 * c2)","Conversion of a height of 25 metres into US survey feet:
For the metre (EPSG UoM code 9001) factor b = 1 and factor c = 1, so U1 has a value of 1.
For the US survey foot (EPSG UoM code 9003) factor b = 12 and factor c = 39.37, so U2 has a value of approximately 0.304800609601219 (one US survey foot ? 0.304800609601219 metres).
Then H(ftUS) = (25 * (1 / 1)) / (12 / 39.37) = 82.02 ftUS.
For the reverse conversion of 82.02 ftUS into metres:
U1 and U2 are as for the forward case, with values of 1/1 and 12/39.37 respectively.
Then Hm = (82.02 * (12 / 39.37) / (1 / 1)) = 82.02 m.",,"EPSG Guidance Note #7-2.","IOGP",1/19/2018 0:00:00,"2014.027 2018.004",0
1070,"Point motion by grid (Canada NTv2_Vel)",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The NTv2_Vel grid contains point velocities resolved into north, east, and up components.
First the three velocities at a point are interpolated within the NTv2_Vel grid using bi-linear interpolation within the source and target CRS.
Then ellipsoidal coordinates of a point at coordinate epoch t1 may be calculated at any other coordinate epoch t2 from:
[Lat(t2)] = [Lat(t1)] + (t2-t1) * [V_lat]
[Lon(t2)] = [Lon(t1)] + (t2-t1) * [V_lon]
[h(t2)] = [h(t1)] + (t2-t1) * [V_up]
where V describes the velocities of the ellipsoidal coordinates. These are given as a rate in linear units (metres per year) resolved into north, east and up components VN, VE and Vh. The north and east components are converted into latitude and longitude components by:
V_lat = VN / (rho+h)
V_lon = VE / [(nu+h) cos(lat)]
where
rho is radius of curvature of the CRS's ellipsoid in the plane of the meridian at latitude lat
rho = a(1 ? e^2)/(1 ? e^2sin^2(lat))^3/2
nu is radius of curvature of the ellipsoid perpendicular to the meridian at latitude lat
nu = a /(1 ? e^2sin^2(lat))^1/2","For geodetic station Centennial Monument NCC100 with ellipsoidal coordinates referenced to NAD83(CSRS)v6 at coordinate epoch 2010.00:
lat = 45?25'45.714920""N
lon = 75?42'05.960075""W
h = 39.524 m
t = 2010.00 (years)
whose coordinates are required at coordinate epoch 2002.00.
First interpolate within the grid file for velocities at lat = 45?25'45.714920""N, lon = 75?42'05.960075""W:
VN = -0.00156 m/yr
VE = 0.00177 m/yr
V_up = 0.00202 m/yr
NAD83(CSRS)v6 uses the GRS 1980 ellipsoid for which a = 6378137.0m and e^2 = 0.006694380.
Then:
V_lat = -2.44979E-10 rad/yr
V_lon = 3.94759E-10 rad/yr
and
lat = 0.792892001 + (2002.00-2010.00) * -2.44979E-10 = 0.792892003 radians = 45?25'45.715324""N
lon = -1.321243139 + (2002.00-2010.00) * 3.94759E-10 = -1.321243142 radians = 75?42'05.960726""W
h = 39.524 + (2002.00-2010.00) * 0.00202 = 39.508m
t = 2002.00","Interpolation within the grid is in the horizontal component of the source CRS.","Natural Resources Canada.","IOGP",8/29/2018 0:00:00,"2016.060 2018.034",0
1071,"Vertical Offset by Grid Interpolation (NZLVD)",1,"The relationship between NZVD16 height and thirteen local gravity-related coordinate reference systems (LVDs) for New Zealand is available through gridded data files of offsets (sometimes called height differences). The vertical offset at a point is first interpolated within the grid of values using bi-linear interpolation. For interpolation within the offset grids, horizontal positions in the NZGD2000 geographic 2D CRS are used.
The interpolated value is then applied as an offset:
? If the height in a LVD is required when a NZVD2016 height is given, add the interpolated offset to the NZVD2016 height (LVD height = NZVD2016 height + interpolated offset).
? If the NZVD2016 height is required when a LVD height is given, subtract the interpolated offset from the LVD height (NZVD2016 height = LVD height - interpolated offset).
Care! In the EPSG Dataset, offsets are additive (for the forward transformation). See EPSG Guidance Note 7-2, Implementation Notes - Offsets. The LINZ convention assumes that the offset is subtracted, and grid file names reflect that convention. The offset files therefore have names which conflict with the source and target CRSs of the EPSG transformation.","For a point in the Dunedin 1958 height area at 44.42?S, 168.92?E (NZGD2000) with NZVD2016 height of 50.000 metres, to find its Dunedin 1958 height:
First obtain the offsets at each of the surrounding grid nodes:
NW corner 44.4000?S, 168.9000?E, offset = 0.3028m
NE corner 44.4000?S, 168.9333?E, offset = 0.3029m
SE corner 44.4333?S, 168.9333?E, offset = 0.3055m
SW corner 44.4333?S, 168.9000?E, offset = 0.3051m
Then using bi-linear interpolation for 44.42?S, 168.92?E, offset = 0.304m
Then Dunedin 1958 height = 50.000 + 0.304 = 50.304m.
For the reverse calculation to find the NZVD2016 height from a point with Dunedin 1958 height of 50.304m:
First obtain the offsets at each of the surrounding grid nodes:
NW corner 44.4000?S, 168.9000?E, offset = 0.3028m
NE corner 44.4000?S, 168.9333?E, offset = 0.3029m
SE corner 44.4333?S, 168.9333?E, offset = 0.3055m
SW corner 44.4333?S, 168.9000?E, offset = 0.3051m
Then using bi-linear interpolation for 44.42?S, 168.92?E, offset = 0.304m
Then NZVD2016 height = 50.304 - 0.304 = 50.000m.",,"IOGP","IOGP",3/22/2019 0:00:00,"2016.031 2019.011",0
1072,"Geographic3D to GravityRelatedHeight (OSGM15-Ire)",0,"This transformation involves the application of a height difference interpolated from a height correction model. The model provides height difference values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the height difference is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the height correction (C) of the target datum above the ellipsoid of the source Geographic 3D CRS. C differs from the geoid-ellipsoid separation N because a vertical datum is a realisation of the geoid surface, not the geoid itself.
Then:
H = h - C
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","Transformation of the vertical component of a Geographic 3D CRS to a Vertical CRS.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",11/22/2016 0:00:00,"2016.049",0
1073,"Geographic3D to GravityRelatedHeight (IGN2009)",0,"This transformation involves the application of a height difference interpolated from a height correction model. The model provides height difference values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the height difference is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the height correction (C) of the target datum above the ellipsoid of the source Geographic 3D CRS. C differs from the geoid-ellipsoid separation N because a vertical datum is a realisation of the geoid surface, not the geoid itself.
Then:
H = h - C
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","Transformation of the vertical component of a Geographic 3D CRS to a Vertical CRS. File header of method code 9664 (4 lines) has changed in this method (1 line); recommended interpolation method now in a separate XML file with same name as the grid.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",1/6/2018 0:00:00,"2017.045",0
1074,"NADCON5 (2D)",1,"The latitude and longitude offsets at a point are derived by interpolation within the gridded data. Separate grid files are given for latitude and longitude offsets. The grid file format is given in documentation available from the information source. Bi-quadratic interpolation is used to derive the offset values. For the forward calculation the interpolated value of the offset is then added to the source CRS coordinate value to give the coordinates in the target CRS.
Reversibility
Iteration is required for the reverse transformation. The coordinate reference system for the coordinates of the grid nodes is the source coordinate reference system for the forward transformation. Then in forward transformations the offset is obtained through straightforward interpolation of the grid file. But for the reverse transformation the first grid interpolation entry will be the value of the point in the second coordinate reference system, the offsets are interpolated and applied with sign reversed, and the result used in further iterations of interpolation and application of offset until the difference between results from successive iterations is insignificant.","(none)","Geodetic transformation operating on geographic coordinate differences by bi-quadratic interpolation. Input expects longitudes to be positive east in range 0-360? (0? = Greenwich).","National Geodetic Survey - http://www.ngs.noaa.gov","IOGP",4/16/2018 0:00:00,"2018.017",0
1075,"NADCON5 (3D)",1,"The latitude, longitude and ellipsoidal height offsets at a point are derived by interpolation within the gridded data. Separate grid files are given for latitude, longitude and ellipsoidal height offsets. The grid file format is given in documentation available from the information source. Bi-quadratic interpolation is used to derive the offset values. For the forward calculation the interpolated value of the offset is then added to the source CRS coordinate value to give the coordinates in the target CRS.
Reversibility
Iteration is required for the reverse transformation. The coordinate reference system for the coordinates of the grid nodes is the source coordinate reference system for the forward transformation. Then in forward transformations the offset is obtained through straightforward interpolation of the grid file. But for the reverse transformation the first grid interpolation entry will be the value of the point in the second coordinate reference system, the offsets are interpolated and applied with sign reversed, and the result used in further iterations of interpolation and application of offset until the difference between results from successive iterations is insignificant.","(none)","Geodetic transformation operating on geographic coordinate differences by bi-quadratic interpolation. Input expects longitudes to be positive east in range 0-360? (0? = Greenwich).","National Geodetic Survey - http://www.ngs.noaa.gov","IOGP",4/16/2018 0:00:00,"2018.017",0
1076,"Wellbore local to geographic (LMP)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
LMP requires that the input local horizontal coordinates are referenced to true north, and that depths (D) are referenced to the reference surface of the vertical CRS. Depths are required as input but are not changed by the algorithm. They are used to compute the depth correction factor and therefore a depth of zero must be close to Mean Sea Level (strictly speaking to the reference ellipsoid).
Required inputs are:
Latitude of the Well Reference Point (WRP) in the target 2D geographic CRS;
Longitude of the Well Reference Point in the target 2D geographic CRS;
dcf flag: 0 = no depth correction is applied; 2 = a variable depth correction factor along the path is applied;
n[]: Array of length num_points containing local northing coordinates, oriented to true north;
e[]: Array of length num_points containing local easting coordinates, oriented to true east;
D[]: Array of length num_points containing depths below the Vertical Reference Surface at survey stations down the wellbore (referred to as TVD in the SPE paper).
The output of LMP is ellipsoidal coordinates of the wellbore trajectory in a geographic 2D CRS. If required, projected coordinates of the wellbore may be calculated from these.
The following equations assume that the local coordinates (n,e) and depth D are in metres. If these use a different unit of measure, then as a first step they should be converted to metres.
When dcf flag = 2, output arrays LAT, LON (in radians) are computed as:
for i=1:num_points
? = a (1 ? e^2) / (1 ? e^2 sin^2(LAT[i-1]))^3/2
? = a / (1 ? e^2 sin^2(LAT[i-1]))^1/2
LAT[i] = LAT[i-1] + (n[i] ? n[i-1]) / (? ? D[i])
LON[i] = LON[i-1] + (e[i] ? e[i-1]) / (? ? D[i]) / cos(LAT[i-1])
where for i=1, for the first station: LAT[0] and LON[0] are the geographic coordinates of the WRP in radians; n[0] = 0, e[0] = 0.
and
a is the semi-major axis of the reference ellipsoid;
e is the eccentricity of the reference ellipsoid (not to be confused with local easting);
? is the radius of curvature in the meridian;
? is the radius of curvature in the prime vertical.
When dcf flag = 0, the equations above are modified to:
LAT[i] = LAT[i-1] + (n[i] ? n[i-1]) / ?
LON[i] = LON[i-1] + (e[i] ? e[i-1]) / {? cos(LAT[i-1])}
Reverse case:
To compute (n,e) from (LAT,LON,D):
For i=1:num_points
? = a (1 ? e^2) / (1 ? e^2 sin^2(LAT[i-1]))^3/2
? = a / (1 ? e^2 sin^2(LAT[i-1]))^1/2
n[i] = n[i-1] + (LAT[i] ? LAT[i-1]) * (? ? D[i])
e[i] = e[i-1] + (LON[i] ? LON[i-1]) * (? ? D[i]) * cos(LAT[i-1])
where for the first station i=0 variables are initialised as in the forward case. Note that LAT[i-1], the latitude of the previous point, is used in the reverse formulas to compute the radii of curvature in order to be internally consistent in round trip calculations.
When dcf flag = 0, the equations above are modified to:
n[i] = n[i-1] + (LAT[i] ? LAT[i-1]) * ?
e[i] = e[i-1] + (LON[i] ? LON[i-1]) * ? * cos(LAT[i-1])","An example computation for the LMP method is provided below. This is not a realistic wellbore survey but demonstrates the computation steps. A straight path is modelled to a bottom hole at 5 km depth, approximately 14 km away from the surface location, using a constant inclination 70.54002?, azimuth 51.3178? from true north, and survey stations every 30.0167 metre.
Parameters:
Ellipsoid: WGS 84, a = 6378137 metres, 1/f = 298.257223563
then e = 0.0818191908 and e^2 = 0. 00669437999
Forward calculation for a well with surface coordinates (origin of the local coordinates)
Latitude of Well Reference Point 9.0364081? = 0.1577150739 rad
Longitude of Well Reference Point -30.2713073? = -0.5283339813 rad
Depth Correction dcf flag = 2
Zero Depth Point MSL height H = 25 m (not used in equations)
Local northing (n) and easting (e) given in the table are the unscaled output of an algorithm such as minimum curvature as commonly applied in wellbore survey (described as cubical coordinates in the SPE paper). Axes n and e are aligned with true north and true east, respectively.
Initialisation: LAT[0] = 0.1577150739 rad; LON[0] = -0.5283339813 rad.
i D(m) n(m) e(m) ?(m) ?(m) LAT(rad) LON(rad)
1 0.00 0.00 0.00 6337009.005 6378663.709 0.1577150739 -0.5283339813
2 10.00 15.00 24.00 6337009.005 6378663.709 0.1577174409 -0.5283301714
3 20.00 30.00 48.00 6337009.052 6378663.724 0.1577198080 -0.5283263616
4 30.00 45.00 72.00 6337009.099 6378663.740 0.1577221751 -0.5283225517
5 40.00 60.00 96.00 6337009.145 6378663.756 0.1577245421 -0.5283187419
...
497 4960.00 7440.00 11904.00 6337032.231 6378,671.502 0.1588895878 -0.5264433944
498 4970.00 7455.00 11928.00 6337032.278 6378,671.517 0.1588919567 -0.5264395808
499 4980.00 7470.00 11952.00 6337032.325 6378,671.533 0.1588943256 -0.5264357673
500 4990.00 7485.00 11976.00 6337032.373 6378,671.549 0.1588966945 -0.5264319538
501 5000.00 7500.00 12000.00 6337032.420 6378,671.565 0.1588990634 -0.5264281403","For wellbore survey.","SPE paper 96813 ""Accounting for Earth Curvature in Directional Drilling"", 2005.","IOGP",2/17/2019 0:00:00,"2017.040",0
1077,"Wellbore local to projected (GNL)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users
are encouraged to use that document rather than the text which follows as reference
because limitations in the transcription will be avoided.
The method has options for the application of two corrections:
i) point scale factor (psf) is used to scale horizontal offsets to the local scale for the projected CRS, resolved into east and north components (when a conformal map projection is used, psf_E = psf_N): (i) no psf correction is applied, (ii) the point scale factor at the well reference point is applied to the complete wellbore, and (iii) the point scale factor at each point along the wellbore is applied.
ii) depth correction factor (dcf) is used to scale horizontal offsets according to the depth of the point in the wellbore: (i) no dcf correction is applied, or (ii) a dcf correction is applied at each point along the wellbore.
The inputs are:
WRP_E : Easting of the Well Reference Point in the target projected CRS;
WRP_N : Northing of the Well Reference Point in the target projected CRS;
psf flag : 0 for none; 1 for constant at first point; 2 to apply a variable point scale factor;
dcf flag : 0 for none; 2 to apply a variable depth correction factor along the path;
n[] : Array of length num_points containing local northing coordinates, oriented to Grid North of the target projected CRS;
e[] : Array of length num_points containing local easting coordinates, oriented to Grid East of the target projected CRS;
D[] : Array of length num_points containing depths below the Vertical Reference Surface at survey stations down the wellbore.
Care must be taken to convert local easting and northing to the same unit as used by the target projected CRS before applying the following equations.
If psf flag=0 and dcf flag=0 then the target coordinates are computed using a simple 2D translation:
E = WRP_E + e
N = WRP_N + n
If psf flag=1 and dcf flag=0 then the target coordinates are computed using a scaled offset as:
E = WRP_E + e * psf_E
N = WRP_N + n * psf_N
If psf flag=2 and dcf flag=2 then an iterative method is needed to compute the variable point scale factor and depth correction factor along the wellbore. The LMP method is recommended in this situation, but if the GNL method is used then map convergence should also be calculated variable along-hole to reference the local coordinates to grid north.
Output arrays E and N of length num_points are computed as follows:
For i=1:num_points
for psf_E and psf_N, compute_psf(E[i-1], N[i-1])
dcf = R / (R ? D[i])
E[i] = E[i-1] + (e[i] ? e[i-1]) * psf_E * dcf
N[i] = N[i-1] + (n[i] ? n[i-1]) * psf_N * dcf
where for i=1, for the first station: E[0] = WRP_E; N[0] = WRP_N; e[0] = 0; n[0] = 0, and:
compute_psf() is a function that returns the local point scale factors in easting and northing directions at the given location;
R = 6371000 m. Note: Using a constant rather than the ellipsoidal local radii of curvature for the depth correction factors in easting and northing directions at the given location results in an insignificantly small error (e.g., for a lateral of 10 km at 10 km depth the worst case difference is 8 centimetres: 6371/(6371-10)=1.001572 vs. 6340/(6340-10)=1.001580).
Reverse case: to compute (n,e) from given (E,N):
For psf_flag=0 and dcf_flag=0:
n = N ? WRP_N
e = E ? WRP_E
For psf_flag=1 and dcf_flag=0:
n = (N ? WRP_N / psf_N
e = (E ? WRP_E) / psf_E
For psf_flag=2 and dcf_flag=2 (additionally requiring D as input):
For i=1:num_points
for psf_E and psf_N, compute_psf(E[i-1], N[i-1])
dcf = R / (R ? D[i])
n[i] = n[i-1] + (N[i] ? N[i-1]) / (psf_N * dcf_N)
e[i] = e[i-1] + (E[i] ? E[i-1]) / (psf_E * dcf_E)
where for the first station i=0 is initialised as in the forward case.","The example here is not a realistic wellbore survey but demonstrates the magnitude of differences in computed coordinates when different corrections are applied. A straight path is modelled to a bottom hole at 5 km depth, approximately 14 km away from the surface location, using a constant inclination 70.54002?, bearing 51.3178? from grid north, and survey stations every 30.0167 metre.
Forward calculation for a well with surface coordinates in WGS 84 / UTM zone 25N, EPSG CRS code 32625. Since this is a conformal map projection, psf_E = psf_N.
Easting of Well Reference Point WRP_E = 800,000 m
Northing of Well Reference Point WRP_N = 1,000,000 m
Point scale factor at WRP psf_E = psf_N = 1.00071
Zero Depth Point MSL height H = 25 m (not used in equations)
Local axes n and e are aligned with grid north and grid east, respectively.
Initialisation: E[0] = 800,000 m; N[0] = 1,000,000 m.
psf_flag=0;dcf_flag=0 psf_flag=1;dcf_flag=0 psf_flag=2;dcf_flag=2
i D(m) n(m) e(m) psfE=psfN dcf E(m) N(m) E(m) N(m) E(m) N(m)
1 0.00 0.00 0.00 1.000714 1.000000 800000.00 1000000.00 800000.00 1000000.00 800000.00 1000000.00
2 10.00 15.00 24.00 1.000714 1.000002 800000.00 1000000.00 800000.00 1000000.00 800000.00 1000000.00
3 20.00 30.00 48.00 1.000714 1.000003 800024.00 1000015.00 800024.02 1000015.01 800024.02 1000015.01
4 30.00 45.00 72.00 1.000714 1.000005 800048.00 1000030.00 800048.03 1000030.02 800048.03 1000030.02
5 40.00 60.00 96.00 1.000715 1.000006 800072.00 1000045.00 800072.05 1000045.03 800072.05 1000045.03
?
497 4960.00 7440.00 11904.00 1.000804 1.000779 811904.00 1007440.00 811912.50 1007445.31 811917.68 1007448.55
498 4970.00 7455.00 11928.00 1.000804 1.000781 811928.00 1007455.00 811936.52 1007460.32 811941.72 1007463.57
499 4980.00 7470.00 11952.00 1.000804 1.000782 811952.00 1007470.00 811960.53 1007475.33 811965.76 1007478.60
500 4990.00 7485.00 11976.00 1.000804 1.000784 811976.00 1007485.00 811984.55 1007490.34 811989.79 1007493.62
501 5000.00 7500.00 12000.00 1.000804 1.000785 812000.00 1007500.00 812008.57 1007505.35 812013.83 1007508.65","For wellbore survey.","IOGP GN7-2.","IOGP",2/17/2019 0:00:00,"2017.040",0
1078,"Equal Earth",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The Equal Earth projection is an equal-area pseudocylindrical projection. It is appropriate for mapping global phenomena or for any other thematic world map that requires areas at their correct relative sizes. Its key features are its resemblance to the Robinson projection and continents with a visually pleasing appearance similar to those found on a globe.
Forward:
To derive the projected coordinates of a point, geodetic latitude (lat) is converted to authalic latitude (?). The formulae to convert geodetic latitude and longitude (lat,lon) to Easting (E) and Northing (N) are:
Easting, E = FE + Rq 2 (lon ? lonO) cos(?) / {v3 [1.340264 ? 0.243318 ?^2 + ?^6 (0.006251 + 0.034164 ?^2)]}
Northing, N = FN + Rq ? [1.340264 ? 0.081106 ?^2 + ?^6 (0.000893 + 0.003796 ?^2)]
where
? = asin [sin(?) v3 / 2]
Rq = a (qP / 2)^0.5
? = asin(q / qP)
q = (1 ? e^2) ({sin(lat)/ [1 ? e^2 sin^2(lat)]} ? 1/(2e) ln{[1 ? e sin(lat)] / [1 + e sin(lat)]})
qP = (1 ? e^2) ({sin(latP)/ [1 ? e^2 sin^2(latP)]} ? 1/(2e) ln{[1 ? e sin(latP)] / [1 + e sin(latP)]})
where latP = p/2 radians, thus
qP = (1 ? e^2) ([1 / (1 ? e^2)] ? {[1/(2e)] ln [(1 ? e) / (1 + e)]})
Reverse:
The reverse conversion from easting and northing to latitude and longitude requires iteration of northing (N) equation to obtain ?. ?0 = (N?FN) / Rq is used as the first trial ?. A correction ??n is calculated and subtracted from the trial ?n value to obtain the next trial ?n+1. n is the suffix number of iteration.
?0 = (N?FN) / Rq
??0 = { ?0 [1.340264 ? 0.081106 ?0^2 + ?0^6 (0.000893 + 0.003796 ?0^2)] ? [N?FN] / Rq} /
{1.340264 ? 0.243318 ?0^2 + ?0^6 [0.006251 + 0.034164 ?0^2]}
?1 = ?0 ? ??0
??1 = {?1 [1.340264 ? 0.081106 ?1^2 + ?1^6 (0.000893 + 0.003796 ?1^2)] ? [N?FN] / Rq} /
{1.340264 ? 0.243318 ?1^2 + ?1^6 [0.006251 + 0.034164 ?1^2]}
?2 = ?1 ? ??1
etc.
The calculation is repeated until ??n is less than a predetermined convergence value. Then, using the final ?n+1 as ?, the geodetic latitude and longitude of a point are determined as follows:
lat = ? + {[(e^2/3 + 31e^4/180 + 517e^6/5040) sin(2?)] + [(23e^4/360 + 251e^6/3780) sin(4?)]
+ [(761e^6/45360) sin(6?)]}
lon = lonO + v3 (E?FE) {1.340264 ? 0.243318 ?^2 + ?^6 (0.006251 + 0.034164 ?^2)} / {2 Rq cos(?) }
where ? = asin{ 2 sin(?) / v3} and Rq is defined as in the forward equations.
Sphere:
For the spherical form of the projection, ? = lat and Rq = R, where R is the radius of the sphere.","Parameters:
Ellipsoid: WGS 1984 a = 6378137.0 metres 1/f =298.257223563
then e = 0.08181919084262
Longitude of natural origin (lonO) = 90?00'00.00""W = -1.5707963268 rad
False easting (FE) = 0.000 m
False northing (FN) = 0.000 m
Forward calculation for:
Latitude = 34?03'27.169"" N = 0.5944163293 rad
Longitude = 117?11'48.349"" W = -2.0454693977 rad
First gives
? = 0.5923399644 Rq = 6371007.181
? = 0.5046548375
whence
E = -2390749.042 m
N = 4242849.758 m
Reverse calculation for the same Easting and Northing (-2390749.042 E, 4242849.758 N) starts with an iteration to obtain ?:
(N?FN) / Rq = 0.665962169
Step 1: ?0 = 0.665962169 ??0 = 0.164312383
Step 2: ?1 = 0.501649786 ??1 = -0.003004201
Step 3: ?2 = 0.504653987 ??2 = -8.512742e-07
Step 4: ?3 = 0.504654838 ??3 = -6.859963e-14
? = ?4 = 0.504654838
This gives:
? = 0.592339965
Then Latitude = 34?03'27.169"" N
Longitude = 117?11'48.349"" W",,"?avric et al, International Journal of Geographical Iinformation Science, August 2018.","IOGP",5/3/2019 0:00:00,"2018.048 2019.018",0
1079,"New Zealand Deformation Model",1,"fNZGD2000 = fITRF96 + dft (all in decimal degrees)
?NZGD2000 = ?ITRF96 + d?t (all in decimal degrees)
hNZGD2000 = hITRF96 + dht (all in metres)
where
dft = dnt from the deformation model, converted from metres to degrees;
d?t = det from the deformation model, converted from metres to degrees;
dht = dut from the deformation model, in metres.","See information source.","Zip final contains a set of CSV files documentation for a secular tectonic plate velocity model and patches which account for movements due to specific deformation events (earthquakes).","""NZGD2000 Deformation Model"", Land Information New Zealand (LINZ).","IOGP",6/28/2019 0:00:00,"2019.037",0
9601,"Longitude rotation",1,"Target_longitude = Source_longitude + longitude_offset.","(none)","This transformation allows calculation of the longitude of a point in the target system by adding the parameter value to the longitude value of the point in the source system.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"1999.790 2017.018",0
9602,"Geographic/geocentric conversions",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Latitude, P, and Longitude, L, in terms of Geographic Coordinate Reference System A may
be expressed in terms of a geocentric (earth centred) Cartesian coordinate reference system X, Y, Z
with the Z axis corresponding with the Polar axis positive northwards, the X axis through
the intersection of the Greenwich meridian and equator, and the Y axis through the
intersection of the equator with longitude 90 degrees E. If the prime meridian for geogCRS A is not
Greewich, longitudes must first be transformed to their Greenwich equivalent. If the earth's
spheroidal semi major axis is a, semi minor axis b, and inverse flattening 1/f, then
X = (nu + h) cos P cos L
Y = (nu + h) cos P sin L
Z = ((1 - e^2) nu + h) sin P
where nu is the prime vertical radius of curvature at latitude P and is equal to
nu = a /(1 - e^2*sin^2(P))^0.5,
P and L are respectively the latitude and longitude (related to Greenwich) of the point
h is height above the ellipsoid, (topographic height plus geoidal height), and
e is the eccentricity of the ellipsoid where e^2 = (a^2 -b^2)/a^2 = 2f -f^2
Cartesian coordinates in geocentric coordinate reference system B may be used to derive geographical coordinates in terms of geographic coordinate reference system B by:
P = atan2[(Z + eta b sin^3 q) , (p ? e^2 a cos^3 q)]
L = atan2 (Y,X)
(see implementation notes in GN7-2 preface for atan2 convention)
where
eta = e^2 / (1 ? e^2)
b = a(1 ? f)
p = (X^2 + Y^2)^0.5
q = atan[(Z a) / (p b)]
and where L is relative to Greenwich. If the geographic system has a non Greenwich prime meridian, the Greenwich value of the local prime meridian should be applied to longitude.
Then
h (p / cos P) ? nu
(Note that h is the height above the ellipsoid. This is the height value which is delivered by Transit and GPS satellite observations but is not the topographic height value which is normally used for national mapping and levelling operations. The topographic height is usually the height above mean sea level or an alternative
level reference for the country. If one starts with a topographic height, it will be necessary to convert it to an ellipsoid height before using the above transformation formulas. h = N + H, where N is the geoid height above the ellipsoid at the point and is sometimes negative, and H is the height of the point above the geoid. The height above the geoid is often taken to be that above mean sea level, perhaps with a constant correction applied. Geoid heights of points above the nationally used ellipsoid may not be readily available. For the WGS84 ellipsoid the value of N,
representing the height of the geoid relative to the ellipsoid, can vary between values of -100m in the Sri Lanka area to +80m in the North Atlantic.)","Consider a North Sea point with coordinates derived by GPS satellite in the WGS 84 geographical coordinate system with coordinates of:
latitude 53 deg 48 min 33.82 sec N,
longitude 02 deg 07 min 46.38 sec E,
and ellipsoidal height 73.0m,
whose coordinates are required in terms of the ED50 geographical coordinate system which takes the International 1924 ellipsoid. The three parameter datum shift from WGS 84 to ED50 for this North Sea area is given as dX = +84.87m, dY = +96.49m, dZ = +116.95m.
The WGS 84 geographical coordinates convert to the following geocentric values using the above formulas for X, Y, Z:
XA = 3771 793.97m
YA = 140 253.34m
ZA = 5124 304.35m
Applying the quoted datum shifts to these, we obtain new geocentric values now related to ED50:
XB = 3771 878.84m
YB = 140 349.83m
ZB = 5124 421.30m
These convert to ED50 values on the International 1924 ellipsoid as:
latitude 53 deg 48 min 36.565 sec N,
longitude 02 deg 07 min 51.477 sec E,
and ellipsoidal height 28.02 m,
Note that the derived height is referred to the International 1924 ellipsoidal surface and will need a further correction for the height of the geoid at this point in order to relate it to Mean Sea Level.","This is a parameter-less conversion. In applications it is often concatenated with the 3- 7- or 10-parameter transformations 9603, 9606, 9607 or 9636 to form a geographic to geographic transformation.","EPSG guidance note #7-2, http://www.epsg.org, from ""Datums and Map Projections""; Iliffe and Lott (2007).","IOGP",8/29/2018 0:00:00,"1997.290 2002.510 2004.330 2007.058 2017.018 2017.024",0
9603,"Geocentric translations (geog2D domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Coordinate Frame Rotation (geog2D domain) transformation has 5 steps:
(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;
(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Geocentric translations(geocentric domain) method, EPSG method code 1031;
(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","(For the geocentric step only). Given a three parameter datum shift from WGS 84 to ED50 for this North Sea area is given as
dX = +84.87m, dY = +96.49m, dZ = +116.95m.
The WGS84 geographical coordinates convert to the following WGS 84 geocentric values using
the above formulas for X, Y, Z:
XA = 3771 793.97m
YA = 140 253.34m
ZA = 5124 304.35m
Applying the given datum shifts to these, we obtain new geocentric values now related
to ED50:
XB = 3771 878.84m
YB = 140 349.83m
ZB = 5124 421.30m","See methods 1031 and 1035 for similar tfms operating between other CRSs types.","EPSG guidance note #7-2, http://www.epsg.org","OGP",11/24/2009 0:00:00,"2009.083",0
9604,"Molodensky",1,"See information source.","(none)","See Abridged Molodensky.","EPSG guidance note #7-2, http://www.epsg.org","OGP",8/25/2004 0:00:00,"2004.470",0
9605,"Abridged Molodensky",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
As an alternative to the computation of the new latitude, longitude and height above ellipsoid in discrete steps through geocentric coordinates, the changes in these geographic coordinates may be derived directly by formulas derived by Molodenski. Abridged versions of these formulas, which are quite satisfactory for three parameter transformations, are as follows:
dlat "" = [(-dX*sin(lat)*cos(lon)) - (dY*sin(lat)*sin(lon)) + (dZ*cos(lat)) + (((a*Df) + (f*Da))*sin(2*lat))] / (rho * sin(1""))
dlon "" = (-dX*sin(lon) + dY*cos(lon)) / ((nu*cos(lat)) * sin(1""))
dh = (dX*cos(lat)*cos(lon)) + (dY*cos(lat)*sin(lon)) + (dZ*sin(lat)) + ((a*Df + f*Da)*(sin(lat)^2)) - da
where the dX, dY and dZ terms are the geocentric translation parameters, and rho and nu are the meridian and prime vertical radii of curvature at the given latitude (lat) on the first ellipsoid, da is the difference in the semi-major axes (a1 - a2) of the first and second ellipsoids and df is the difference in the flattening of the two ellipsoids.
The formulas for dlat and dlon indicate changes in latitude and longitude in arc-seconds.","For a North Sea point with coordinates derived by GPS satellite in the WGS84 geographical coordinate reference system, with coordinates of:
latitude lat_s =53?48'33.82""N,
longitude lon_s = 2?07'46.38""E,
and ellipsoidal height h_s = 73.0m,
whose coordinates are required in terms of the ED50 geographical coordinate reference system which takes the International 1924 ellipsoid.
The three geocentric translations parameter values from WGS84 to ED50 for this North Sea area are given as dX = +84.87m, dY = +96.49m, dZ = +116.95m.
Ellipsoid Parameters are:
WGS 84 a = 6378137.0 metres 1/f = 298.2572236
International 1924 a = 6378388.0 metres 1/f = 297.0
Then
da = 6378388 ? 6378137 = 251
df = 0.003367003 - 0.003352811 = 1.41927E-05
whence
dlat = 2.543""
dlon = 5.097""
dh = ? 44.909 m
ED50 values on the International 1924 ellipsoid are then:
latitude lat_t = 53?48'36.563""N,
longitude lon_t = 2?07'51.477""E,
and ellipsoidal height h_t = 28.091 m.","This transformation is a truncated Taylor series expansion of a transformation between two geographic coordinate systems, modelled as a set of geocentric translations.","EPSG guidance note #7-2, http://www.epsg.org","OGP",6/12/2006 0:00:00,"1999.010 2004.470 2006.420",0
9606,"Position Vector transformation (geog2D domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Position Vector transformation (geog2D domain) has 5 steps:
(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;
(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Position Vector(geocentric domain) method, EPSG method code 1033;
(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","Input point:
Coordinate reference system: WGS 72 (geographic 2D)
Latitude = 55 deg 00 min 00 sec N
Longitude = 4 deg 00 min 00 sec E
This is taken to be geographic 3D with an assumed Ellipsoidal height hS = 0 m
This transforms to Cartesian geocentric coords:
Xs = 3 657 660.66 (m)
Ys = 255 768.55 (m)
Zs = 5 201 382.11 (m)
Transformation parameters WGS 72 to WGS 84:
tX (m) = 0.000
tY (m) = 0.000
tZ (m) = 4.5
rX ("") = 0.000 = 0.0 radians
rY ("") = 0.000 = 0.0 radians
rZ ("") = 0.554 = 0.000002685868 radians
dS (ppm) = 0.219
from which M = 1.000000219
Application of the 7 parameter Position Vector transformation results in WGS 84 coordinates of:
Xt = 3 657 660.78 (m)
Yt = 255 778.43 (m)
Zt = 5 201 387.75 (m)
This converts into:
Latitude = 55 deg 00 min 00.090 sec N
Longitude = 4 deg 00 min 00.554 sec E
Ellipsoidal height = +3.22 m
on the WGS 84 geographic 3D coordinate reference system. For the 2D equivalent the height is ignored.","Note the analogy with the Coordinate Frame rotation (code 9607) but beware of the differences! The Position Vector convention is used by IAG and recommended by ISO 19111. See methods 1033 and 1037 for similar tfms operating between other CRS types.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"1998.160 2009.083 2013.021 2018.001 2019.006",0
9607,"Coordinate Frame rotation (geog2D domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Coordinate Frame rotation (geog2D domain) transformation has 5 steps:
(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;
(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Coordinate Frame rotation (geocentric domain) method, EPSG method code 1032;
(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","The same example as for the Position Vector transformation (coordinate operation method 9606) can be calculated, however the following transformation parameters have to be applied to achieve the same input and output in terms of coordinate values:
Transformation parameters Coordinate Frame rotation convention:
tX (m) = 0.000
tY (m) = 0.000
tZ (m) = 4.5
rX ("") = 0.000
rY ("") = 0.000
rZ ("") = -0.554 = -0.000002685868 radians
dS (ppm) = 0.219
from which M = 1.000000219
Please note that only the rotation has changed sign as compared to the Position Vector transformation.","Note the analogy with the Position Vector tfm (code 9606) but beware of the differences! The Position Vector convention is used by IAG and recommended by ISO 19111. See methods 1032 and 1038 for similar tfms operating between other CRS types.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2004.141 2009.083 2013.021 2018.001 2019.006",0
9613,"NADCON",1,"The latitude and longitude offsets at a point are derived by interpolation within the gridded data. Separate grid files are given for latitude and longitude offsets. The grid file format is given in documentation available from the information source. Bi-linear interpolation is used to derive the offset values. For the forward calculation the interpolated value of the offset is then added to the source CRS coordinate value to give the coordinates in the target CRS.
Reversibility
Iteration is required for the reverse transformation. The coordinate reference system for the coordinates of the grid nodes is the source coordinate reference system for the forward transformation. Then in forward transformations the offset is obtained through straightforward interpolation of the grid file. But for the reverse transformation the first grid interpolation entry will be the value of the point in the second coordinate reference system, the offsets are interpolated and applied with sign reversed, and the result used in further iterations of interpolation and application of offset until the difference between results from successive iterations is insignificant.","(none)","Geodetic transformation operating on geographic coordinate differences by bi-linear interpolation. Input expects longitudes to be positive west.","US Coast and geodetic Survey - http://www.ngs.noaa.gov","IOGP",6/13/2017 0:00:00,"2004.190 2008.027 2017.018",0
9614,"NTv1",1,"The latitude and longitude offsets at a point are derived by interpolation within the gridded data. One grid file contains both latitude and longitude offsets. The grid file format is given in documentation available from the information source. Bi-linear interpolation is used to derive the offset values. For the forward calculation the interpolated value of the offset is then added to the source CRS coordinate value to give the coordinates in the target CRS.
Reversibility
Iteration is required for the reverse transformation. The coordinate reference system for the coordinates of the grid nodes is the source coordinate reference system for the forward transformation. Then in forward transformations the offset is obtained through straightforward interpolation of the grid file. But for the reverse transformation the first grid interpolation entry will be the value of the point in the second coordinate reference system, the offsets are interpolated and applied with sign reversed, and the result used in further iterations of interpolation and application of offset until the difference between results from successive iterations is insignificant.","(none)","Geodetic transformation operating on geographic coordinate differences by bi-linear interpolation. Superseded in 1997 by NTv2 (transformation method code 9615). Input expects longitudes to be positive west.","Geomatics Canada - Geodetic Survey Division.","IOGP",6/13/2017 0:00:00,"2004.190 2008.027 2017.018",0
9615,"NTv2",1,"The latitude and longitude offsets at a point are derived by interpolation within the gridded data. One grid file contains both latitude and longitude offsets. The grid file format is given in documentation available from the information source. Bi-linear interpolation is used to derive the offset values. For the forward calculation the interpolated value of the offset is then added to the source CRS coordinate value to give the coordinates in the target CRS.
Reversibility
Iteration is required for the reverse transformation. The coordinate reference system for the coordinates of the grid nodes is the source coordinate reference system for the forward transformation. Then in forward transformations the offset is obtained through straightforward interpolation of the grid file. But for the reverse transformation the first grid interpolation entry will be the value of the point in the second coordinate reference system, the offsets are interpolated and applied with sign reversed, and the result used in further iterations of interpolation and application of offset until the difference between results from successive iterations is insignificant.","(none)","Geodetic transformation operating on geographic coordinate differences by bi-linear interpolation. Supersedes NTv1 (transformation method code 9614). Input expects longitudes to be positive west.","Geomatics Canada - Geodetic Survey Division, http://www.geod.nrcan.gc.ca/. Also GDAit software documentation from University of Melbourne Geomatics Department, http://www.sli.unimelb.edu.au/gda94/.","IOGP",6/13/2017 0:00:00,"2004.190 2008.027 2017.018",0
9616,"Vertical Offset",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
A vertical offset allows calculation of coordinates in the target vertical coordinate reference system by adding a correction parameter A to the coordinate values of the point in the source system:
X2 = X1 + A1>2
where
X2 = value in the forward target vertical coordinate reference system.
X1 = value in the forward source vertical coordinate reference system.
A1>2 is the offset to be applied for the transformation from CRS 1 to CRS 2. Its value for the forward calculation is the value of the origin of the source CRS 1 in the target CRS 2.
For the reverse transformation from CRS 2 to CRS 1 the same formula is used but with the sign of the offset A1>2 reversed:
X1 = X2 + (?A1>2)
The above formulas apply only when the positive direction of the axis of the source and target CRS is the same and where the two vertical CRSs share the same unit of measure. If either of these are to be changed then a concatenated operation including one or more conversions through intermediate vertical CRSs is required. See Height-depth reversal (code 1068) and Change of vertical axis unit (code 1069) for the required conversion methods.","For coordinate transformation: Baltic height to Black Sea height (1), code 5447:
Transformation Parameter: Vertical Offset A1>2 = 0.4m
Consider a point having a gravity-related height in the Baltic height system of 2.55m. Its value in the Black Sea height system is
2.55m + 0.4m = 2.95m.
For the reverse calculation to transform the Black Sea height of 2.95m to Baltic height:
H(Baltic) = 2.95m + (?(0.4m)) = 2.55m.","This transformation allows calculation of height (or depth) in the target system by adding the parameter value to the height (or depth)-value of the point in the source system.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",10/16/2017 0:00:00,"1999.790 2008.010 2014.027",0
9617,"Madrid to ED50 polynomial",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The polynomial expressions are:
dLat seconds = A0 + (A1*lat) + (A2*lon) + (A3*H)
dLon seconds = B00 + B0 + (B1*lat) + (B2*lon) + (B3*H)
where latitude lat and longitude lon are in decimal degrees referred to the Madrid 1870 (Madrid) geographic coordinate reference system and H is gravity-related height in metres. B00 is the longitude (in seconds) of the Madrid meridian measured from the Greenwich meridian; it is the value to be applied to a longitude relative to the Madrid meridian to transform it to a longitude relative to the Greenwich meridan.
The results of these expressions are applied through the formulae:
Lat(ED50) = Lat(M1870(M)) + dLat
and Lon(ED50) = Lon(M1870(M)) + dLon.","Input point coordinate system: Madrid 1870 (Madrid) (geographic 3D)
Latitude = 42 deg 38 min 52.77 sec N
= 42.647992 degrees
Longitude = 3 deg 39 min 34.57 sec E of Madrid
= +3.659603 degrees from the Madrid meridian.
Height = 0 m
For the north zone transformation:
A0 = 11.328779
A1 = -0.1674
A2 = -0.03852
A3 = 0.0000379
B00 = -13276.58
B0 = 2.5079425
B1 = 0.08352
B2 = -0.00864
B3 = -0.0000038
dLat = +4.05 seconds
Then ED50 latitude = 42 deg 38 min 52.77 sec N + 4.05sec = 42 deg 38 min 56.82 sec N
dLon = -13270.54 seconds = -3 deg 41 min 10.54 sec
Then ED50 longitude = 3 deg 39 min 34.57 sec E - 3 deg 41 min 10.54 sec = 0 deg 01 min 35.97 sec W of Greenwich.",,"EPSG guidance note #7-2, http://www.epsg.org, after Institut de Geomatica; Barcelona.","OGP",7/10/2011 0:00:00,"1999.284 1999.820 1999.640 2006.910 2011.003 2011.057",0
9618,"Geographic2D with Height Offsets",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Lat_T = Lat_S + latitude_offset
Lon_T = Lon_S + longitude_offset
EllipsoidHeight_T = GravityHeight_S + gravity-related_to_ellipsoid_height_offset.","(none)","This transformation allows calculation of coordinates in the target system by adding the parameter value to the coordinate values of the point in the source system.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"1999.790 2004.330 2017.018",0
9619,"Geographic2D offsets",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Lat_T = Lat_S + latitude_offset
Lon_T = Lon_S + longitude_offset.","A position with coordinates of 38?08'36.565""N, 23?48'16.235""E referenced to the old Greek geographic 2D coordinate reference system (EPSG CRS code 4120) is to be transformed to the newer GGRS87 system (EPSG CRS code 4121). Transformation parameters from Greek to GGRS87 are:
dLat = -5.86""
dLon = +0.28""
Then Lat(GGRS87) = 38?08'3656 5"" N (5.8 6"") = 38?08'30.705""N
and Lon(GGRS87) = 23?48'16.23 5""E + 0. 28"" = 23?48'16.515""E
For the reverse transformation for the same point,
Lat(GREEK) = 38?08'30.705 "" N + 5. 86"" = 38?08'36.565""N
Lon(GREEK) = 23?48'16.515"" E + (-0. 28 "") = 23?48'16.235""E","This transformation allows calculation of coordinates in the target system by adding the parameter value to the coordinate values of the point in the source system.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"1999.790 2004.330 2017.018",0
9620,"Norway Offshore Interpolation",0,"See information source.","(none)","Although in principle this method is not reversible, in practice reversibility is better than 10 cm. For the applications for which it was designed it may be considered reversible.","Norwegian Mapping Authority note of 13-Feb-1991 ""Om Transformasjon mellom Geodetiske Datum i Norge"".","IOGP",6/13/2017 0:00:00,"2005.230 2017.018",0
9621,"Similarity transformation",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The similarity transformation in algebraic form is:
XT = XT0 + XS * M * cos q + YS * M * sin q
YT = YT0 ? XS * M * sin q + YS * M * cos q
where:
XT0 , YT0 = the coordinates of the origin point of the source coordinate reference system expressed in the target coordinate reference system;
M = the length of one unit in the source coordinate reference system expressed in units of the target coordinate reference system;
q = the angle about which the axes of the source coordinate reference system need to be rotated to coincide with the axes of the target coordinate reference system, counter-clockwise being positive. Alternatively, the bearing of the source coordinate reference system Y-axis measured relative to target coordinate reference system north.
The similarity transformation can also be described as a special case of the parametric affine transformation where coefficients A1 = B2 and A2 = - B1.
Reversibility
The reverse formula for the Similarity Transformation is:
XS = [(XT ? XTO) * cos q ? (YT ? YTO) * sin q ] / [M ]
YS = [(XT ? XTO) * sin q + (YT ? YTO) * cos q] / [M ]
An alternative approach for the reverse operation is to use the same formula as for the forward computation but with different parameter values:
XT = XTO' + XS * M' * cos theta' + YS * M' * sin theta'
YT = YTO' ? XS * M' * sin theta' + YS * M' * cos theta'
The reverse parameter values, indicated by a prime ('), can be calculated from those of the forward operation as follows:
XTO' = (YTO sin theta ? XTO cos theta) / M
YTO' = ?(YTO cos theta + XTO sin theta) / M
M' = 1/M
theta' = ?theta","ED50 / UTM zone 31N to ETRS89 / UTM zone 31N
Parameters of the Similarity Transformation:
XTO = -129.549 metres
YTO = -208.185 metres
M = 1.00000155
theta = 1.56504"" = 0.000007588 rad
Forward computation for source coordinates 300000m E, 4500000m N:
E(ETRS89) = ?129.549 + 300000.465 + 34.144
= 299905.060 m E
N(ETRS89) = ?208.185 ?2.276 + 4500006.977
= 4499796.515m N
Reverse computation of ETRS89 / UTM 31N coordinates 299905.060m E, 4499796.515m N:
E(ED50) = (300034.609 ? 34.144) / 1.00000155
= 300000.000m E
N(ED50) = (2.276 + 4500004.700) / 1.00000155
= 4500000.000m N
Alternative reverse computation:
First calculate new parameter values:
XTO' = 129.5472 m
YTO' = 208.1857 m
M' = 0.99999845
theta' = ?0.000007588 rad
Then apply these values to forward formula:
E(ED50) = 129.547 + 299904.595 + (?34.142)
= 300000.000 m
N(ED50) = 208.186 ? (?2.276) + 4499789.539
= 4500000.000 m","Defined for two-dimensional coordinate systems.","EPSG guidance note #7-2, http://www.epsg.org","OGP",6/29/2010 0:00:00,"2000.830 2004.670 2010.053",0
9622,"Affine orthogonal geometric transformation",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
XT = XT0 + XS . k . dSX . cos q + YS . k . dSY . sin q
YT = YT0 ? XS . k . dSX . sin q + YS . k . dSY . cos q
where:
XT0 ,YT0 = the coordinates of the origin point of the source coordinate reference system, expressed in the target coordinate reference system;
dSX , dSY = the length of one unit of the source axis, expressed in units of the target axis, for the X axes and the Y- axes respectively;
k = point scale factor of the target coordinate reference system in a chosen reference point;
q = the angle through which the source coordinate reference system axes must be rotated to coincide with the target coordinate refderence system axes (counter-clockwise is positive). Alternatively, the bearing (clockwise positive) of the source coordinate reference system Y-axis measured relative to target coordinate reference system north.","Source coordinate system: imaginary 3D seismic acquisition bin grid. The two axes are orthogonal, but the unit on the I-axis is 25 metres, whilst the unit on the J-axis is 12.5 metres.
The target projected coordinate system is WGS 84 / UTM Zone 31N and the origin of the bin grid (centre of bin 0,0) is defined at E = 456781.0, N = 5836723.0. The projected coordinate system point scale factor at the bin grid origin is 0.99984.
The map grid bearing of the I and J axes are 110* and 20* respectively. Thus the angle through which both the positive I and J axes need to be rotated to coincide with the positive Easting axis and Northing axis respectively is +20 degrees.
Hence:
XT0 , = 456 781.0 m
YT0 = 5 836 723.0 m
dSX = 25
dSY = 12.5
k = 0.99984
q = +20 degrees
Forward calculation for centre of bin with coordinates: I = 300, J = 247:
XT = Easting = XT0 + XS . k . dSX . cos q + YS . k . dSY . sin q = 464 855.62 m.
YT = Northing = YT0 ? XS . k . dSX . sin q + YS . k . dSY . cos q = 5 837 055.90 m
Reverse calculation for this point:
XS = [( XT ? XT0) . cos qY ? (YT ? YT0) . sin qY ] / [k . dSX . cos (qX ? qY)] = 230 bins
YS = [(XT ? XT0) . sin qX + (YT ? YT0) . cos qX ] / [k . dSY . cos (qX ? qY)] = 162 bins",,"EPSG guidance note #7, http://www.epsg.org","OGP",6/10/2000 0:00:00,,1
9623,"Affine geometric transformation",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
General case.
The geometric representation of the affine transformation is:
XT = XT0 + XS * k * MX * cos qX + YS * k * MY * sin qY
YT = YT0 ? XS * k * MX * sin qX + YS * k * MY * cos qY
where:
XT0 ,YT0 = the coordinates of the origin point of the source coordinate reference system, expressed in the target coordinate reference system;
MX , MY = the length of one unit of the source axis, expressed in units of the target axis, for the first and second source and target axis pairs respectively;
qX , qY = the angles about which the source coordinate reference system axes XS and YS must be rotated to coincide with the target coordinate reference system axes XT and YT respectively (counter-clockwise being positive).
k = point scale factor of the target coordinate reference system in a chosen reference point;
Comparing the algebraic representation with the parameters of the parameteric form (code 9624) it can be seen that the parametric and geometric forms of the affine transformation are related as follows:
A0 = XT0
A1 = k * MX * cos qX
A2 = k * MY * sin qY
B0 = YT0
B1 = ? k * MX * sin qX
B2 = k *MY * cos qY
Reversibility
For the Affine Geometric Transformation, the reverse operation can be described by a different formula, as shown below, in which the same parameter values as the forward transformation may be used:
XS = [( XT ? XT0) . cos qY ? (YT ? YT0) . sin qY ] / [k * MX * cos (qX ? qY)]
YS = [(XT ? XT0) . sin qX + (YT ? YT0) . cos qX ] / [k * MY * cos (qX ? qY)]
Orthogonal case
If the source coordinate reference system happens to have orthogonal axes, that is both axes are rotated through the same angle to bring them into the direction of the orthogonal target coordinate reference system axes, i.e. qX = qY = q, then the Affine Geometric Transformation can be simplified to:
XT = XT0 + XS . k . MX . cos q + YS . k . MY . sin q
YT = YT0 ? XS . k . MX . sin q + YS . k . MY . cos q
where:
q = the angle through which the source coordinate reference system axes must be rotated to coincide with the target coordinate refderence system axes (counter-clockwise is positive). Alternatively, the bearing (clockwise positive) of the source coordinate reference system Y-axis measured relative to target coordinate reference system north.
The reverse formulas of the general case can also be simplified by replacing qX and qY with q:
XS = [(XT ? XTO) * cos q ? (YT ? YTO) * sin q ] / [k * MX ]
YS = [(XT ? XTO) * sin q + (YT ? YTO) * cos q] / [k * MY ]
In the EPSG dataset this orthogonal case (code 9622) has been deprecated. The formulas for the general case should be used, inserting q for both qX and qY. The case has been documented here as part of the progression through increasing constraints on the degrees of freedom between the general case and the Similarity Transformation.","(none)",,"EPSG guidance note #7-2, http://www.epsg.org","OGP",8/26/2005 0:00:00,"2004.670",0
9624,"Affine parametric transformation",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
XT = A0 + A1 * XS + A2 * YS
YT = B0 + B1 * XS + B2 * YS
where
XT , YT are the coordinates of a point P in the target coordinate reference system;
XS , YS are the coordinates of P in the source coordinate reference system.
Reversibility
The reverse transformation is another affine transformation using the same formulas but with different parameter values. The reverse parameter values, indicated by a prime (?), can be calculated from those of the forward transformation as follows:
D = A1 * B2 ? A2 * B1
A0? = (A2 * B0 ? B2 * A0) / D
B0? = (B1 * A0 ? A1 * B0) / D
A1? = +B2 / D
A2? = ? A2 / D
B1? = ? B1 / D
B2? = +A1 / D
Then
XS = A0' + A1' * XT + A2' * YT
YS = B0' + B1' * XT + B2' * YT","Coordinate transformation: Jamaica 1875 / Jamaica (Old Grid) to JAD69 / Jamaica National Grid
Coordinate transformation parameter values:
A0 = 82357.457 metres
A1 = 0.304794369
A2 = 0.000015417425
B0 = 28091.324 metres
B1 = -0.000015417425
B2 = 0.304794369
The units for the Jamaica 1875 / Jamaica (Old Grid) are feet, those for the JAD69 / Jamaica National Grid are metres. The foot-metre conversion factor is embedded within the transformation, together with the scale and orientation differences between the two coordinate reference systems.
Jamaica 1875 / Jamaica (Old Grid) input point:
Easting E = XS = 553,900.00 feet
Northing N = YS = 482,500.00 feet
Then on JAD69 / Jamaica National Grid
Easting E = XT = 251,190.497 metres
Northing N = XT = 175,146.067 metres
For the reverse calculation, first calculate derived parameter values:
D = 0.092899608
A0' = -270201.960 Clarke's foot
A1' = 3.280900499
A2' = -0.000165958
B0' = -92178.507 Clarke's foot
B1' = 0.000165958
B2' = 3.280900499
Then for the same point
Easting E = XT = 251,190.497 metres
Northing N = XT = 175,146.067 metres
gives
Easting E = XS = 553,900.000 feet
Northing N = YS = 482,500.000 feet",,"EPSG guidance note #7-2, http://www.epsg.org","OGP",6/12/2013 0:00:00,"2004.670 2012.075",0
9625,"General polynomial (2nd-order)",0,"The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the input parameters, usually coordinate offsets U and V relative to a central evaluation point, to ?manageable? numbers, between ?10 and +10 at most.
U = XS - XS0 in defined units (which may not be those of the coordinate reference system),
V = YS - YS0
Then (XT - XT0) = (XS - XS0) + dX
(YT - YT0) = (YS - YS0) + dY
or
XT = XS - XS0 + XT0 + dX
YT = YS - YS0 + YT0 + dY
where
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system.
and where
dX = A0 + A1.U + A2.V + A3.U2 + A4.U.V + A5.V2
dY = B0 + B1.U + B2.V +B3.U2 +B4.U.V +B5.V2",,,"EPSG guidance note #7.","OGP",3/7/2000 0:00:00,,1
9626,"General polynomial (3rd-order)",0,"The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the input parameters, usually coordinate offsets U and V relative to a central evaluation point, to ?manageable? numbers, between ?10 and +10 at most.
U = XS - XS0 in defined units (which may not be those of the coordinate reference system),
V = YS - YS0
Then (XT - XT0) = (XS - XS0) + dX
(YT - YT0) = (YS - YS0) + dY
or
XT = XS - XS0 + XT0 + dX
YT = YS - YS0 + YT0 + dY
where
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system.
and where
dX = A0 + A1.U + A2.V + A3.U2 + A4.U.V + A5.V2 + A6.U3 + A7.U2.V + A8.U.V2 + A9.V3
dY = B0 + B1.U + B2.V +B3.U2 +B4.U.V +B5.V2 + B6.U3 +B7.U2.V +B8.U.V2 +B9.V3",,,"EPSG guidance note #7.","OGP",3/7/2000 0:00:00,,1
9627,"General polynomial (4th-order)",0,"The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the input parameters, usually coordinate offsets U and V relative to a central evaluation point, to ?manageable? numbers, between ?10 and +10 at most.
U = XS - XS0 in defined units (which may not be those of the coordinate reference system),
V = YS - YS0
Then (XT - XT0) = (XS - XS0) + dX
(YT - YT0) = (YS - YS0) + dY
or
XT = XS - XS0 + XT0 + dX
YT = YS - YS0 + YT0 + dY
where
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system.
and where
dX = A0 + A1.U + A2.V + A3.U2 + A4.U.V + A5.V2 + A6.U3 + A7.U2.V + A8.U.V2 + A9.V3 + A10.U4 + A11.U3.V + A12.U2.V2 + A13.U.V3 + A14.V4
dY = B0 + B1.U + B2.V +B3.U2 +B4.U.V +B5.V2 + B6.U3 +B7.U2.V +B8.U.V2 +B9.V3 + B10.U4 + B11.U3.V + B12.U2.V2 + B13.U.V3 + B14.V4",,,"EPSG guidance note #7.","OGP",3/7/2000 0:00:00,,1
9628,"Reversible polynomial (2nd-order)",1,"See EPSG Guidance Note 7.",,"Reversibility is subject to constraints. See Guidance Note 7 for clarification.","EPSG guidance note #7.","OGP",3/7/2000 0:00:00,"1999.640",1
9629,"Reversible polynomial (3rd-order)",1,"See EPSG Guidance Note 7.",,"Reversibility is subject to constraints. See Guidance Note 7 for clarification.","EPSG guidance note #7.","OGP",3/7/2000 0:00:00,"1999.640",1
9630,"Reversible polynomial (4th-order)",1,"See EPSG Guidance Note 7.","For geodetic transformation ED50 to ED87 (1)
Offset unit: degree
Ordinate 1 of evaluation point X0 = 55? 00' 00.000""N = +55 degrees
Ordinate 2 of evaluation point Y0 = 0? 00' 00.000""E = +0 degrees
Parameters:
A0 = -5.56098E-06 A1 = -1.55391E-06 ... A14 = -4.01383E-09
B0 = +1.48944E-05 B2 = +2.68191E-05 ... B14 = +7.62236E-09
Forward calculation for:
ED50 Latitude = Xs =52* 30?30""N = +52.508333333 degrees
ED50 Longitude = Ys = 2*E= +2.0 degrees
U = XS - X0 = * ED50 - X0 = 52.508333333 - 55.0 = -2.491666667 degrees
V = YS - Y0 = * ED50 - Y0 = 2.0 - 0.0 = 2.0 degrees
dX = A0 + A1.U + ... + A14.V4
= -5.56098E-06 + (-1.55391E-06 * -2.491666667) + ... + (-4.01383E-09 * 2.0^4)
= -3.12958E-06 degrees
dY = B0 + B1.U + ... + B14.V4
= +1.48944E-05 + (2.68191E-05 * -2.491666667) + ... + (7.62236E-09 * 2.0^4)
= +9.80126E-06 degrees
Then ED87 Latitude = XT = XS + dX
= 52.508333333 - 3.12958E-06 degrees
= 52* 30? 29.9887"" N
ED87 Longitude = YT = YS + dY
= 2* 00? 00.0353"" E
Reverse calculation for transformation ED50 to ED87 (1).
The transformation method for the ED50 to ED87 (1) transformation, 4th-order reversible polynomial, is reversible. The same formulas may be applied for the reverse calculation, but coefficients A0 through A14 and B0 through B14 are applied with reversal of their signs. Sign reversal is not applied to the coordinates of the evaluation point. Thus:
Ordinate 1 of evaluation point X0 = 55? 00' 00.000""N = +55 degrees
Ordinate 2 of evaluation point Y0 = 0? 00' 00.000""E = +0 degrees
A0 = +5.56098E-06 A1 = +1.55391E-06 ... A14 = +4.01383E-09
B0 = -1.48944E-05 B1 = -2.68191E-05 ... B14 = -7.62236E-09
Reverse calculation for:
ED87 Latitude = XS = 52? 30?29.9887""N = +52.5083301944 degrees
ED87 Longitude = YS = 2? 00? 00.0353"" E = +2.0000098055 degrees
U = 52.5083301944 - 55.0 = -2.4916698056 degrees
V = 2.0000098055 - 0.0 = 2.0000098055 degrees
dX = A0 + A1.U + ... + A14.V4
= +5.56098E-06 + (1.55391E-06 * -2.491666667) + ... + (4.01383E-09 * 2.0000098055^4)
= +3.12957E-06 degrees
dY = B0 + B1.U + ... + B14.V4
= -1.48944E-05 + (-2.68191E-05 * -2.491666667) + ... + (-7.62236E-09 * 2.0000098055^4)
= -9.80124E-06 degrees
Then ED50 Latitude = XT = XS + dX
= 52.5083301944 + 3.12957E-06 degrees
= 52? 30? 30.000"" N
ED50 Longitude = YT = YS + dY
= 2? 00? 00.000"" E","Reversibility is subject to constraints. See Guidance Note 7 for clarification.","EPSG guidance note #7.","OGP",3/7/2000 0:00:00,"1999.640",1
9631,"Complex polynomial (3rd-order)",0,"The relationship between two projected coordinate reference systems may be approximated more elegantly by a single polynomial regression formula written in terms of complex numbers. The advantage is that the dependence between the ?A? and ?B? coefficients (for U and V) is taken into account in the formula, resulting in fewer coefficients for the same order polynomial. A third-order polynomial in complex numbers is used in Belgium. A fourth-order polynomial in complex numbers is used in The Netherlands for transforming coordinates referenced to the Amersfoort / RD system to and from ED50 / UTM.
(dX + i. dY) = (A1 + i. A2).(U + i.V) + (A3 + i. A4).(U + i.V)^2 + (A5 + i. A6).(U + i.V)^3
where U = (XS - XS0).10-5
and V = (YS - YS0).10-5
Then
XT = XS - XS0 + XT0 + dX
YT = YS - YS0 + YT0 + dY
where
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system.
Note that the zero order coefficients of the general polynomial, A0 and B0, have apparently disappeared. In reality they are absorbed by the different coordinates of the source and of the target evaluation point, which in this case, are numerically very different because of the use of two different projected coordinate reference systems for source and target.
The transformation parameter values (the coefficients) are not reversible. For the reverse transformation a different set of parameter values are required, used within the same formulas as the forward direction","For transformation Belge Lambert 72 to ED50 / UTM zone 31N,
Eo1 = 0
No1 = 0
Eo2 = 449681.702
No2 = 5460505.326
A1 = -71.3747
A2 = 1858.8407
A3 = -5.4504
A4 = -16.9681
A5 = 4.0783
A6 = 0.2193
For source coordinate system E1=200000 N1=100000, then
E2 = 647737.377 N2 = 5564124.227.","Coordinate pairs treated as complex numbers. This exploits the correlation between the polynomial coefficients and leads to a smaller number of coefficients than the regular 3rd-order polynomial.","EPSG guidance note #7.","OGP",3/7/2000 0:00:00,,1
9632,"Complex polynomial (4th-order)",0,"The relationship between two projected coordinate reference systems may be approximated more elegantly by a single polynomial regression formula written in terms of complex numbers. The advantage is that the dependence between the ?A? and ?B? coefficients (for U and V) is taken into account in the formula, resulting in fewer coefficients for the same order polynomial. A third-order polynomial in complex numbers is used in Belgium. A fourth-order polynomial in complex numbers is used in The Netherlands for transforming coordinates referenced to the Amersfoort / RD system to and from ED50 / UTM.
(dX + i. dY) = (A1 + i. A2).(U + i.V) + (A3 + i. A4).(U + i.V)^2 + (A5 + i. A6).(U + i.V)^3 + (A7 + i.A8).(U + i.V)^4
where U = (XS - XS0).10-5
and V = (YS - YS0).10-5
Then
XT = XS - XS0 + XT0 + dX
YT = YS - YS0 + YT0 + dY
where
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system.
Note that the zero order coefficients of the general polynomial, A0 and B0, have apparently disappeared. In reality they are absorbed by the different coordinates of the source and of the target evaluation point, which in this case, are numerically very different because of the use of two different projected coordinate reference systems for source and target.
The transformation parameter values (the coefficients) are not reversible. For the reverse transformation a different set of parameter values are required, used within the same formulas as the forward direction.","For transformation RD / Netherlands New to ED50 / UTM zone 31N,
Eo1 = 155000
No1 = 463000
Eo2 = 663395.607
No2 = 5781194.380
A1 = -51.681
A2 = 3290.525
A3 = 20.172
A4 = 1.133
A5 = 2.075
A6 = 0.251
A7 = 0.075
A8 = -0.012
For source coordinate system E1=200000 N1=500000, then
E2 =707155.557 N2 = 5819663.128.","Coordinate pairs treated as complex numbers. This exploits the correlation between the polynomial coefficients and leads to a smaller number of coefficients than the regular 4th-order polynomial.","EPSG guidance note #7.","OGP",3/7/2000 0:00:00,,1
9633,"Ordnance Survey National Transformation",1,"This method is used for transformation between ETRS89 (or WGS 84) geographic and OSGB36 / National Grid projected coordinate reference systems. It uses a gridded data set with nodes in a temporary projected CRS (ETRS89 / National Grid) for bi-linear interpolation of grid coordinate differences.
The offsets at a point are derived by interpolation within the gridded data. One grid file contains both easting and northing offsets. The grid file format is given in documentation available from the information source. The coordinate reference system for the coordinates of the grid fiule nodes is an intermediary projected CRS, ETRS89 / British National Grid.
For the transformation of ETRS89 coordinates to OSGB36 / British National Grid coordinates, the ETRS89 latitude and longitude are first converted into ETRS89 / National Grid values using the Transverse Mercator formulas given in EPSG method code 9807. Then bi-linear interpolation is used to derive easting and northing offset values. These are added to the temporary ETRS89 / National Grid coordinate values to give OSGB36 / National Grid coordinate values.
For the transformation of OSGB36 / British National Grid coordinates to ETRS89 latitude and longitude, iteration is required. For the first iteration the OSGB36 / British National Grid easting and northing values are assumed to be in the ETRS89 / National Grid. Bi-linear interpolation is used to derive the easting and northing offset values which are applied with sign reversed to the input coordinates.The result is used in further iterations of interpolation and application of offset until the difference between results from successive iterations is insignificant. Finally these Transverse Mercator grid coordinates are converted to ETRS89 latitude and longitude using the formulae given in EPSG method code 9807.","See information source.","Geodetic transformation between ETRS89 (or WGS 84) and OSGB36 / National Grid. Uses ETRS89 / National Grid as an intermediate coordinate system for bi-linear interpolation of gridded grid coordinate differences.","http://www.gps.gov.uk/gpssurveying.asp","IOGP",6/13/2017 0:00:00,"2004.190 2008.027 2017.018",0
9634,"Maritime Provinces polynomial interpolation",0,"The transformation makes use of a residual file for each Canadian maritime province. The process of residual interpolation accounts for local variations in the coordinate reference system and provides a transformation accuracy of +/- 5 cm.
By using a second residual file, the transformation may be reversed. Only one residual file is in use by the method during any given execution.","(none)","This transformation is an executable module within the application NBGeocalc. It is an adaptation of the ESTPM program developed by Geodetic Survey of Canada.","Survey of New Brunswick","IOGP",6/13/2017 0:00:00,"2017.018",0
9635,"Geog3D to Geog2D+GravityRelatedHeight (US .gtx)",1,"This is a complex, multi-step transformation, involving the application of a geoid height difference interpolated at a point in a ""geoid model"". The geoid model should be available as a regular grid of latitude and longitude with the height of the geoid above the ellipsoid at each grid node. Only the height is affected by this transformation; the geodetic latitude and longitude are not.
The transformation involves the following sequence of steps:
? Selection of a subset of the geoid file covering the extent of the points to be transformed.
? If the geoid file is not based on the source or target CRS, it needs to be transformed first. This involves transformation of the chosen subset of the geoid file from its orignal Geographic 3D CRS to the Geographic 3D CRS that is the source or the target of this transformation.
? Calculation of the height of the geoid above the ellipsoid (""geoid undulation"") at the relevant point(s). This is achieved through a bi-linear interpolation of the geoid undulation, using the latitude and longitude to locate the point in the sub-grid. This step results in the height of the geoid above the ellipsoid (N) of the Geographic 3D CRS, whether source or target.
? At each point, the application of the calculated geoid undulation to the height to be transformed.
H=h-N for Geographic3D to Geographic2D+GravityRelatedHeight
h=H+N for Geographic2D+GravityRelatedHeight to Geographic3D
where h = the ellipsoidal height (height above the ellipsoid in a geographic 3D CRS)
and H = the gravity-related height (vertical CRS) component of the compound CRS.","(none)","Transformation from a Geographic 3D CRS to a Compound CRS consisting of a Geographic 2D CRS and a Vertical CRS, or vice versa. The Geographic 3D and the Geographic 2D CRS must be based on the same Geodetic Datum.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/16/2018 0:00:00,"2004.190 2004.340 2017.037",0
9636,"Molodensky-Badekas (CF geog2D domain)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Transformation of coordinates from one geographic coordinate reference system into another is carried out as a concatenation of the following operations:
(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)
The Molodensky-Badekas (CF geog2D domain) transformation has 5 steps:
(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;
(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;
(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Molodensky-Badekas (geocentric domain) method, EPSG method code 1034;
(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","Input point:
Coordinate reference system: La Canoa (geographic 2D)
Latitude = 9 deg 35 min 00.386 sec N
Longitude = 66 deg 04 min 48.091 sec W
This is taken to be geographic 3D with an assumed Ellipsoidal height hs = 201.465 m
This transforms to Cartesian geocentric coords:
Xs = 2 550 408.965 m
Ys = -5 749 912.266 m
Zs = 1 054 891.114 m
Transformation parameters La Canoa to REGVEN:
tX = -270.933 m
tY = +115.599 m
tZ = -360.226 m
rX = -5.266 sec = -0.000025530288 radians
rY = -1.238 sec = -0.000006001993 radians
rZ = +2.381 sec = 0.000011543414 radians
dS = -5.109 ppm
Coordinate 1 of evaluation point = 2464351.59 m
Coordinate 2 of evaluation point = -5783466.61 m
Coordinate 3 of evaluation point = 974809.81 m
from which M = 0.999994891
Application of the 10 parameter Molodenski-Badekas Transformation results in REGVEN geocentric coordinates of:
Xt = 2 550 138.467 m
Yt = -5 749 799.862 m
Zt = 1 054 530.826 m
This converts into:
Latitude = 9 deg 34 min 49.001 sec N
Longitude = 66 deg 04 min 54.705 sec W
Ellipsoidal height = -18.10 m
on the REGVEN geographic 3D coordinate reference system.
Because the source coordinate reference system was 2D, the target system ellipsoidal height is ignored and the results treated as a geographic 2D coordinate reference system:
Latitude = 9 deg 34 min 49.001 sec N
Longitude = 66 deg 04 min 54.705 sec W","See method codes 1034 and 1039 for this operation in other coordinate domains and method code 1063 for the opposite rotation convention in geographic 2D domain.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/21/2019 0:00:00,"2002.510 2004.470 2009.083 2013.021 2014.039 2014.065 2017.018 2018.001 2019.006",0
9637,"Degree representation conversion: deg to DMSH",1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:
deg decimal degrees
adeg absolute value of decimal degrees
ideg integer degrees
sdeg signed integer degree
min real-number minutes
imin integer minutes
sec real-number seconds
lathem, lonhem hemisphere abbreviation
Forward calculation from decimal degree representation to DMSH representation:
adeg = ABS(deg)
ideg = INT(adeg)
min = (adeg - ideg) * 60
imin = INT(min)
sec = (min - imin) * 60
Then for latitude, if deg < 0, lathem = S else lathem = N
For longitude, if deg < 0, lonhem = W else lonhem = E
Reverse calculation from DMSH representation to decimal degree representation:
deg = (ideg + imin/60 + sec/3600) * H
where for latitude H = 1 if lathem = N and H = -1 if lathem = S
and for longitude H = 1 if lonhem = E and H = -1 if lonhem = W","Source CRS = WGS 84 (deg) (CRS code 63266405).
Latitude = 35.75255, longitude = -85.20415
Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35?45?09.18""N, longitude = 85?12?14.94""W","Applies to 2D and the horizontal component of 3D ellipsoidal systems.","EPSG guidance note #7.","OGP",11/22/2002 0:00:00,,1
9638,"Degree representation conversion: degH to DMSH",1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:
deg decimal degrees
adeg absolute value of decimal degrees
ideg integer degrees
sdeg signed integer degree
min real-number minutes
imin integer minutes
sec real-number seconds
lathem, lonhem hemisphere abbreviation
In this conversion (both forward and reverse) the hemisphere parameter remains unchanged and retains its position in the respective coordinate strings.
Forward calculation from degH representation to DMSH representation:
ideg = INT(adeg)
min = (adeg - ideg) * 60
imin = INT(min)
sec = (min - imin) * 60
Reverse calculation from DMSH representation to decimal degree representation:
adeg = (ideg + imin/60 + sec/3600)","Source CRS = WGS 84 (degH) (CRS code 63266406).
Latitude = 35.75255N, longitude = 85.20415W
Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35?45?09.18""N, longitude = 85?12?14.94""W","Applies to 2D and the horizontal component of 3D ellipsoidal systems.","EPSG guidance note #7.","OGP",11/22/2002 0:00:00,,1
9639,"Degree representation conversion: Hdeg to DMSH",1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:
deg decimal degrees
adeg absolute value of decimal degrees
ideg integer degrees
sdeg signed integer degree
min real-number minutes
imin integer minutes
sec real-number seconds
lathem, lonhem hemisphere abbreviation
In this conversion the hemisphere parameters retain their values but change their positions in their respective coordinate strings from the end of the strings to the beginnings (both forward and reverse).
Forward calculation from Hdeg representation to DMSH representation:
First, re-order fields from
lathem, lat_adeg and lonhem, lon_adeg
to lat_adeg, lathem and lon_adeg, lonhem
Then
ideg = INT(adeg)
min = (adeg - ideg) * 60
imin = INT(min)
sec = (min - imin) * 60
Reverse calculation from DMSH representation to Hdeg representation:
adeg = (ideg + imin/60 + sec/3600)
Then re-order fields from
lat_adeg, lathem and lon_adeg, lonhem
to lathem, lat_adeg and lonhem, lon_adeg","Source CRS = WGS 84 (Hdeg) (CRS code 63266407).
Latitude = N35.75255, longitude = W85.20415
Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35?45?09.18""N, longitude = 85?12?14.94""W","Applies to 2D and the horizontal component of 3D ellipsoidal systems.","EPSG guidance note #7.","OGP",11/22/2002 0:00:00,,1
9640,"Degree representation conversion: DM to DMSH",1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:
deg decimal degrees
adeg absolute value of decimal degrees
ideg integer degrees
sdeg signed integer degree
min real-number minutes
imin integer minutes
sec real-number seconds
lathem, lonhem hemisphere abbreviation
Forward calculation from DM representation to DMSH representation:
ideg = ABS(sdeg)
imin = INT(min)
sec = (min - imin) * 60
If lat_sdeg < 0, lathem = S else lathem = N
If lon_sdeg < 0, lonhem = W else lathem = E
Reverse calculation from DMSH representation to DM representation:
sdeg = ideg * H
where for latitude, H = 1 if lathem = N and H = -1 if lathem = S
and for longitude H = 1 if lonhem = E and H = -1 if lonhem = W
Then
min = imin + (sec / 60)","Source CRS = WGS 84 (DM) (CRS code 63266408).
Latitude = 35?45.153?, longitude = -85?12.249?
Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35?45?09.18""N, longitude = 85?12?14.94""W","Applies to 2D and the horizontal component of 3D ellipsoidal systems.","EPSG guidance note #7.","OGP",11/22/2002 0:00:00,,1
9641,"Degree representation conversion: DMH to DMSH",1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:
deg decimal degrees
adeg absolute value of decimal degrees
ideg integer degrees
sdeg signed integer degree
min real-number minutes
imin integer minutes
sec real-number seconds
lathem, lonhem hemisphere abbreviation
The degree and hemisphere parameters remain unchanged in this conversion (both forward and reverse) and also retain their position in their respective coordinate strings.
Forward calculation from DMH representation to DMSH representation:
imin = INT(min)
sec = (min - imin) * 60
Reverse calculation from DMSH representation to DMH representation:
min = imin + (sec / 60)","Source CRS = WGS 84 (DMH) (CRS code 63266409).
Latitude = 35?45.153? N, longitude = 85?12.249? W
Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35?45?09.18""N, longitude = 85?12?14.94""W","Applies to 2D and the horizontal component of 3D ellipsoidal systems.","EPSG guidance note #7.","OGP",11/22/2002 0:00:00,,1
9642,"Degree representation conversion: HDM to DMSH",1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:
deg decimal degrees
adeg absolute value of decimal degrees
ideg integer degrees
sdeg signed integer degree
min real-number minutes
imin integer minutes
sec real-number seconds
lathem, lonhem hemisphere abbreviation
In this conversion the degree parameters remain unchanged in this conversion (both forward and reverse). The hemisphere parameters retain their values but change their positions in their respective coordinate strings.
Forward calculation from HDM representation to DMSH representation:
imin = INT(min)
sec = (min - imin) * 60
Then reorder fields from hem, ideg, imin, sec to ideg, imin, sec, hem.
Reverse calculation from DMSH representation to HDM representation:
min = imin + (sec / 60)
Then re-order fields from ideg, min, hem to hem, ideg, min.","Source CRS = WGS 84 (HDM) (CRS code 63266410).
Latitude = N35?45.153?, longitude = W85?12.249?
Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35?45?09.18""N, longitude = 85?12?14.94""W","Applies to 2D and the horizontal component of 3D ellipsoidal systems.","EPSG guidance note #7.","OGP",11/22/2002 0:00:00,,1
9643,"Degree representation conversion: DMS to DMSH",1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:
deg decimal degrees
adeg absolute value of decimal degrees
ideg integer degrees
sdeg signed integer degree
min real-number minutes
imin integer minutes
sec real-number seconds
lathem, lonhem hemisphere abbreviation
In this conversion (both forward and reverse) the minute and second parameters remain unchanged.
Forward calculation from DMS representation to DMSH representation:
ideg = ABS(sdeg)
If lat_sdeg < 0, lathem = S else lathem = N
If lon_sdeg < 0, lonhem = W else lathem = E
Reverse calculation from DMSH representation to DMS representation:
sdeg = ideg * H
where for latitude, H = 1 if lathem = N and H = -1 if lathem = S
and for longitude H = 1 if lonhem = E and H = -1 if lonhem = W","Source CRS = WGS 84 (DMS) (CRS code 63266411).
Latitude = 35?45?09.18"", longitude = -85?12?14.94""
Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35?45?09.18""N, longitude = 85?12?14.94""W","Applies to 2D and the horizontal component of 3D ellipsoidal systems.","EPSG guidance note #7.","OGP",11/22/2002 0:00:00,,1
9644,"Degree representation conversion: HDMS to DMSH",1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:
deg decimal degrees
adeg absolute value of decimal degrees
ideg integer degrees
sdeg signed integer degree
min real-number minutes
imin integer minutes
sec real-number seconds
lathem, lonhem hemisphere abbreviation
In this conversion the parameter values remain unchanged but are re-ordered.
For the forward calculation from HDMS representation to DMSH representation, for each of latitude and longitude re-order the fields:
from hem, ideg, imin, sec
to ideg, imin, sec, hem
For the reverse calculation from DMSH representation to HDMS representation, for each of latitude and longitude re-order the fields:
from ideg, imin, sec, hem
to hem, ideg, imin, sec","Source CRS = WGS 84 (HDMS) (CRS code 63266412).
Latitude = N35?45?09.18"", longitude = W85?12?14.94""
Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35?45?09.18""N, longitude = 85?12?14.94""W","Applies to 2D and the horizontal component of 3D ellipsoidal systems.","EPSG guidance note #7.","OGP",11/22/2002 0:00:00,,1
9645,"General polynomial of degree 2",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the coordinate values in both the source and the target coordinate reference system to ?manageable? numbers, between ?10 and +10 at most. This is achieved by working with offsets relative to a central evaluation point, scaled to the desired number range by applying a scaling factor to the coordinate offsets.
Hence an evaluation point is chosen in the source coordinate reference system (XS0, YS0) and in the target coordinate reference system (XT0, YT0). Often these two sets of coordinates do not refer to the same physical point but two points are chosen that have the same coordinate values in both the source and the target coordinate reference system. (When the two points have identical coordinates, these parameters are conveniently eliminated from the formulas, but the general case where the coordinates differ is given here).
The selection of an evaluation point in each of the two coordinate reference systems allows the point coordinates in both to be reduced as follows:
XS - XS0
YS - YS0
and
XT ? XT0
YT ? YT0
These coordinate differences are expressed in their own unit of measure, which may not be the same as that of the corresponding coordinate reference system. )
A further reduction step is usually necessary to bring these coordinate differences into the desired numerical range by applying a scaling factor to the coordinate differences in order to reduce them to a value range that may be applied to the polynomial formulae below without introducing numerical precision errors:
U = mS.(XS - XS0)
V = mS.(YS - YS0)
where
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
mS is the scaling factor applied the coordinate differences in the source coordinate reference system.
The normalised coordinates U and V of the point whose coordinates are to be transformed are used as input to the polynomial transformation formula. In order to control the numerical range of the polynomial coefficients An and Bn the output coordinate differences dX and dY are multiplied by a scaling factor, mT.
mT.dX = A0 + A1.U + A2.V + A3.U^2 + A4.U.V + A5.V^2
mT.dY = B0 + B1.U + B2.V + B3.U^2 + B4.U.V + B5.V^2
from which dX and dY are evaluated. These will be in the units of the target coordinate reference system.
The polynomial coefficients are given as parameters of the form Aumvn and Bumvn, where m is the power to which U is raised and n is the power to which V is raised. For example, A3 is represented as coordinate operation parameter Au2v0.
The relationship between the two coordinate reference systems can now be written as follows:
(XT - XT0) = (XS ? XS0) + dX
(YT - YT0) = (YS ? YS0) + dY
or
XT = XS ? XS0 + XT0 + dX
YT = YS ? YS0 + YT0 + dY
where:
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system,
dX, dY are derived through the scaled polynomial formulas.","(none)",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9646,"General polynomial of degree 3",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the coordinate values in both the source and the target coordinate reference system to ?manageable? numbers, between ?10 and +10 at most. This is achieved by working with offsets relative to a central evaluation point, scaled to the desired number range by applying a scaling factor to the coordinate offsets.
Hence an evaluation point is chosen in the source coordinate reference system (XS0, YS0) and in the target coordinate reference system (XT0, YT0). Often these two sets of coordinates do not refer to the same physical point but two points are chosen that have the same coordinate values in both the source and the target coordinate reference system. (When the two points have identical coordinates, these parameters are conveniently eliminated from the formulas, but the general case where the coordinates differ is given here).
The selection of an evaluation point in each of the two coordinate reference systems allows the point coordinates in both to be reduced as follows:
XS - XS0
YS - YS0
and
XT ? XT0
YT ? YT0
These coordinate differences are expressed in their own unit of measure, which may not be the same as that of the corresponding coordinate reference system. )
A further reduction step is usually necessary to bring these coordinate differences into the desired numerical range by applying a scaling factor to the coordinate differences in order to reduce them to a value range that may be applied to the polynomial formulae below without introducing numerical precision errors:
U = mS.(XS - XS0)
V = mS.(YS - YS0)
where
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
mS is the scaling factor applied the coordinate differences in the source coordinate reference system.
The normalised coordinates U and V of the point whose coordinates are to be transformed are used as input to the polynomial transformation formula. In order to control the numerical range of the polynomial coefficients An and Bn the output coordinate differences dX and dY are multiplied by a scaling factor, mT.
mT.dX = A0 + A1.U + A2.V + A3.U^2 + A4.U.V + A5.V^2 + A6.U^3 + A7.U^2.V + A8.U.V^2 + A9.V^3
mT.dY = B0 + B1.U + B2.V + B3.U^2 + B4.U.V + B5.V^2 + B6.U^3 + B7.U^2.V + B8.U.V^2 + B9.V^3
from which dX and dY are evaluated. These will be in the units of the target coordinate reference system.
The polynomial coefficients are given as parameters of the form Aumvn and Bumvn, where m is the power to which U is raised and n is the power to which V is raised. For example, A7 is represented as coordinate operation parameter Au2v1.
The relationship between the two coordinate reference systems can now be written as follows:
(XT - XT0) = (XS ? XS0) + dX
(YT - YT0) = (YS ? YS0) + dY
or
XT = XS ? XS0 + XT0 + dX
YT = YS ? YS0 + YT0 + dY
where:
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system,
dX, dY are derived through the scaled polynomial formulas.","(none)",,"EPSG guidance note #7-2, http://www.epsg.org","OGP",12/21/2002 0:00:00,,0
9647,"General polynomial of degree 4",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the coordinate values in both the source and the target coordinate reference system to ?manageable? numbers, between ?10 and +10 at most. This is achieved by working with offsets relative to a central evaluation point, scaled to the desired number range by applying a scaling factor to the coordinate offsets.
Hence an evaluation point is chosen in the source coordinate reference system (XS0, YS0) and in the target coordinate reference system (XT0, YT0). Often these two sets of coordinates do not refer to the same physical point but two points are chosen that have the same coordinate values in both the source and the target coordinate reference system. (When the two points have identical coordinates, these parameters are conveniently eliminated from the formulas, but the general case where the coordinates differ is given here).
The selection of an evaluation point in each of the two coordinate reference systems allows the point coordinates in both to be reduced as follows:
XS - XS0
YS - YS0
and
XT ? XT0
YT ? YT0
These coordinate differences are expressed in their own unit of measure, which may not be the same as that of the corresponding coordinate reference system. )
A further reduction step is usually necessary to bring these coordinate differences into the desired numerical range by applying a scaling factor to the coordinate differences in order to reduce them to a value range that may be applied to the polynomial formulae below without introducing numerical precision errors:
U = mS.(XS - XS0)
V = mS.(YS - YS0)
where
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
mS is the scaling factor applied the coordinate differences in the source coordinate reference system.
The normalised coordinates U and V of the point whose coordinates are to be transformed are used as input to the polynomial transformation formula. In order to control the numerical range of the polynomial coefficients An and Bn the output coordinate differences dX and dY are multiplied by a scaling factor, mT.
mT.dX = A0 + A1.U + A2.V + A3.U^2 + A4.U.V + A5.V^2 + A6.U^3 + A7.U^2.V + A8.U.V^2 + A9.V^3
+ A10.U^4 + A11.U^3.V + A12.U^2.V^2 + A13.U.V^3 + A14.V^4
mT.dY = B0 + B1.U + B2.V + B3.U^2 + B4.U.V + B5.V^2 + B6.U^3 + B7.U^2.V + B8.U.V^2 + B9.V^3
+ B10.U^4 + B11.U^3.V + B12.U^2.V^2 + B13.U.V^3 + B14.V^4
from which dX and dY are evaluated. These will be in the units of the target coordinate reference system.
The polynomial coefficients are given as parameters of the form Aumvn and Bumvn, where m is the power to which U is raised and n is the power to which V is raised. For example, A13 is represented as coordinate operation parameter Au1v3.
The relationship between the two coordinate reference systems can now be written as follows:
(XT - XT0) = (XS ? XS0) + dX
(YT - YT0) = (YS ? YS0) + dY
or
XT = XS ? XS0 + XT0 + dX
YT = YS ? YS0 + YT0 + dY
where:
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system,
dX, dY are derived through the scaled polynomial formulas.","(none)",,"EPSG guidance note #7-2, http://www.epsg.org","OGP",12/21/2002 0:00:00,,0
9648,"General polynomial of degree 6",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as limitations in the transcription will be avoided.
The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the coordinate values in both the source and the target coordinate reference system (CRS) to ?manageable? numbers, between ?10 and +10 at most. This is achieved by working with offsets relative to a central evaluation point, scaled to the desired number range by applying a scaling factor to the coordinate offsets.
Hence an evaluation point is chosen in the source CRS (XS0, YS0) and in the target CRS (XT0, YT0). Often these two sets of coordinates do not refer to the same physical point but two points are chosen that have the same coordinate values in both the source and the target CRS. (When the two points have identical coordinates, these parameters are conveniently eliminated from the formulas, but the general case where the coordinates differ is given here).
The selection of an evaluation point in each of the two CRSs allows the point coordinates in both to be reduced as follows:
XS - XS0
YS - YS0
and
XT ? XT0
YT ? YT0
These coordinate differences are expressed in their own unit of measure, which may not be the same as that of the corresponding CRS.)
A further reduction step is usually necessary to bring these coordinate differences into the desired numerical range by applying a scaling factor to the coordinate differences in order to reduce them to a value range that may be applied to the polynomial formulae below without introducing numerical precision errors:
U = mS.(XS - XS0)
V = mS.(YS - YS0)
where
XS , YS are coordinates in the source CRS,
XS0 , YS0 are coordinates of the evaluation point in the source CRS,
mS is the scaling factor applied the coordinate differences in the source CRS.
The normalised coordinates U and V of the point whose coordinates are to be transformed are used as input to the polynomial transformation formula. In order to control the numerical range of the polynomial coefficients An and Bn the output coordinate differences dX and dY are multiplied by a scaling factor, mT.
mT.dX = A0 + A1.U + A2.V + A3.U^2 + A4.U.V + A5.V^2 + A6.U^3 + A7.U^2.V + A8.U.V^2 + A9.V^3
+ A10.U^4 + A11.U^3.V + A12.U^2.V^2 + A13.U.V^3 + A14.V^4
+ A15.U^5 + A16.U^4.V + A17.U^3.V^2 + A18.U^2.V^3 + A19.U.V^4 + A20.V^5
+ A21.U^6 + A22.U^5.V + A23.U^4.V^2 + A24.U^3.V^3 + A25.U^2.V^4 + A26.U.V^5 + A27.V^6
mT.dY = B0 + B1.U + B2.V + B3.U^2 + B4.U.V + B5.V^2 + B6.U^3 + B7.U^2.V + B8.U.V^2 + B9.V^3
+ B10.U^4 + B11.U^3.V + B12.U^2.V^2 + B13.U.V^3 + B14.V^4
+ B15.U^5 + B16.U^4.V + B17.U^3.V^2 + B18.U^2.V^3 + B19.U.V^4 + B20.V^5
+ B21.U^6 + B22.U^5.V + B23.U^4.V^2 + B24.U^3.V^3 + B25.U^2.V^4 + B26.U.V^5 + B27.V^6
from which dX and dY are evaluated. These will be in the units of the target CRS.
The polynomial coefficients are given as parameters of the form Aumvn and Bumvn, where m is the power to which U is raised and n is the power to which V is raised. For example, A17 is represented as coordinate operation parameter Au3v2.
The relationship between the two CRSs can now be written as follows:
(XT - XT0) = (XS ? XS0) + d
(YT - YT0) = (YS ? YS0) + dY
or
XT = XS ? XS0 + XT0 + d
YT = YS ? YS0 + YT0 + dY
where:
XT, YT are coordinates in the target CRS,
XS, YS are coordinates in the source CRS,
XS0, YS0 are coordinates of the evaluation point in the source CRS,
XT , YT0 are coordinates of the evaluation point in the target CRS,
dX, dY are derived through the scaled polynomial formulas.","See EPSG Guidance Note 7-2.",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9649,"Reversible polynomial of degree 2",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
See method code 9645 for description of general polynomial formula.
A general polynomial transformation is reversible only when the following conditions are met.
1. The co-ordinates of source and target evaluation point are (numerically) the same.
2. The unit of measure of the coordinate differences in source and target coordinate reference system are the same.
3. The scaling factors applied to source and target coordinate differences are the same.
4. The spatial variation of the differences between the coordinate reference systems around any given location is sufficiently small.
Clarification on conditions for polynomial reversibility:
Regarding 1 and 2 - In the reverse transformation the roles of the source and target coordinate reference systems are reversed. Consequently, the co-ordinates of the evaluation point in the source coordinate reference system become those in the target coordinate reference system in the reverse transformation. Usage of the same transformation parameters for the reverse transformation will therefore only be valid if the evaluation point coordinates are numerically the same in source and target coordinate reference system and in the same units. That is, XS0 = XT0 = X0 and YS0 = YT0 = Y0.
Re 3 - The same holds for the scaling factors of the source and target coordinate differences and for the units of measure of the coordinate differences. That is, mS = mT = m.
Re 4 - If conditions 1, 2 and 3 are all satisfied it then may be possible to use the forward polynomial algorithm with the forward parameters for the reverse transformation. This is the case if the spatial variations in dX and dY around any given location are sufficiently constant. The signs of the polynomial coefficients are then reversed but the scaling factor and the evaluation point coordinates retain their signs. If these spatial variations in dX and dY are too large, for the reverse transformation iteration would be necessary. It is therefore not the algorithm that determines whether a single step solution is sufficient or whether iteration is required, but the desired accuracy combined with the degree of spatial variability of dX and dY.
An example of a reversible polynomial is transformation is ED50 to ED87 (1) for the North Sea. The suitability of this transformation to be described by a reversible polynomial can easily be explained. In the first place both source and target coordinate reference systems are of type geographic 2D. The typical difference in coordinate values between ED50 and ED87 is in the order of 2 metres (approximately 10E-6 degrees) in the area of application. The polynomial functions are evaluated about central points with coordinates of 55 deg N, 0 deg E in both coordinate reference systems. The reduced coordinate differences (in degrees) are used as input parameters to the polynomial functions. The output coordinate differences are corrections to the input coordinate offsets of about 10E-6 degrees. This difference of several orders of magnitude between input and output values is the property that makes a polynomial function reversible in practice (although not in a formal mathematical sense).
The error made by the polynomial approximation formulas in calculating the reverse correction is of the same order of magnitude as the ratio of output versus input:
(output error / input error) = ( output valu/ input value) which is approximately 10E-6
As long as the input values (the coordinate offsets from the evaluation point) are orders of magnitude larger than the output (the corrections), and provided the coefficients are used with changed signs, the polynomial transformation may be considered to be reversible.","See Reversible polynomial of degree 4, code 9651, for general methodology.","Reversibility is subject to constraints. See Guidance Note 7 for conditions and clarification.","EPSG guidance note #7-2, http://www.epsg.org","OGP",12/21/2002 0:00:00,,0
9650,"Reversible polynomial of degree 3",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
See method code 9646 for description of general polynomial formula.
A general polynomial transformation is reversible only when the following conditions are met.
1. The co-ordinates of source and target evaluation point are (numerically) the same.
2. The unit of measure of the coordinate differences in source and target coordinate reference system are the same.
3. The scaling factors applied to source and target coordinate differences are the same.
4. The spatial variation of the differences between the coordinate reference systems around any given location is sufficiently small.
Clarification on conditions for polynomial reversibility:
Regarding 1 and 2 - In the reverse transformation the roles of the source and target coordinate reference systems are reversed. Consequently, the co-ordinates of the evaluation point in the source coordinate reference system become those in the target coordinate reference system in the reverse transformation. Usage of the same transformation parameters for the reverse transformation will therefore only be valid if the evaluation point coordinates are numerically the same in source and target coordinate reference system and in the same units. That is, XS0 = XT0 = X0 and YS0 = YT0 = Y0.
Re 3 - The same holds for the scaling factors of the source and target coordinate differences and for the units of measure of the coordinate differences. That is, mS = mT = m.
Re 4 - If conditions 1, 2 and 3 are all satisfied it then may be possible to use the forward polynomial algorithm with the forward parameters for the reverse transformation. This is the case if the spatial variations in dX and dY around any given location are sufficiently constant. The signs of the polynomial coefficients are then reversed but the scaling factor and the evaluation point coordinates retain their signs. If these spatial variations in dX and dY are too large, for the reverse transformation iteration would be necessary. It is therefore not the algorithm that determines whether a single step solution is sufficient or whether iteration is required, but the desired accuracy combined with the degree of spatial variability of dX and dY.
An example of a reversible polynomial is transformation is ED50 to ED87 (1) for the North Sea. The suitability of this transformation to be described by a reversible polynomial can easily be explained. In the first place both source and target coordinate reference systems are of type geographic 2D. The typical difference in coordinate values between ED50 and ED87 is in the order of 2 metres (approximately 10E-6 degrees) in the area of application. The polynomial functions are evaluated about central points with coordinates of 55 deg N, 0 deg E in both coordinate reference systems. The reduced coordinate differences (in degrees) are used as input parameters to the polynomial functions. The output coordinate differences are corrections to the input coordinate offsets of about 10E-6 degrees. This difference of several orders of magnitude between input and output values is the property that makes a polynomial function reversible in practice (although not in a formal mathematical sense).
The error made by the polynomial approximation formulas in calculating the reverse correction is of the same order of magnitude as the ratio of output versus input:
(output error / input error) = ( output valu/ input value) which is approximately 10E-6
As long as the input values (the coordinate offsets from the evaluation point) are orders of magnitude larger than the output (the corrections), and provided the coefficients are used with changed signs, the polynomial transformation may be considered to be reversible.","See Reversible polynomial of degree 4, code 9651, for general methodology.","Reversibility is subject to constraints. See Guidance Note 7 for conditions and clarification.","EPSG guidance note #7-2, http://www.epsg.org","OGP",12/21/2002 0:00:00,,0
9651,"Reversible polynomial of degree 4",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
See method code 9647 for description of general polynomial formula.
A general polynomial transformation is reversible only when the following conditions are met.
1. The co-ordinates of source and target evaluation point are (numerically) the same.
2. The unit of measure of the coordinate differences in source and target coordinate reference system are the same.
3. The scaling factors applied to source and target coordinate differences are the same.
4. The spatial variation of the differences between the coordinate reference systems around any given location is sufficiently small.
Clarification on conditions for polynomial reversibility:
Regarding 1 and 2 - In the reverse transformation the roles of the source and target coordinate reference systems are reversed. Consequently, the co-ordinates of the evaluation point in the source coordinate reference system become those in the target coordinate reference system in the reverse transformation. Usage of the same transformation parameters for the reverse transformation will therefore only be valid if the evaluation point coordinates are numerically the same in source and target coordinate reference system and in the same units. That is, XS0 = XT0 = X0 and YS0 = YT0 = Y0.
Re 3 - The same holds for the scaling factors of the source and target coordinate differences and for the units of measure of the coordinate differences. That is, mS = mT = m.
Re 4 - If conditions 1, 2 and 3 are all satisfied it then may be possible to use the forward polynomial algorithm with the forward parameters for the reverse transformation. This is the case if the spatial variations in dX and dY around any given location are sufficiently constant. The signs of the polynomial coefficients are then reversed but the scaling factor and the evaluation point coordinates retain their signs. If these spatial variations in dX and dY are too large, for the reverse transformation iteration would be necessary. It is therefore not the algorithm that determines whether a single step solution is sufficient or whether iteration is required, but the desired accuracy combined with the degree of spatial variability of dX and dY.
An example of a reversible polynomial is transformation is ED50 to ED87 (1) for the North Sea. The suitability of this transformation to be described by a reversible polynomial can easily be explained. In the first place both source and target coordinate reference systems are of type geographic 2D. The typical difference in coordinate values between ED50 and ED87 is in the order of 2 metres (approximately 10E-6 degrees) in the area of application. The polynomial functions are evaluated about central points with coordinates of 55 deg N, 0 deg E in both coordinate reference systems. The reduced coordinate differences (in degrees) are used as input parameters to the polynomial functions. The output coordinate differences are corrections to the input coordinate offsets of about 10E-6 degrees. This difference of several orders of magnitude between input and output values is the property that makes a polynomial function reversible in practice (although not in a formal mathematical sense).
The error made by the polynomial approximation formulas in calculating the reverse correction is of the same order of magnitude as the ratio of output versus input:
(output error / input error) = ( output valu/ input value) which is approximately 10E-6
As long as the input values (the coordinate offsets from the evaluation point) are orders of magnitude larger than the output (the corrections), and provided the coefficients are used with changed signs, the polynomial transformation may be considered to be reversible.","For geodetic transformation ED50 to ED87 (1)
Offset unit: degree
Ordinate 1 of evaluation point X0 = 55? 00' 00.000""N = +55 degrees
Ordinate 2 of evaluation point Y0 = 0? 00' 00.000""E = +0 degrees
Parameters:
A0 = -5.56098E-06 A1 = -1.55391E-06 ... A14 = -4.01383E-09
B0 = +1.48944E-05 B2 = +2.68191E-05 ... B14 = +7.62236E-09
Forward calculation for:
ED50 Latitude = Xs =52* 30?30""N = +52.508333333 degrees
ED50 Longitude = Ys = 2*E= +2.0 degrees
U = XS - X0 = * ED50 - X0 = 52.508333333 - 55.0 = -2.491666667 degrees
V = YS - Y0 = * ED50 - Y0 = 2.0 - 0.0 = 2.0 degrees
dX = A0 + A1.U + ... + A14.V4
= -5.56098E-06 + (-1.55391E-06 * -2.491666667) + ... + (-4.01383E-09 * 2.0^4)
= -3.12958E-06 degrees
dY = B0 + B1.U + ... + B14.V4
= +1.48944E-05 + (2.68191E-05 * -2.491666667) + ... + (7.62236E-09 * 2.0^4)
= +9.80126E-06 degrees
Then ED87 Latitude = XT = XS + dX
= 52.508333333 - 3.12958E-06 degrees
= 52* 30? 29.9887"" N
ED87 Longitude = YT = YS + dY
= 2* 00? 00.0353"" E
Reverse calculation for transformation ED50 to ED87 (1).
The transformation method for the ED50 to ED87 (1) transformation, 4th-order reversible polynomial, is reversible. The same formulas may be applied for the reverse calculation, but coefficients A0 through A14 and B0 through B14 are applied with reversal of their signs. Sign reversal is not applied to the coordinates of the evaluation point. Thus:
Ordinate 1 of evaluation point X0 = 55? 00' 00.000""N = +55 degrees
Ordinate 2 of evaluation point Y0 = 0? 00' 00.000""E = +0 degrees
A0 = +5.56098E-06 A1 = +1.55391E-06 ... A14 = +4.01383E-09
B0 = -1.48944E-05 B1 = -2.68191E-05 ... B14 = -7.62236E-09
Reverse calculation for:
ED87 Latitude = XS = 52? 30?29.9887""N = +52.5083301944 degrees
ED87 Longitude = YS = 2? 00? 00.0353"" E = +2.0000098055 degrees
U = 52.5083301944 - 55.0 = -2.4916698056 degrees
V = 2.0000098055 - 0.0 = 2.0000098055 degrees
dX = A0 + A1.U + ... + A14.V4
= +5.56098E-06 + (1.55391E-06 * -2.491666667) + ... + (4.01383E-09 * 2.0000098055^4)
= +3.12957E-06 degrees
dY = B0 + B1.U + ... + B14.V4
= -1.48944E-05 + (-2.68191E-05 * -2.491666667) + ... + (-7.62236E-09 * 2.0000098055^4)
= -9.80124E-06 degrees
Then ED50 Latitude = XT = XS + dX
= 52.5083301944 + 3.12957E-06 degrees
= 52? 30? 30.000"" N
ED50 Longitude = YT = YS + dY
= 2? 00? 00.000"" E","Reversibility is subject to constraints. See Guidance Note 7 for conditions and clarification.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9652,"Complex polynomial of degree 3",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The relationship between two projected coordinate reference systems may be approximated more elegantly by a single polynomial regression formula written in terms of complex numbers. The advantage is that the dependence between the ?A? and ?B? coefficients (for U and V) is taken into account in the formula, resulting in fewer coefficients for the same order polynomial. A third-order polynomial in complex numbers is used in Belgium.
mT.(dX + i. dY) = (A1 + i. A2).(U + i.V) + (A3 + i. A4).(U + i.V)^2 + (A5 + i. A6).(U + i.V)^3
where U = mS.(XS - XS0)
V = mS.(YS - YS0)
and mS, mT are the scaling factors for the coordinate differences in the source and target coordinate reference systems.
The polynomial to degree 4 can alternatively be expressed in matrix form.
Then
XT = XS - XS0 + XT0 + dX
YT = YS - YS0 + YT0 + dY
where
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system.
Note that the zero order coefficients of the general polynomial, A0 and B0, have apparently disappeared. In reality they are absorbed by the different coordinates of the source and of the target evaluation point, which in this case, are numerically very different because of the use of two different projected coordinate reference systems for source and target.
The transformation parameter values (the coefficients) are not reversible. For the reverse transformation a different set of parameter values are required, used within the same formulas as the forward direction.","For transformation Belge Lambert 72 to ED50 / UTM zone 31N,
Eo1 = 0
No1 = 0
Eo2 = 449681.702
No2 = 5460505.326
A1 = -71.3747
A2 = 1858.8407
A3 = -5.4504
A4 = -16.9681
A5 = 4.0783
A6 = 0.2193
For source coordinate system E1=200000 N1=100000, then
E2 = 647737.377 N2 = 5564124.227.","Coordinate pairs treated as complex numbers. This exploits the correlation between the polynomial coefficients and leads to a smaller number of coefficients than the general polynomial of degree 3.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9653,"Complex polynomial of degree 4",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The relationship between two projected coordinate reference systems may be approximated more elegantly by a single polynomial regression formula written in terms of complex numbers. The advantage is that the dependence between the ?A? and ?B? coefficients (for U and V) is taken into account in the formula, resulting in fewer coefficients for the same order polynomial. A fourth-order polynomial in complex numbers is used in The Netherlands for transforming coordinates referenced to the Amersfoort / RD system to and from ED50 / UTM.
mT.(dX + i. dY) = (A1 + i. A2).(U + i.V) + (A3 + i. A4).(U + i.V)^2 + (A5 + i. A6).(U + i.V)^3 + (A7 + i.A8).(U + i.V)^4
where U = mS.(XS - XS0)
V = mS.(YS - YS0)
and mS, mT are the scaling factors for the coordinate differences in the source and target coordinate reference systems.
The polynomial to degree 4 can alternatively be expressed in matrix form.
Then
XT = XS - XS0 + XT0 + dX
YT = YS - YS0 + YT0 + dY
where
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system.
Note that the zero order coefficients of the general polynomial, A0 and B0, have apparently disappeared. In reality they are absorbed by the different coordinates of the source and of the target evaluation point, which in this case, are numerically very different because of the use of two different projected coordinate reference systems for source and target.
The transformation parameter values (the coefficients) are not reversible. For the reverse transformation a different set of parameter values are required, used within the same formulas as the forward direction.","For transformation RD / Netherlands New to ED50 / UTM zone 31N,
Eo1 = 155000
No1 = 463000
Eo2 = 663395.607
No2 = 5781194.380
A1 = -51.681
A2 = 3290.525
A3 = 20.172
A4 = 1.133
A5 = 2.075
A6 = 0.251
A7 = 0.075
A8 = -0.012
For source coordinate system E1=200000 N1=500000, then
E2 =707155.557 N2 = 5819663.128.","Coordinate pairs treated as complex numbers. This exploits the correlation between the polynomial coefficients and leads to a smaller number of coefficients than the general polynomial of degree 4.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9654,"Reversible polynomial of degree 13",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
See method code 9648 for description of general polynomial formula.
A general polynomial transformation is reversible only when the following conditions are met.
1. The co-ordinates of source and target evaluation point are (numerically) the same.
2. The unit of measure of the coordinate differences in source and target coordinate reference system are the same.
3. The scaling factors applied to source and target coordinate differences are the same.
4. The spatial variation of the differences between the coordinate reference systems around any given location is sufficiently small.
Clarification on conditions for polynomial reversibility:
Regarding 1 and 2 - In the reverse transformation the roles of the source and target coordinate reference systems are reversed. Consequently, the co-ordinates of the evaluation point in the source coordinate reference system become those in the target coordinate reference system in the reverse transformation. Usage of the same transformation parameters for the reverse transformation will therefore only be valid if the evaluation point coordinates are numerically the same in source and target coordinate reference system and in the same units. That is, XS0 = XT0 = X0 and YS0 = YT0 = Y0.
Re 3 - The same holds for the scaling factors of the source and target coordinate differences and for the units of measure of the coordinate differences. That is, mS = mT = m.
Re 4 - If conditions 1, 2 and 3 are all satisfied it then may be possible to use the forward polynomial algorithm with the forward parameters for the reverse transformation. This is the case if the spatial variations in dX and dY around any given location are sufficiently constant. The signs of the polynomial coefficients are then reversed but the scaling factor and the evaluation point coordinates retain their signs. If these spatial variations in dX and dY are too large, for the reverse transformation iteration would be necessary. It is therefore not the algorithm that determines whether a single step solution is sufficient or whether iteration is required, but the desired accuracy combined with the degree of spatial variability of dX and dY.
An example of a reversible polynomial is transformation is ED50 to ED87 (1) for the North Sea. The suitability of this transformation to be described by a reversible polynomial can easily be explained. In the first place both source and target coordinate reference systems are of type geographic 2D. The typical difference in coordinate values between ED50 and ED87 is in the order of 2 metres (approximately 10E-6 degrees) in the area of application. The polynomial functions are evaluated about central points with coordinates of 55 deg N, 0 deg E in both coordinate reference systems. The reduced coordinate differences (in degrees) are used as input parameters to the polynomial functions. The output coordinate differences are corrections to the input coordinate offsets of about 10E-6 degrees. This difference of several orders of magnitude between input and output values is the property that makes a polynomial function reversible in practice (although not in a formal mathematical sense).
The error made by the polynomial approximation formulas in calculating the reverse correction is of the same order of magnitude as the ratio of output versus input:
(output error / input error) = ( output valu/ input value) which is approximately 10E-6
As long as the input values (the coordinate offsets from the evaluation point) are orders of magnitude larger than the output (the corrections), and provided the coefficients are used with changed signs, the polynomial transformation may be considered to be reversible.","See Reversible polynomial of degree 4, code 9651, for general methodology.",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9655,"France geocentric interpolation",1,"The transformation is made in the geocentric domain using the geocentric translations method (code 9603). However the translation parameter values are derived by bilinear interpolation in a gridded data set. The arguments in accessing the grid are geodetic latitude and longitude of the forward transformation source coordinate reference system.
The method is reversible. Iteration for latitude and longitude is required to obtain the correct geocentric translation values. The geocentric translations are applied with sign reversal.","See information source.",,"IGN document NTG_88.pdf, ""Grille de parametres de transformation de coordonnees"". http://www.ign.fr","IOGP",6/13/2017 0:00:00,"2008.027 2017.018",0
9656,"Cartesian Grid Offsets",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Easting_T = Easting_S + easting_offset
Northing_T = Northingn_S + northing_offset.","(none)","This transformation allows calculation of coordinates in the target system by adding the parameter value to the coordinate values of the point in the source system.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9657,"Vertical Offset and Slope",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
In Europe, national height systems are related to the pan-European height system through three transformation parameters and the formula:
Ht = Hs + A + [IncLat * rhoO * (Lat ? LatO)] + [IncLong * nuO * (Long ? LongO) * cos(Lat)]
where
Ht = gravity-related height value in the target vertical coordinate reference system.
Hs = gravity-related height value in the source vertical coordinate reference system.
A is the value of the vertical offset to be applied.
IncLat is the value in radians of the inclination parameter in the latitude domain, i.e. in the plane of the meridian, derived at an evaluation point with coordinates of LatO , LongO.
IncLon is the value of the inclination parameter in the longitude domain, i.e. perpendicular to the plane of the meridian.
rhoO is the radius of curvature of the meridian at latitude LatO, where rhoO = a(1 ? e^2)/(1 ? e^2 * sin^2(LatO))^1.5
nuO is the radius of curvature on the prime vertical (i.e. perpendicular to the meridian) at latitude LatO, wh ere nuO = a /(1 ? e^2 * sin^2(LatO))^0.5
Lat , Long are the horizontal coordinates of the point in the ETRS89 coordinate reference system, in radians.
LatO , LongO are the coordinates of the evaluation point in the ETRS89 coordinate reference system, in radians.
The horizontal location of the point must always be given in ETRS89 terms. Care is required where compound coordinate reference systems are in use: if the horizontal coordinates of the point are known in the local CRS they must first be transformed to ETRS89 values. The method is reversible.","For coordinate transformation LN02 to EVRF2000 (1)
Ordinate 1 of evaluation point: 46deg 55min N = 0.818850307 radians
Ordinate 2 of evaluation point: 8deg 11min E of Greenwich = 0.142826110 radians
Transformation Parameters:
A = -0.245m
IncLat = -0.210"" = -0.000001018 radians
IncLong = -0.032"" = -0.000000155 radians
Consider a point having a gravity-related height in the LN02 system (Hs) of 473.0m and with horizontal coordinates in the ETRS89 geographical coordinate reference system of:
ETRS89 latitude: 47deg 20 min N = 0.826122513 radians
ETRS89 longitude: 9 deg 40min E of Greenwich = 0.168715161 radians
Then rhoO = 6369526.88 m
IncLat term = -0.047 m
nuO = 6389555.64 m
incLong term = -0.017 m
whence EVRF2000 height (Ht) = 473.0 +(-0.245) + (-0.047) + (-0.017) = 472.690 m.","This transformation allows calculation of height in the target system by applying the parameter values to the height value of the point in the source system.","EPSG guidance note #7-2, http://www.epsg.org","OGP",4/14/2004 0:00:00,"2011.108",1
9658,"VERTCON",1,"The relationship between NGVD29 height and NAVD88 height vertical coordinate reference systems for the coterminous US is available through three gridded data files of offsets (sometimes called height differences). The vertical offset at a point is first interpolated within the grid of values using bi-linear interpolation. The interpolated offset is then applied as an offset:
? If a NAVD88 height is desired when a NGVD29 height is given, add the interpolated offset to the NGVD29 height.
? If a NGVD29 height is desired when a NAVD88 height is given, subtract the interpolated offset from the NAVD88 height.
Most horizontal positions of the bench marks used to generate the VERTCON grids were scaled from USGS topographic maps. The estimated uncertainty of the scaled positions, 6 arc-seconds, is greater than the differences between NAD27 and NAD83 coordinates. Therefore the latitude and longitude used for interpolation of the grids can be referenced to either NAD27 or to NAD83(1986) or to any of the NAD83(NSRS) realisations (HARN, NSRS2007 or 2011).","For a point in the Conus central grid area at 29?28'00.443""N, 98?28'49.346""W (NAD83) with NGVD29 height of 247.47 meters (811.9 feet).
First reduce the sexagesimal degree values to decimal degrees:
29?28'00.443""N = 29.4667897?N
98?28'49.346""W = 98.4803739?W
Then obtain the offsets at each of the surrounding grid nodes:
NW corner 29.50?N, 98.50?W, offset = 0.144m
NE corner 29.50?N, 98.45?W, offset = 0.132m
SE corner 29.45?N, 98.45?W, offset = 0.121m
SW corner 29.45?N, 98.50?W, offset = 0.125m
Then using bi-linear interpolation for 29.4667897?N, 98.4803739?W, offset = 0.129m
Then NAVD88 height = 247.47 + 0.129 = 247.599m.
For the reverse calculation of a NAVD88 height of 247.599m to NGVD29 height:
First obtain the offsets at each of the surrounding grid nodes:
NW corner 29.50?N, 98.50?W, offset = 0.144m
NE corner 29.50?N, 98.45?W, offset = 0.132m
SE corner 29.45?N, 98.45?W, offset = 0.121m
SW corner 29.45?N, 98.50?W, offset = 0.125m
Then using bi-linear interpolation for 29.4667897?N, 98.4803739?W, offset = +0.129m
Then NGVD29 height = 247.599 - 0.129 = 247.470m.","Any NAD realization may be used as the Interpolation CRS; bi-linear interpolation is used. Input expects longitudes to be positive west.","US National Geodetic Survey - http://www.ngs.noaa.gov","IOGP",12/15/2016 0:00:00,"2016.046",0
9659,"Geographic3D to 2D conversion",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The forward case is trivial. A 3-dimensional geographic coordinate reference system comprising of geodetic latitude, geodetic longitude and ellipsoidal height is converted to its 2-dimensional subset by the simple expedient of dropping the height. The axis order change operates on coordinates of a point whose order change is described by the following matrix operation:
[Derived CRS 1st coordinate] [1 0 0] [Base CRS 1st coordinate]
[Derived CRS 2nd coordinate] = [0 1 0] * [Base CRS 2nd coordinate]
[Derived CRS 3rd coordinate] [0 0 0] [Base CRS 3rd coordinate]
The reverse conversion, from 2D to 3D, is indeterminate. It is however a requirement when a geographic 2D coordinate reference system is to be transformed using a geocentric method which is 3-dimensional. In practice an artificial ellipsoidal height is created and appended to the geographic 2D coordinate reference system to create a geographic 3D coordinate reference system referenced to the same geodetic datum. The assumed ellipsoidal height is usually either set to the gravity-related height of a position in a compound coordinate reference system, or set to zero. As long as the height chosen is within a few kilometres of sea level, no error will be induced into the horizontal position resulting from the later geocentric transformation; the vertical coordinate will however be meaningless.","(none)","This is a parameter-less conversion.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",5/22/2015 0:00:00,"2015.018",0
9660,"Geographic3D offsets",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Lat_T = Lat_S + latitude_offset
Lon_T = Lon_S + longitude_offset
EllipsoidHeight_T = EllipsoidHeight_S + ellipsoid_height_offset.","(none)","This transformation allows calculation of coordinates in the target system by adding the parameter value to the coordinate values of the point in the source system.","EPSG guidance note #7-2, http://www.epsg.org","OGP",4/27/2004 0:00:00,,0
9661,"Geographic3D to GravityRelatedHeight (EGM)",0,"This transformation involves the application of a geoid-ellipsoid separation value interpolated from a geoid model. The model provides separation values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the separation is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the geoid-ellipsoid separation (N) above the ellipsoid of the source Geographic 3D CRS.
Then:
H = h - N
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","Applies to EGM84 and EGM96 models. For later model see Geographic3D to GravityRelatedHeight (EGM2008), method code 1025.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",12/15/2016 0:00:00,"2008.010 2008.097 2016.050",0
9662,"Geographic3D to GravityRelatedHeight (Ausgeoid98)",0,"This transformation involves the application of a geoid-ellipsoid separation value interpolated from a geoid model. The model provides separation values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the separation is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the geoid-ellipsoid separation (N) above the ellipsoid of the source Geographic 3D CRS.
Then:
H = h - N
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",12/15/2016 0:00:00,"2008.010 2016.050",0
9663,"Geographic3D to GravityRelatedHeight (OSGM-GB)",0,"This transformation involves the application of a height difference interpolated from a height correction model. The model provides height difference values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the height difference is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the height correction (C) of the target datum above the ellipsoid of the source Geographic 3D CRS. C differs from the geoid-ellipsoid separation N because a vertical datum is a realisation of the geoid surface, not the geoid itself.
Then:
H = h - C
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","Transformation of the vertical component of a Geographic 3D CRS to a Vertical CRS.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",12/15/2016 0:00:00,"2008.010 2010.104 2016.049",0
9664,"Geographic3D to GravityRelatedHeight (IGN1997)",0,"This transformation involves the application of a height difference interpolated from a height correction model. The model provides height difference values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the height difference is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the height correction (C) of the target datum above the ellipsoid of the source Geographic 3D CRS. C differs from the geoid-ellipsoid separation N because a vertical datum is a realisation of the geoid surface, not the geoid itself.
Then:
H = h - C
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","Transformation of the vertical component of a Geographic 3D CRS to a Vertical CRS.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",1/19/2018 0:00:00,"2008.010 2016.050 2017.045",0
9665,"Geographic3D to GravityRelatedHeight (US .gtx)",0,"This transformation involves the application of a height difference interpolated from a height correction model. The model provides height difference values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.
The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.
Calculation of the height difference is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the height correction (C) of the target datum above the ellipsoid of the source Geographic 3D CRS. C differs from the geoid-ellipsoid separation N because a vertical datum is a realisation of the geoid surface, not the geoid itself.
Then:
H = h - C
where h = the height above the ellipsoid in the source geographic 3D CRS
and H = the gravity-related height in the target vertical CRS.","(none)","Transformation of the vertical component of a Geographic 3D CRS to a Vertical CRS. NOAA .gtx grid file format.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",2/16/2018 0:00:00,"2008.010 2016.050 2017.037",0
9666,"P6 (I = J+90?) seismic bin grid transformation",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The P6 (I = J+90?) seismic bin grid transformation is a special case of the Affine Geometric Transformation (method code 9623). In practice bin grid definitions are based on an identified map grid. As such the example bin grids in the EPSG Dataset are described with the map grid (projected CRS) as source CRS. However to retain consistency with the syntax used in the description of the affine geometric transformation, in the formulas below the bin grid is the affine source CRS and the map grid is the affine target CRS.
Map grid Easting = XT = XT0 + XS . k . dSX . cos q + YS . k . dSY . sin q
Map grid Northing = YT = YT0 ? XS . k . dSX . sin q + YS . k . dSY . cos q
Bin grid I = XS = {[( XT ? XTO) * cos q ? (YT ? YTO) * sin q ] * [IncSX / (k * MX)]} + XSO
Bin grid J = YS = {[(XT ? XTO) * sin q + (YT ? YTO) * cos q] * [IncSY / (k * MY)]} + YSO
where:
XT0 ,YT0 = the map grid coordinates (easting, northing) of the origin point of the bin grid;
dSX , dSY = the length of one unit of the bin grid, expressed in units of map grid, for the bin grid I and J axes respectively;
k = point scale factor of the map grid at a chosen reference point;
q = the angle through which the bin grid axes must be rotated to coincide with the map grid axes (counter-clockwise is positive). Alternatively, the bearing (clockwise positive) of the bin grid J-axis measured relative to map grid north.","This example is given in the P6/98 document. The bin grid is based on projected CRS WGS 84 / UTM Zone 31N. The origin of the bin grid is defined to be at msp grid easting = 456781.0mE, northing = 5836723.0mN. The map grid point scale factor at this point is 0.99984.
The bin width on the I-axis (XS axis) is 25 metres, whilst the bin width on the J-axis (YS axis) is 12.5 metres. At its origin the bin grid has bin values of 1,1.
In the map grid, the bearing of the bin grid I and J axes are 110? and 20? respectively. The latter is one of the transformation parameters.
The transformation parameter values are:
Bin grid origin I: 1 bin
Bin grid origin J: 1 bin
Bin grid origin Easting: 456781.00 m
Bin grid origin Northing: 5836723.00 m
Scale factor of bin grid: 0.99984
Bin Width on I-axis: 25 m
Bin Width on J-axis: 12.5 m
Map grid bearing of bin grid J-axis: 20?
Bin node increment on I-axis: 1
Bin node increment on J-axis: 1
Calculation of map grid coordinates for centre of bin with coordinates: I = 300, J = 247:
Easting = XT = XTO + [(XS ? XSO) * cos q * k * MX / IncSX] + [(YS ? YSO) * sin q * k * MY / IncSY]
= 456781.000 + 7023.078 + 1051.544
= 464855.62 m.
Northing = YT = YTO ? [(XS ? XSO) * sin q * k * MX / IncSX] + [(YS ? YSO) * cos q * k * MY / IncSY]
= 5836723.000 - 2556.192 + 2889.092
= 5837055.90 m.
Calculation of bin grid coordinates for this point, map grid location 464 855.62mE, 5 837 055.90mN:
Bin grid I = XS = {[( XT ? XTO) * cos q ? (YT ? YTO) * sin q ] * [IncSX / (k * MX)]} + XSO
= 300 bins,
Bin grid J = YS = {[(XT ? XTO) * sin q + (YT ? YTO) * cos q] * [IncSY / (k * MY)]} + YSO
= 247 bins","Use only when the bin grid I-axis is rotated 90? clockwise from the J-axis when viewed from above the plane containing the two axes (CS code 1033). If the I-axis is rotated 90? counter-clockwise from the J-axis, method 1049 should be used.","UKOOA Data Exchange Format P6/98, Definition of 3D Seismic Binning Grids, revision 3, May 2000, and OGP P6/11 seismic bin grid data exchange format user guide, July 2012.","OGP",7/28/2014 0:00:00,"2011.106 2014.044",0
9801,"Lambert Conic Conformal (1SP)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
To derive the projected Easting and Northing coordinates of a point with geographical coordinates (lat,lon) the formulas for the one standard parallel case are:
E = FE + r sin(theta)
N = FN + r0 - r cos(theta)
where
n = sin lat0
r = a F t^n k0 for r0, and r
m = cos(lat)/(1 - e^2 sin^2(lat))^0.5 for m0, lat0, and m2, lat2 where lat1 and lat2 are the latitudes of the standard parallels.
t = tan(pi/4 - lat/2)/[(1 - e sin(lat))/(1 + e sin(lat))]^(e/2) for t0 and t using lat0 and lat respectively.
F = m0/(n t1^n)
theta = n(lon - lon0)
The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:
lat = pi/2 - 2arctan{t'[(1 - esin(lat))/(1 + esin(lat))]^(e/2)}
lon = theta'/n +lon0
where
theta' = atan2[(E - FE),{r0 -(N - FN)}]
(see implementation notes in GN7-2 preface for atan2 convention)
r' = +/-[(E - FE)^2 + {r0 - (N - FN)}^2]^0.5 taking the sign of n
t' = (r'/(a k0 F))^(1/n)
and n, F, and rF are derived as for the forward calculation.","For Projected Coordinate System JAD69 / Jamaica National Grid
Parameters:
Ellipsoid: Clarke 1866, a = 6378206.400 m., 1/f = 294.97870
then e = 0.08227185 and e^2 = 0.00676866
Latitude Natural Origin 18?00'00""N = 0.31415927 rad
Longitude Natural Origin 77?00'00""W = -1.34390352 rad
Scale factor at origin 1.000000
False Eastings FE 250000.00 m
False Northings FN 150000.00 m
Forward calculation for:
Latitude: 17?55'55.80""N = 0.31297535 rad
Longitude: 76?56'37.26""W = -1.34292061 rad
first gives
m0 = 0.95136402 t0 = 0.72806411
F = 3.39591092 n = 0.30901699
r = 19643955.26 r0 = 19636447.86
theta = 0.00030374 t = 0.728965259
Then Easting E = 255966.58 m
Northing N = 142493.51 m
Reverse calculation for the same easting and northing first gives
theta' = 0.000303736
t' = 0.728965259
m0 = 0.95136402
r' = 19643955.26
Then Latitude = 17?55'55.800""N
Longitude = 76?56'37.260""W",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"2001.080 2017.018 2017.024",0
9802,"Lambert Conic Conformal (2SP)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
To derive the projected Easting and Northing coordinates of a point with geographical coordinates (lat,lon) the formulas for the one standard parallel case are:
E = EF + r sin(theta)
N = NF + rF - r cos(theta)
where
m = cos(lat)/(1 - e^2 sin^2(lat))^0.5 for m1, lat1, and m2, lat2 where lat1 and lat2 are the latitudes of the two standard parallels.
t = tan(pi/4 - lat/2)/[(1 - e sin(lat))/(1 + e sin(lat))]^(e/2) for t1, t2, tF and t using lat1, lat2, latF and lat respectively.
n = (loge(m1) - loge(m2))/(loge(t1) - loge(t2))
F = m1/(n t1^n)
r = a F t^n for rF and r, where rF is the radius of the parallel of latitude of the false origin.
theta = n(lon - lon0)
The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:
lat = pi/2 - 2arctan{t'[(1 - esin(lat))/(1 + esin(lat))]^(e/2)}
lon = theta'/n +lon0
where
r' = +/-[(E - EF)^2 + {rF - (N - NF)}^2]^0.5 , taking the sign of n
t' = (r'/(aF))^(1/n)
theta' = atan2 [(E- EF),(rF - (N- NF))]
(see implementation notes in GN7-2 preface for atan2 convention)
and n, F, and rF are derived as for the forward calculation.
Note that the formula for lat requires iteration. First calculate t' and then a trial value for lat using
lat = p/2-2atan(t'). Then use the full equation for lat substituting the trial value into the right hand side of the equation. Thus derive a new value for lat. Iterate the process until lat does not change significantly. The solution should quickly converge, in 3 or 4 iterations.","For Projected Coordinate System NAD27 / Texas South Central
Parameters:
Ellipsoid Clarke 1866, a = 6378206.400 metres = 20925832.16 US survey feet
1/f = 294.97870
then e = 0.08227185 and e^2 = 0.00676866
First Standard Parallel 28?23'00""N = 0.49538262 rad
Second Standard Parallel 30?17'00""N = 0.52854388 rad
Latitude False Origin 27?50'00""N = 0.48578331 rad
Longitude False Origin 99?00'00""W = -1.72787596 rad
Easting at false origin 2000000.00 US survey feet
Northing at false origin 0.00 US survey feet
Forward calculation for:
Latitude 28?30'00.00""N = 0.49741884 rad
Longitude 96?00'00.00""W = -1.67551608 rad
first gives :
m1 = 0.88046050 m2 = 0.86428642
t = 0.59686306 tF = 0.60475101
t1 = 0.59823957 t2 = 0.57602212
n = 0.48991263 F = 2.31154807
r = 37565039.86 rF = 37807441.20
theta = 0.02565177
Then Easting E = 2963503.91 US survey feet
Northing N = 254759.80 US survey feet
Reverse calculation for same easting and northing first gives:
theta' = 0.025651765 r' = 37565039.86
t' = 0.59686306
Then Latitude = 28?30'00.000""N
Longitude = 96?00'00.000""W",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"1999.280 2001.080 2017.018 2017.024",0
9803,"Lambert Conic Conformal (2SP Belgium)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
For the Lambert Conic Conformal (2 SP Belgium), the formulas for the regular two standard parallel case (coordinate operation method code 9802) are used except for easting, northing in the forward formula and lon in the rverse formula.
To derive the projected Easting and Northing coordinates of a point with geographical coordinates (lat,lon) the formulas for the one standard parallel case are:
Easting, E = EF + r sin (theta - alpha)
Northing, N = NF + rF - r cos (theta - alpha)
where
m = cos(lat)/(1 - e^2 sin^2(lat))^0.5 for m1, lat1, and m2, lat2 where lat1 and lat2 are the latitudes of the two standard parallels.
t = tan(pi/4 - lat/2)/[(1 - e sin(lat))/(1 + e sin(lat))]^(e/2) for t1, t2, tF and t using lat1, lat2, latF and lat respectively.
n = (loge(m1) - loge(m2))/(loge(t1) - loge(t2))
F = m1/(n t1^n)
r = a F t^n for rF and r, where rF is the radius of the parallel of latitude of the false origin.
theta = n(lon - lon0)
alpha = 29.2985 seconds.
The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:
lat = pi/2 - 2arctan{t'[(1 - esin(lat))/(1 + esin(lat))]^(e/2)}
lon = ((theta' + alpha)/n) +lon0
where
r' = +/-[(E - EF)^2 + {rF - (N - NF)}^2]^0.5 , taking the sign of n
t' = (r'/(aF))^(1/n)
theta' = atan2 [(E- EF),(rF - (N- NF))]
(see implementation notes in GN7-2 preface for atan2 convention)
alpha = 29.2985 seconds
and n, F, and rF are derived as for the forward calculation.
Note that the formula for lat requires iteration. First calculate t' and then a trial value for lat using
lat = p/2-2atan (t'). Then use the full equation for lat substituting the trial value into the right hand side of the equation. Thus derive a new value for lat. Iterate the process until lat does not change significantly. The solution should quickly converge, in 3 or 4 iterations.","For Projected Coordinate System Belge 1972 / Belge Lambert 72
Parameters:
Ellipsoid International 1924, a = 6378388 metres
1/f = 297
then e = 0.08199189 and e^2 = 0.006722670
First Standard Parallel 49?50'00""N = 0.86975574 rad
Second Standard Parallel 51?10'00""N = 0.89302680 rad
Latitude False Origin 90?00'00""N = 1.57079633 rad
Longitude False Origin 4?21'24.983""E = 0.07604294 rad
Easting at false origin EF 150000.01 metres
Northing at false origin NF 5400088.44 metres
Forward calculation for:
Latitude 50?40'46.461""N = 0.88452540 rad
Longitude 5?48'26.533""E = 0.10135773 rad
first gives :
m1 = 0.64628304 m2 = 0.62834001
t = 0.59686306 tF = 0.00000000
t1 = 0.36750382 t2 = 0.35433583
n = 0.77164219 F = 1.81329763
r = 37565039.86 rF = 0.00
alpha = 0.00014204 theta = 0.01953396
Then Easting E = 251763.20 metres
Northing N = 153034.13 metres
Reverse calculation for same easting and northing first gives:
theta' = 0.01939192 r' = 548041.03
t' = 0.35913403
Then Latitude = 50?40'46.461""N
Longitude = 5?48'26.533""E","In 2000 this modification was replaced through use of the regular Lambert Conic Conformal (2SP) method [9802] with appropriately modified parameter values.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"1999.280 2017.018 2017.024",0
9804,"Mercator (variant A)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The formulas to derive projected Easting and Northing coordinates are:
E = FE + a*ko(lon - lonO)
N = FN + a*ko* ln{tan(pi/4 + lat/2)[(1 - esin(lat))/(1 + esin(lat))]^e/2} where symbols are as listed above and logarithms are natural.
The reverse formulas to derive latitude and longitude from E and N values are:
lat = chi + (esq/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2chi)
+ (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4chi)
+ (7e^6/120 + 81e^8/1120) sin(6chi) + (4279e^8/161280) sin(8chi)
where chi = pi/2 - 2 arctan t
t = B^((FN-N)/(a*ko))
B = base of the natural logarithm, 2.7182818...
and for the 2 SP Case, ko is calculated as for the forward transformation above.
lon = ((E - FE)/(a*ko)) + lonO","For Projected Coordinate System Makassar / NEIEZ
Parameters:
Ellipsoid Bessel 1841 a = 6377397.155 m 1/f = 299.15281
then e = 0.08169683
Latitude of natural origin = 00?00'00""N = 0.0000000 rad
Longitude of natural origin = 110?00'00""E = 1.91986218 rad
Scale factor at natural origin ko = 0.997
False Eastings FE = 3900000.00 m
False Northings FN = 900000.00 m
Forward calculation for:
Latitude = 3?00'00.00""S = -0.05235988 rad
Longitude = 120?00'00.00""E = 2.09439510 rad
gives
Easting E = 5009726.58 m
Northing N = 569150.82 m
Reverse calculation for same easting and northing first gives :
t = 1.0534121
chi = -0.0520110
Latitude = 3?00'00.000""S
Longitude = 120?00'00.000""E","Note that in these formulas the parameter latitude of natural origin (latO) is not used. However for this Mercator (variant A) method the EPSG dataset includes this parameter, which must have a value of zero, for completeness in CRS labelling.","EPSG guidance note #7-2, http://www.epsg.org","OGP",11/2/2010 0:00:00,"2001.080 2004.430 2008.000 2010.058",0
9805,"Mercator (variant B)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The formulas to derive projected Easting and Northing coordinates are:
ko = cos(latSP1)/(1 - e^2*sin^2(latSP1))^0.5
where latSP1 is the absolute value of the first standard parallel (i.e. positive).
E = FE + a*ko(lon - lonO)
N = FN + a*ko* ln{tan(pi/4 + lat/2)[(1 - esin(lat))/(1 + esin(lat))]^e/2} where logarithms are natural.
The reverse formulas to derive latitude and longitude from E and N values are:
lat = chi + (esq/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2chi)
+ (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4chi)
+ (7e^6/120 + 81e^8/1120) sin(6chi) + (4279e^8/161280) sin(8chi)
where chi = pi/2 - 2 arctan t
t = B^((FN-N)/(a*ko))
B = base of the natural logarithm, 2.7182818...
and ko is calculated as for the forward transformation above.
lon = ((E - FE)/(a*ko)) + lonO","For Projected Coordinate System Pulkovo 1942 / Caspian Sea Mercator
Parameters:
Ellipsoid Krassowski 1940 a = 6378245.00m 1/f = 298.300
then e = 0.08181333 and e^2 = 0.00669342
Latitude of first SP = 42?00'00""N = 0.73303829 rad
Longitude of natural origin = 51?00'00""E = 0.89011792 rad
False Eastings FE = 0.00 m
False Northings (at equator) FN = 0.00 m
Forward calculation for:
Latitude = 53?00'00.00""N = 0.9250245 rad
Longitude = 53?00'00.00""E = 0.9250245 rad
gives
ko = 0.744260894
Easting E = 165704.29 m
Northing N = 5171848.07 m
Reverse calculation for same easting and northing first gives :
t = 0.336391288
chi = 0.921795958
Latitude = 53?00'00.000""N
Longitude = 53?00'00.000""E","Used for most nautical charts.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",5/3/2019 0:00:00,"2004.320 2004.430 2010.058 2010.098 2015.021 2019.018",0
9806,"Cassini-Soldner",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The formulas to derive projected Easting and Northing coordinates are:
Easting E = FE + nu[A - TA^3/6 -(8 - T + 8C)TA^5/120]
Northing N = FN + M - M0 + nu*tan(lat)*[A^2/2 + (5 - T + 6C)A^4/24]
where A = (lon - lon0)cos(lat)
T = tan^2(lat)
C = e2 cos2*/(1 - e2) nu = a /(1 - esq*sin^2(lat))^0.5
and M, the distance along the meridian from equator to latitude lat, is given by
M = a[(1 - e^2/4 - 3e^4/64 - 5e^6/256 -....)*lat - (3e^2/8 + 3e^4/32 + 45e^6/1024 +....)sin(2*lat) + (15e^4/256 + 45e^6/1024 +.....)sin(4*lat) - (35e^6/3072 + ....)sin(6*lat) + .....]
with lat in radians.
M0 is the value of M calculated for the latitude of the chosen origin. This may not necessarily be chosen as the equator.
To compute latitude and longitude from Easting and Northing the reverse formulas are:
lat = lat1 - (nu1tan(lat1)/rho1)[D2/2 - (1 + 3*T1)D^4/24]
lon = lon0 + [D - T1*D^3/3 + (1 + 3*T1)T1*D^5/15]/cos(lat1)
where lat1 is the latitude of the point on the central meridian which has the same Northing as the point whose coordinates are sought, and is found from:
lat1 = mu1 + (3*e1/2 - 27*e1^3/32 +.....)sin(2*mu1) + (21*e1^2/16 - 55*e1^4/32 + ....)sin(4*mu1)+ (151*e1^3/96 +.....)sin(6*mu1) + (1097*e1^4/512 - ....)sin(8*mu1) + ......
where
e1 = [1- (1 - esq)^0.5]/[1 + (1 - esq)^0.5]
mu1 = M1/[a(1 - esq/4 - 3e^4/64 - 5e^6/256 - ....)]
M1 = M0 + (N - FN)
T1 = tan^2(lat1)
D = (E - FE)/nu1","For Projected Coordinate System Trinidad 1903 / Trinidad Grid
Parameters:
Ellipsoid Clarke 1858 a = 20926348 ft = 31706587.88 links
b = 20855233 ft
then 1/f = 294.97870 and e^2 = 0.00676866
Latitude Natural Origin 10?26'30""N = 0.182241463 rad
Longitude Natural Origin 61?20'00""W = -1.07046861 rad
False Eastings FE 430000.00 links
False Northings FN 325000.00 links
Forward calculation for:
Latitude 10?00'00.00"" N = 0.17453293 rad
Longitude 62?00'00.00""W = -1.08210414 rad
A = -0.01145876 C = 0.00662550
T = 0.03109120 M = 5496860.24 nu = 31709831.92 M0 = 5739691.12
Then Easting E = 66644.94 links
Northing N = 82536.22 links
Reverse calculation for same easting and northing first gives :
e1 = 0.00170207 D = -0.01145875
T1 = 0.03109544 M1 = 5497227.34
nu1 = 31709832.34 mu1 = 0.17367306
phi1 = 0.17454458 rho1 = 31501122.40
Then Latitude = 10?00'00.000""N
Longitude = 62?00'00.000""W",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9807,"Transverse Mercator",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
For the calculation of easting and northing from latitude and longitude, first calculate constants for the projection:
n = f / (2-f)
B = [a/(1+n)] (1 + n^2/4 + n^4/64)
h1 = n/2 ? (2/3)n^2 + (5/16)n^3 + (41/180)n^4
h2 = (13/48)n^2 ? (3/5)n^3 + (557/1440)n^4
h3 = (61/240)n^3 ? (103/140)n^4
h4 = (49561/161280)n^4
Then the meridional arc distance from equator to the projection origin (Mo) is computed from:
If LatO = 0 then Mo = 0
else if LatO = 90?N = p/2 radians
Mo = B (p/2)
else if LatO = 90?S = -p/2 radians
Mo = B (-p/2)
else
Qo = asinh(tan LatO) ? [e atanh(e sin LatO)]
?o = atan(sinh Qo)
?O0 = asin (sin ?o)
Note: The previous two steps are taken from the generic calculation flow given below for latitude Lat, but here for LatO may be simplified to ?O0 = ?o = atan(sinh Qo).
?O1 = h1 sin(2?Oo)
?O2 = h2 sin(4?Oo)
?O3 = h3 sin(6?Oo)
?O4 = h4 sin(8?Oo)
?O = ?O0+ ?O1+ ?O2+ ?O3+ ?O4
Mo = B ?O
end
Note: if the projection grid origin is very close to the pole (within 2"" or 50m), the tangent function in the equation for Qo is unstable and may fail. Mo may instead be calculated as:
Mo = a[(1 ? e^2/4 ? 3e^4/64 ? 5e^6/256 ?....)LatO ? (3e^2/8 + 3e^4/32 + 45e^6/1024+....)sin2LatO
+ (15e^4/256 + 45e^6/1024 +.....)sin4LatO ? (35e^6/3072 + ....)sin6LatO + .....]
with LatO in radians.
Then
Q = asinh(tan Lat) ? [e atanh(e sin Lat)]
? = atan(sinh Q)
?0 = atanh [cos ? sin(Lon ? LonO)]
?0 = asin (sin ? cosh ?0)
?1 = h1 sin(2?0) cosh(2?0)
?1 = h1 cos(2?0) sinh(2?0)
?2 = h2 sin(4?0) cosh(4?0)
?2 = h2 cos(4?0) sinh(4?0)
?3 = h3 sin(6?0) cosh(6?0)
?3 = h3 cos(6?0) sinh(6?0)
?4 = h4 sin(8?0) cosh(8?0)
?4 = h4 cos(8?0) sinh(8?0)
? = ?0 + ?1 + ?2 + ?3 + ?4
? = ?0 + ?1 + ?2 + ?3 + ?4
and
Easting, E = FE + ko B ?
Northing, N = FN + ko (B ? ? Mo)
For the reverse formulas to convert Easting and Northing projected coordinates to latitude and longitude first calculate constants of the projection where n is as for the forward conversion, as are B and Mo:
h1' = n/2 ? (2/3)n^2 + (37/96)n^3 ? (1/360)n^4
h2' = (1/48)n^2 + (1/15)n^3 ? (437/1440)n^4
h3' = (17/480)n^3 ? (37/840)n^4
h4' = (4397/161280)n^4
Then
?' = (E ? FE) / (B ko)
?' = [(N ? FN) + ko Mo] / (B ko)
?1' = h1' sin(2?') cosh(2?')
?1' = h1' cos(2?') sinh(2?')
?2' = h2' sin(4?') cosh(4?')
?2' = h2' cos(4?') sinh(4?')
?3' = h3' sin(6?') cosh(6?')
?3' = h3' cos(6?') sinh(6?')
?4' = h4' sin(8?') cosh(8?')
?4' = h4' cos(8?') sinh(8?')
?0' = ?' ? (?1' + ?2' + ?3' + ?4')
?0' = ?' ? (?1' + ?2' + ?3' + ?4')
?' = asin(sin ?0' / cosh ?0')
Q' = asinh(tan ?')
Q"" = Q' + [e atanh(e tanh Q')] = Q' + [e atanh(e tanh Q"")] which should be iterated until the change in Q"" is insignificant. Then
Lat = atan(sinh Q"")
Lon = LonO + asin(tanh(?0') / cos ?')","For Projected Coordinate System OSGB 1936 / British National Grid
Parameters:
Ellipsoid Airy 1830 a = 6377563.396 m 1/f = 299.32496
then e'^2 = 0.00671534 and e^2 = 0.00667054
Latitude of natural origin (LatO) = 49?00'00""N = 0.85521133 rad
Longitude of natural origin (LonO) = 2?00'00""W = -0.03490659 rad
Scale factor (ko) = 0.9996013
False Eastings (FE) = 400000.00 m
False Northings (FN) = -100000.00 m
Forward calculation for:
Latitude = 50?30'00.00""N = 0.88139127 rad
Longitude = 00?30'00.00""E = 0.00872665 rad
Constants of the projection:
n = 0.00167322
B = 6366914.609
h1 = 0.0008347452
h2 = 0.0000007554
h3 = 1.18487E-09
h4 = 2.40864E-12
QO = 0.9787671618
?O0 = 0.8518980373
?O1 = 0.0008273732
?O2 = -0.0000001986
?O3 = -1.0918E-09
?O4 = 1.2218E-12
Mo = 5429228.602
Q = 1.0191767215
? = 0.8781064142
?0 = 0.0278629616
?0 = 0.8785743280
?1 = -0.0000086229
?1 = 0.0008215669
?2 = -0.0000000786
?2 = -0.0000002768
?3 = 1.05551E-10
?3 = -1.01855E-09
?4 = 3.97791E-13
?4 = 1.67447E-12
? = 0.0278542603
? = 0.8793956171
Then
Easting E = 577274.99 metres
Northing N = 69740.50 metres
Reverse calculation for same easting and northing first gives:
h1' = 0.0008347455
h2' = 0.0000000586
h3' = 1.65563E-10
h4' = 2.13692E-13
Then
?' = 0.87939562
?' = 0.0278542603
?1' = 0.0008213109
?1' = -0.0000086953
?2' = -0.0000000217
?2' = -0.0000000061
?3' = -1.41881E-10
?3' = 1.486E-11
?4' = 1.49609E-13
?4' = 3.50657E-14
?0' = 0.8785743280
?0' = 0.0278629616
?' = 0.8781064142
Q' = 1.0191767215
Q"" 1st iteration = 1.0243166838
Q"" 2nd iteration = 1.0243306667
Q"" 3rd iteration = 1.0243307046
Q"" 4th iteration = 1.0243307047
Then
Latitude (Lat) = 50?30'00.000""N
Longitude (Lon) = 00?30'00.000""E",,"EPSG guidance note #7-2, http://www.epsg.org","OGP",2/13/2012 0:00:00,"2004.680 2007.029 2011.098",0
9808,"Transverse Mercator (South Orientated)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
For the mapping of southern Africa a south oriented Transverse Mercator projection is used. Here the coordinate axes are called Westings and Southings and increment to the West and South from the origin respectively. The standard Transverse Mercator formulas (coordinate operation method code 9807) need to be modified to cope with this arrangement with
Westing, W = FE - k0 nu[A + (1 - T + C)A^3/6 + (5 - 18*T + T^2 + 72*C - 58*e'^2)A^5/120]
Southing, S = FN - k0{M - M0 + nu*tan(lat)*[A^2/2 + (5 - T + 9*C + 4*C^2)A^4/24 + (61 - 58*T + T^2 + 600*C - 330*e'^2)A^6/720]}
In these formulas the terms FE and FN retain their definition, i.e. in the Transverse Mercator (South Orientated) method they increase the Westing and Southing value at the natural origin. In this method they are effectively false westing (FW) and false southing (FS) respectively.
For the reverse formulas, those for the standard Transverse Mercator above apply, with the exception that:
M1 = M0 - (S - FN)/k0
and D = -(W - FE)/(nu1*k0), with nu1 = nu for lat1","See Transverse Mercator, code 9807, for general methodology.",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2002.510 2017.018",0
9809,"Oblique Stereographic",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Given the geodetic origin of the projection at the tangent point (lat0, lon0), the parameters defining the conformal sphere are:
R= sqrt( rho0 * nu0)
n= {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5
c= [(n+sin(lat0)) (1-sin(chi0))]/[(n-sin(lat0)) (1+sin(chi0))]
where:
sin(chi0) = (w1-1)/(w1+1)
w1 = (S1.(S2)^e)^n
S1 = (1+sin(lat0))/(1-sin(lat0))
S2 = (1-e sin(lat0))/(1+e sin(lat0))
The conformal latitude and longitude (chi0,lambda0) of the origin are then computed from :
chi0 = asin[(w2-1)/(w2+1)]
where S1 and S2 are as above and w2 = c (S1(S2)^e)^n
lambda0 = lon0
For any point with geodetic coordinates (lat, lon) the equivalent conformal latitude and longitude (chi, lambda) are computed from
lambda = n(lon-lambda0) + lambda0
chi = asin[(w-1)/(w+1)]
where w = c (Sa (Sb)^e)^n
Sa = (1+sin(lat))/(1-sin(lat))
Sb = (1-e.sin(lat))/(1+e.sin(lat))
Then B = [1+sin(chi) sin(chi0) + cos(chi) cos(chi0) cos(lambda-lambda0)]
N = FN + 2 R k0 [sin(chi) cos(chi0) - cos(chi) sin(chi0) cos(lambda-lambda0)] / B
E = FE + 2 R k0 cos(chi) sin(lambda-lambda0) / B
The reverse formulae to compute the geodetic coordinates from the grid coordinates involves computing the conformal values, then the isometric latitude and finally the geodetic values.
The parameters of the conformal sphere and conformal latitude and longitude at the origin are computed as above. Then for any point with Stereographic grid coordinates (E,N) :
chi = chi0 + 2 atan[{(N-FN)-(E-FE) tan (j/2)} / (2 R k0)]
lambda = j + 2 i + lambda0
where g = 2 R k0 tan(pi/4 - chi0/2)
h = 4 R k0 tan(chi0) + g
i = atan2[(E-FE) , {h+(N-FN)}]
j = atan2[(E-FE) , (g-(N-FN)] - i
(see GN7-2 implementation notes in preface for atan2 convention)
Geodetic longitude lon = (lambda-lambda0 ) / n + lambda0
Isometric latitude psi = 0.5 ln [(1+ sin(chi)) / { c (1- sin(chi))}] / n
First approximation lat1 = 2 atan(e^psi) - pi/2 where e=base of natural logarithms.
psii = isometric latitude at lati
where psii= ln[{tan(lati/2 + pi/4} {(1-e sin(lati))/(1+e sin(lati))}^(e/2)]
Then iterate lat(i+1) = lati - ( psii - psi ) cos(lati) (1 -e^2 sin^2(lati)) / (1 - e^2)
until the change in lat is sufficiently small.
For Oblique Stereographic projections centred on points in the southern hemisphere, the signs of E, N, lon0, lon, must be reversed to be used in the equations and lat will be negative anyway as a southerly latitude.
An alternative approach is given by Snyder, where, instead of defining a single conformal sphere at the origin point, the conformal latitude at each point on the ellipsoid is computed. The conformal longitude is then always equivalent to the geodetic longitude. This approach is a valid alternative to the above, but gives slightly different results away from the origin point. It is therefore considered by EPSG to be a different coordinate operation method to that described above.","For Projected Coordinate System RD / Netherlands New
Parameters:
Ellipsoid Bessel 1841 a = 6377397.155 m 1/f = 299.15281
then e = 0.08169683
Latitude Natural Origin 52?09'22.178""N = 0.910296727 rad
Longitude Natural Origin 5?23'15.500""E = 0.094032038 rad
Scale factor k0 0.9999079
False Eastings FE 155000.00 m
False Northings FN 463000.00 m
Forward calculation for:
Latitude 53?N = 0.925024504 rad
Longitude 6?E = 0.104719755 rad
first gives the conformal sphere constants:
rho0 = 6374588.71 nu0 = 6390710.613
R = 6382644.571 n = 1.000475857 c = 1.007576465
where S1 = 8.509582274 S2 = 0.878790173 w1 = 8.428769183
sin chi0 = 0.787883237
w = 8.492629457 chi0 = 0.909684757 D0 = d0
for the point chi = 0.924394997 D = 0.104724841
hence B = 1.999870665 N = 557057.739 E = 196105.283
reverse calculation for the same Easting and Northing first gives:
g = 4379954.188 h = 37197327.96 i = 0.001102255 j = 0.008488122
then D = 0.10472467 Longitude = 0.104719584 rad = 6 deg E
chi = 0.924394767 psi = 1.089495123
phi1 = 0.921804948 psi1 = 1.084170164
phi2 = 0.925031162 psi2 = 1.089506925
phi3 = 0.925024504 psi3 = 1.089495505
phi4 = 0.925024504
Then Latitude = 53?00'00.000""N
Longitude = 6?00'00.000""E","This is not the same as the projection method of the same name in USGS Professional Paper no. 1395, ""Map Projections - A Working Manual"" by John P. Snyder.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"1999.811 2006.200 2017.018 2017.024",0
9810,"Polar Stereographic (variant A)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
For the forward conversion from latitude and longitude, for the south pole case
E = FE + rho * sin(lon ? lonO)
N = FN + rho * cos(lon ? lonO)
where
t = tan(pi/4 + lat/2) / {[(1 + e sin(lat)) / (1 ? e sin(lat))]^(e/2)}
rho = 2*a*ko*t / {[(1+e)^(1+e) (1?e)^(1?e)]^0.5}
For the north pole case,
rho and E are found as for the south pole case but
t = tan(pi/4 ? lat/2) * {[(1 + e sin(lat)) / (1 ? e sin(lat))]^(e/2)}
N = FN ? rho * cos(lon ? lonO)
For the reverse conversion from easting and northing to latitude and longitude,
lat = chi + (e^2/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2 chi)
+ (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4 chi)
+ (7e^6/120 + 81e^8/1120) sin(6 chi) + (4279e^8/161280) sin(8 chi)
where rho' = [(E-FE)^2 + (N ? FN)^2]^0.5
t' =rho' {[(1+e)^(1+e) * (1? e)^(1-e)]^0.5} / (2 a ko)
and for the south pole case
chi = 2 atan(t' ) ? pi/2
but for the north pole case
chi = pi/2 - 2 atan(t')
Then for for both north and south cases if E = FE, lon = lonO
else for the south pole case
lon = lonO + atan2[(E ? FE),(N ? FN)]
and for the north pole case
lon = lonO + atan2[(E ? FE),(FN ? N)] (see GN7-2 implementation notes in preface for atan2 convention)","For Projected Coordinate Reference System: WGS 84 / UPS North
Parameters:
Ellipsoid: WGS 84
a = 6378137.0 metre
1/f = 298.2572236
then e = 0.081819191
Latitude of natural origin (latO): 90?00'00.000""N =1.570796327 rad
Longitude of origin (longO): 0?00'00.000""E=0.0 rad
Scale factor at natural origin (ko): 0.994
False easting (FE) 2000000.00 metre
False northing (FN) 2000000.00 metre
Forward calculation for:
Latitude (lat) =73?N =1.274090354 rad
Longitude (lon) =44?E =0.767944871 rad
t = 0.150412808
rho = 1900814.564
whence
E = 3320416.75 m
N = 632668.43 m
Reverse calculation for the same Easting and Northing (3320416.75 E, 632668.43 N) first gives:
rho' = 1900814.566
t' = 0.150412808
chi = 1.2722090
Then
Latitude (lat) = 73?00'00.000""N
Longitude (lon) = 44?00'00.000""E","Latitude of natural origin must be either 90 degrees or -90 degrees (or equivalent in alternative angle unit).","EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"2003.220 2010.054 2017.024",0
9811,"New Zealand Map Grid",1,"See information source.","See information source.",,"New Zealand Department of Lands technical circular 1973/32","IOGP",6/13/2017 0:00:00,"2017.018",0
9812,"Hotine Oblique Mercator (variant A)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The following constants for the projection may be calculated :
B = {1 + [esq * cos^4(latc) / (1 - esq )]}^0.5
A = a * B * kc *(1 - esq )^0.5 / ( 1 - esq * sin^2(latc))
to = tan(pi/4 - latc/2) / ((1 - e*sin(latc)) / (1 + e*sin(latc)))^(e/2)
D = B (1 - esq)^0.5 / (cos(latc) * ( 1 - esq*sin^2(latc))^0.5)
if D < 1 to avoid problems with computation of F make D^2 = 1
F = D + (D^2 - 1)^0.5 * SIGN(latc)
H = F*(to)^B
G = (F - 1/F) / 2
gammao = asin(sin(alphac) / D)
lonO = lonc - (asin(G*tan(gammao))) / B
Forward case: To compute (E,N) from a given (lat,lon) :
t = tan(pi/4 - lat/2) / ((1 - e sin (lat)) / (1 + e sin (lat)))^(e/2)
Q = H / t^B
S = (Q - 1 / Q) / 2
T = (Q + 1 / Q) / 2
V = sin(B (lon - lonO))
U = (- V cos(gammao) + S sin(gammao)) / T
v = A ln((1 - U) / (1 + U)) / 2 B
u = A atan2((S cos(gammao) + V sin(gammao)) , cos(B (lon - lonO))) / B (see GN7-2 implementation notes in preface for atan2 convention)
The rectified skew co-ordinates are then derived from:
E = v cos(gammac) + u sin(gammac) + FE
N = u cos(gammac) - v sin(gammac) + FN
Reverse case: Compute (lat,lon) from a given (E,N) :
v? = (E - FE) cos(gammac) - (N - FN) sin(gammac)
u? = (N - FN) cos(gammac) + (E - FE) sin(gammac)
Q? = e^- (B v ?/ A) where e is the base of natural logarithms.
S' = (Q? - 1 / Q?) / 2
T? = (Q? + 1 / Q?) / 2
V? = sin (B u? / A)
U? = (V? cos(gammac) + S? sin(gammac)) / T?
t? = (H / ((1 + U?) / (1 - U?))^0.5)^(1 / B)
chi = pi / 2 - 2 atan(t?)
lat = chi + sin(2chi).( e^2 / 2 + 5*e^4 / 24 + e^6 / 12 + 13*e^8 / 360) + sin(4*chi).( 7*e^4 /48 + 29*e^6 / 240 + 811*e8 / 11520) + sin(6chi).( 7*e^6 / 120 + 81*e8 / 1120) + sin(8chi).(4279 e^8 / 161280)
lon = lonO - atan2 ((S? cos(gammao) - V? sin(gammao)) , cos(B*u? / A)) / B","For Projected Coordinate System Timbalai 1948 / R.S.O. Borneo (m)
Parameters:
Ellipsoid: Everest 1830 (1967 Definition)
a = 6377298.556 metres 1/f = 300.8017
then e = 0.081472981and e2 = 0.006637847
Latitude Projection Centre fc = 4?00'00""N = 0.069813170 rad
Longitude Projection Centre lc = 115?00'00""E = 2.007128640 rad
Azimuth of central line ac = 53?18'56.9537"" = 0.930536611 rad
Rectified to skew gc= 53?07'48.3685"" = 0.927295218 rad
Scale factor ko= 0.99984
False Eastings FE = 0.00 m
False Northings FN = 0.00 m
Forward calculation for:
Latitude lat = 5?23'14.1129""N = 0.094025313 rad
Longitude lon = 115?48'19.8196""E = 2.021187362 rad
B = 1.003303209 F = 1.072121256
A =6376278.686 H = 1.000002991
to = 0.932946976 go = 0.927295218
D = 1.002425787 lon0 = 1.914373469
D2 =1.004857458
uc =738096.09 vc =0.00
t = 0.910700729 Q = 1.098398182
S = 0.093990763 T = 1.004407419
V = 0.106961709 U = 0.010967247
v = -69702.787 u = 901334.257
Then Easting E = 679245.73 m
Northing N = 596562.78 m
Reverse calculations for same easting and northing first gives :
v? = -69702.787 u? =901334.257
Q? = 1.011028053
S? = 0.010967907 T? = 1.000060146
V? = 0.141349378 U? = 0.093578324
t? = 0.910700729 c = 0.093404829
Then Latitude = 5?23'14.113""N
Longitude = 115?48'19.820""E",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"1997.620 1999.811 2004.430 2004.600 2007.044 2010.058 2017.024",0
9813,"Laborde Oblique Mercator",1,"Note : these formulas have been transcribed from IGN Document NT/G 74. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
From the defining parameters the following constants for the map projection may be calculated:
B = {1+[e^2 cos^4(phiC)]/(1? e^2)}^0.5
phiS = asin[sin(phiC) / B]
R = a kC {(1?e^2)^0.5 / [1?e^2 sin^2(phiC)]}
C = ln[tan(pi/4+phiS /2)] ? B. ln{tan(pi/4+phiC /2) ([1 ? e sin(phiC)]/[1+e sin(phiC)])^(e/2)}
Forward case: To compute (E,N) from a given (lat,lon)
L = B.(lon?lonC)
q = C + B . ln{tan(pi/4+lat/2) ([1?e sin(lat)] / [1+e sin(lat)])^(e/2)}
P = 2.atan[e ^q] ? pi/2 where e is the base of natural logarithms
U = cos(P).cos(L).cos(phiS) + sin(P).sin(phiS)
V = cos(P).cos(L).sin(phiS) - sin(P).cos(phiS)
W = cos(P).sin(L)
d = (U^2+V^2)^0.5
if d <> 0 then L' = 2.atan(V/(U+d)) and P' = atan(W/d)
if d = 0 then L' = 0 and P' = sign(W).pi/2
H = ?L' + i.ln(tan(pi/4+P'/2)) where i^2 = ?1
G = (1 ? cos(2.alphaC) + i.sin(2.alphaC))/12
E = FE + R . IMAGINARY(H + G.H^3)
N = FN + R . REAL(H + G.H^3)
Reverse case: To compute (lat, lon) from a given (E,N):
G = (1?cos(2.alphaC) + i.sin(2.alphaC))/12 where i^2 = ?1
To solve for Latitude and Longitude, a re-iterative solution is required, where the first two elements are
H0 = (N?FN)/R + i.(E?FE)/R ie k = 0
H1 = H0/(H0 + G.H0^3), i.e. k = 1,
and in subsequent reiterations, k increments by 1
Hk+1 = (H0+2.G.Hk^3)/(3.G.Hk^2+1)
Re-iterate until ABSOLUTE(REAL([H0-Hk-G.Hk^3)])) < 1E-11
L' = ?1.REAL(Hk)
P' = 2.atan{ e ^[IMAGINARY(Hk)]} ? pi/2 where e is the base of natural logarithms.
U' = cos(P').cos(L').cos(phiS) + cos(P').sin(L').sin(phiS)
V' = sin(P')
W' = cos(P').cos(L').sin(phiS) ? cos(P').sin(L').cos(phiS)
d = (U'^2+ V'^2)^0.5
if d <> 0 then L = 2 atan[V'/( U'+d)] and P = atan(W'/d)
if d = 0 then L = 0 and P = SIGN(W') . pi/2
lon = lonC + (L/B)
q' = {ln[tan(pi/4+P/2)] ? C}/B
The final solution for latitude requires a second re-iterative process, where the first element is
lat'(0) = 2.atan(e ^q') ? pi/2 where e is the base of natural logarithms.
And the subsequent elements are
lat'(k) = 2.atan{({1+e.sin[lat(k-1)]} / {1?e.sin[lat(k-1)]})^(e/2).e ^q'} ? pi/2 for K =1 ?
Iterate until ABSOLUTE(lat(k)-lat(k-1)) < 1E-11
lat = lat(k)","See information source.",,"""La nouvelle projection du Service Geographique de Madagascar""; J. Laborde; 1928. Also IGN Paris technical note NT/G 74.","OGP",11/2/2010 0:00:00,"1997.613 2006.960 2007.040 2010.058",0
9814,"Swiss Oblique Cylindrical",1,"See information source.","See information source.","Can be accommodated by Oblique Mercator method (code 9815), for which this method is an approximation (see BfL document swissprojectionen.pdf at www.swisstopo.com).","""Die projecktionen der Schweizerischen Plan und Kartenwerke""; J Bollinger; 1967","OGP",11/23/2008 0:00:00,"1997.612 2008.106",0
9815,"Hotine Oblique Mercator (variant B)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The following constants for the projection may be calculated :
B = {1 + [e^2 * cos^4(latc) / (1 - e^2 )]}^0.5
A = a * B * kc *(1 - e^2 )^0.5 / ( 1 - e^2 * sin^2(latc))
to = tan(pi/4 - latc/2) / ((1 - e*sin(latc)) / (1 + e*sin(latc)))^(e/2)
D = B (1 - e^2)^0.5 / (cos(latc) * ( 1 - e^2*sin^2(latc))^0.5)
if D < 1 to avoid problems with computation of F make D^2 = 1
F = D + (D^2 - 1)^0.5 * SIGN(latc)
H = F*(to)^B
G = (F - 1/F) / 2
gammao = asin(sin(alphac) / D)
lonO = lonc - (asin(G*tan(gamma0))) / B
vc = 0
In general: uc = (A / B) atan2((Dsq - 1)^0.5 , cos (alphac) ) * SIGN(latc) (see Gn7-2 implementation notes in preface for atan2 convention)
but for the special cases where alphac = 90 degrees (e.g. Hungary, Switzerland) then
uc = A*(lonc - lonO)
Forward case: To compute (E,N) from a given (lat,lon) :
t = tan(pi/4 - lat/2) / ((1 - e sin (lat)) / (1 + e sin (lat)))^(e/2)
Q = H / t^B
S = (Q - 1 / Q) / 2
T = (Q + 1 / Q) / 2
V = sin(B (lon - lonO))
U = (- V cos(gammao) + S sin(gammao)) / T
v = A ln((1 - U) / (1 + U)) / 2 B
In general:
u = (A atan2((S cos(gammao) + V sin(gammao)) , cos(B (lon - lonO))) / B) - (ABS(uc) . SIGN(latc))
but when alphac = pi/2 rad
if lon = lonc, u = 0
else u = (A atan((S cos(gammao) + V sin(gammao)) / cos(B (lon - lonO))) / B) - (ABS(uc) . SIGN(latc) . SIGN(lonc ? lon))
The rectified skew co-ordinates are then derived from:
E = v cos(gammac) + u sin(gammac) + Ec
N = u cos(gammac) - v sin(gammac) + Nc
Reverse case: Compute (lat,lon) from a given (E,N) :
v? = (E - Ec) cos(gammac) - (N - Nc) sin(gammac)
u? = (N - Nc) cos(gammac) + (E - Ec) sin(gammac) + (ABS(uc) . SIGN(latc))
Q? = e- (B v ?/ A) where e is the base of natural logarithms.
S' = (Q? - 1 / Q?) / 2
T? = (Q? + 1 / Q?) / 2
V? = sin (B u? / A)
U? = (V? cos(gammac) + S? sin(gammac)) / T?
t? = (H / ((1 + U?) / (1 - U?))^0.5)^(1 / B)
chi = pi / 2 - 2 atan(t?)
lat = chi + sin(2chi).( e^2 / 2 + 5*e^4 / 24 + e^6 / 12 + 13*e^8 / 360) + sin(4*chi).( 7*e^4 /48 + 29*e^6 / 240 + 811*e8 / 11520) + sin(6chi).( 7*e^6 / 120 + 81*e8 / 1120) + sin(8chi).(4279 e^8 / 161280)
lon= lonO - atan2 ((S? cos(gammao) , V? sin(gammao)) / cos(B*u? / A)) / B","For Projected Coordinate System Timbalai 1948 / R.S.O. Borneo (m)
Parameters:
Ellipsoid: Everest 1830 (1967 Definition)
a = 6377298.556 metres 1/f = 300.8017
then e = 0.081472981and e^2 = 0.006637847
Latitude Projection Centre fc = 4?00'00""N = 0.069813170 rad
Longitude Projection Centre lc = 115?00'00""E = 2.007128640 rad
Azimuth of central line ac = 53?18'56.9537"" = 0.930536611 rad
Rectified to skew gc= 53?07'48.3685"" = 0.927295218 rad
Scale factor ko= 0.99984
Easting at projection centre Ec = 590476.87 m
Northing at projection centre Nc = 442857.65 m
Forward calculation for:
Latitude lat = 5?23'14.1129""N = 0.094025313 rad
Longitude lon = 115?48'19.8196""E = 2.021187362 rad
B = 1.003303209 F = 1.072121256
A =6376278.686 H = 1.000002991
to = 0.932946976 g0 = 0.927295218
D = 1.002425787 lon0 = 1.914373469
D2 =1.004857458
uc =738096.09 vc =0.00
t =0.910700729 Q =1.098398182
S =0.093990763 T = 1.004407419
V =0.106961709 U = 0.010967247
v =-69702.787 u = 163238.163
Then Easting E = 679245.73 m
Northing N = 596562.78 m
Reverse calculations for same easting and northing first gives :
v? = -69702.787 u? = 901334.257
Q? = 1.011028053
S? = 0.010967907 T? = 1.000060146
V? = 0.141349378 U? = 0.093578324
t? = 0.910700729 c = 0.093404829
Then Latitude = 5?23'14.113""N
Longitude = 115?48'19.820""E",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"1999.811 2004.430 2004.600 2007.044 2010.058 2017.024",0
9816,"Tunisia Mining Grid",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
This grid is used as the basis for mineral leasing in Tunsia. Lease areas are approximately 2 x 2 km or 400 hectares. The southwest corner of these blocks is defined through a six figure grid reference where the first three digits are an easting in kilometres and the last three digits are a northing. The origin of the grid is Djebel Kebar at 38.81973gN, 7.83445gE of Paris at which the grid reference is 270582. The latitudes and longitudes for block corners at 2 km intervals are tabulated in a mining decree dated 1st January 1953. From this tabulation in which geographical coordinates are given to 5 decimal places it can be seen that:
a) the minimum easting is 94 km, on which the longitude is 5.68989 grads east of Paris.
b) the maximum easting is 490 km, on which the longitude is 10.51515 grads east of Paris.
c) each 2 km grid easting interval equals 0.02437 grads.
d) the minimum northing is 40 km, on which the latitude is 33.39 grads.
e) the maximum northing is 860 km, on which the latitude is 41.6039 grads.
f) between 40 km N and 360 km N, each 2 km grid northing interval equals 0.02004 grads.
g) between 360 km N and 860 km N, each 2 km grid northing interval equals 0.02003 grads.
From the above the latitude of N=360 can be determined to be 36.5964gN.
Formulae are:
Grads from Paris
Lat = LatO + [(N - Nf) * A]
where Lat and LatO are in grads, N and Nf are in kilometres and A = 0.010015 if N > Nf, else A = 0.01002.
Lon = LonO + [(E - Ef) * 0.012185]
where Lon and LonO are in in grads relative to the Paris meridian and E and Ef are in kilometres.
The reverse formulae are:
E = Ef + [(Lon - LonO) / 0.012185]
N = Nf + [(Lat - LatO) / A]
where A = 0.010015 if Lat>LatO else A = 0.01002.
Degrees from Greenwich.
Modern practice in Tunisia is to quote latitude and longitude in degrees with longitudes referenced to the Greenwich meridian. The formulae required in addition to the above are:
Lat (degrees) = (Latg * 0.9) where Latg is in grads.
LonGreenwich (degrees) = [(LonParis + 2.5969213) * 0.9] where LonParis is in grads.
Lat (grads) = (Latd / 0.9) where Latd is in decimal degrees.
LonParis (grads) = [(LonGreenwich / 0.9) - 2.5969213)] where LonGreenwich is in decimal degrees.","Defining parameters are:
LatO = 36.5964gN
LonO = 7.83445gE of Paris
Ef = 270 km
Nf = 360 km
For grid location 302598, first derive the easting (302 km) and northing (598 km) from the grid reference. Then:
Lat = 36.5964 + [(598 - 360) * A]. As N > 360, A = 0.010015.
Lat = 38.97997 grads N = 35.08197 degrees N.
Lon = 7.83445 + [(302 - 270) * 0.012185
= 8.22437 grads east of Paris = 9.73916 degrees east of Greenwich.",,"EPSG guidance note #7-2, http://www.epsg.org","OGP",6/12/2013 0:00:00,"1999.811 2000.080 2012.052",0
9817,"Lambert Conic Near-Conformal",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
To compute the Lambert Conic Near-Conformal the following formulae are used. First compute constants for the projection:
n = f / (2-f)
A = 1 / (6 rhoO nuO)
A? = a [ 1- n + 5 (n^2 - n^3 ) / 4 + 81 ( n^4 - n^5 ) / 64]*pi /180
B? = 3 a [ n - n^2 + 7 ( n^3 - n^4 ) / 8 + 55 n^5 / 64] / 2
C? = 15 a [ n^2 -n^3 + 3 ( n^4 - n^5 ) / 4 ] / 16
D? = 35 a [ n^3 - n^4 + 11 n^5 / 16 ] / 48
E? = 315 a [ n^4 - n^5 ] / 512
r0 = ko nu0 / tan(lat0)
s0 = A? latO - B? sin(2 latO) + C? sin(4 latO) - D? sin(6 latO) + E? sin(8 latO) where in the first term latO is in degrees, in the other terms latO is in radians.
Then for the computation of easting and northing from latitude and longitude:
s = A? lat - B? sin(2 lat) + C? sin(4 lat) - D? sin(6 lat) + E? sin(8 lat) where in the first term latO is in degrees, in the other terms latO is in radians.
M = s - sO
M = ko ( m + A m^3)
r = rO - M
theta = (lon - lonO) sin(latO)
and
E = FE + r sin(theta)
N = FN + M + r sin(theta) tan(theta/2)
The reverse formulas for latitude and longitude from Easting and Northing are:
theta' = atan2 {(E ? FE) , [rO ? (N ? FN)]} (see GN7-2 implementation notes in preface for atan2 convention)
r' = +/- {(E ? FE)^2 + [rO ? (N ? FN)]}^2}^0.5, taking the sign of latO
M' = rO ? r'
If an exact solution is required, it is necessary to solve for m and lat using iteration of the two equations:
m'= m' ? [M' ? ko m' ? ko A (m')^3] / [? ko ? 3 ko A (m')^2]
using M' for m' in the first iteration. This will usually converge (to within 1mm) in a single iteration. Then
lat' = lat' +{m' + sO ? [A' lat' (180/pi) ? B' sin(2 lat') + C' sin(4 lat') ? D' sin(6lat') + E' sin(8 lat')]}/A' (pi/180)
first using lat' = latO + m'/A' (pi/180).
However the following non-iterative solution is accurate to better than 0.001"" (3mm) within 5 degrees latitude of the projection origin and should suffice for most purposes:
m' = M' ? [M' ko M' ? ko A (M')^3] / [? ko ? 3 ko A (M')^2]
lat' = latO + m'/A' (pi/180)
s' = A ' lat' ? B' sin(2 lat') + C' sin(4 lat') ? D' sin(6 lat') + E' sin(8 lat')
where in the first term lat' is in degrees, in the other terms lat' is in radians.
Ds' = A'(180 / pi) ? 2B' cos(2 lat') + 4C' cos(4 lat') ? 6D' cos(6 lat') + 8E' cos(8 lat')
lat = lat' ? [(m' + sO ? s') / (?ds')] radians
Then after solution of lat using either method above
lon = lonO + theta' / sin(latO) where lonO and lon are in radians.","For Projected Coordinate System: Deir ez Zor / Levant Zone
Parameters:
Ellipsoid Clarke 1880 (IGN) a = 6378249.2 m 1/f = 293.46602
then b = 6356515.000 n = 0.001706682563
Latitude Natural Origin = 34?39'00""N = 0.604756586 rad
Longitude Natural Origin = 37?21'00""E= 0.651880476 rad
Scale factor at origin ko = 0.99962560
False Eastings FE = 300000.00 m
False Northings FN = 300000.00 m
Forward calculation for:
Latitude of 37?31'17.625""N = 0.654874806 rad
Longitude of 34?08'11.291""E = 0.595793792 rad
first gives
A = 4.1067494 * 10e-15 A?=111131.8633
B?= 16300.64407 C?= 17.38751 D?= 0.02308 E?= 0.000033
so = 3835482.233 s = 4154101.458 m = 318619.225
M = 318632.72 Ms = 30.82262319
q = -0.03188875 ro = 9235264.405 r = 8916631.685
Then Easting E = 15707.96 m (c.f. E = 15708.00 using full formulae)
Northing N = 623165.96 m (c.f. N = 623167.20 using full formulae)
Reverse calculation for the same easting and northing first gives
q' = -0.03188875
r? = 8916631.685
M?= 318632.72
Latitude = 0.654874806 rad = 37?31'17.625""N
Longitude = 0.595793792 rad = 34?08'11.291""E","The Lambert Near-Conformal projection is derived from the Lambert Conformal Conic projection by truncating the series expansion of the projection formulae.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"1999.811 2004.610 2005.390 2008.029 2010.024 2017.024",0
9818,"American Polyconic",1,"See information source.","See information source.","See information source for formula and example.","US Geological Survey Professional Paper 1395; ""Map Projections - A Working Manual""; J. Snyder","IOGP",6/13/2017 0:00:00,"1999.550 2017.018",0
9819,"Krovak",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
From the defining parameters the following constants for the projection may be calculated :
A = a(1 - e^2)^0.5 / [1 - e^2 sin^2.(latC)]
B = {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5
gammao = asin[sin(latC) / B]
to = tan(pi/4 + gammao/2).[(1 + e sin(latC)) / (1 - e sin(latC))]^(e.B/2) / [tan(pi/4 + latC/2)]^B
n = sin(latp)
ro = kp.A / tan(latp)
To derive the projected Southing and Westing coordinates of a point with geographical coordinates (lat, lon) the formulas for the Krovak are:
U = 2(atan{to.tan^B(lat/2 + pi/4) / [(1 + e sin(lat)) / (1 - e sin(lat))]^[e.B/2]} - pi/4)
V = B(lonO - lon) where lonO and lon must both be referenced to the same prime meridian.
T = asin[cos(alphaC).sin(U) + sin(alphaC).cos(U). cos(V)]
D = asin[cos(U).sin(V)/cos(T)]
theta = n.D
r = ro.tan^n(pi/4 + latp/2) / tan^n(T/2 + pi/4)
Xp = r.cos(theta)
Yp = r.sin(theta)
Then
Southing (X) = Xp + FN
Westing (Y) = Yp + FE
Note also that the formula for D is satisfactory for the normal use of the projection within the pseudo-longitude range on the conformal sphere of ?90 degrees from the central line of the projection. Should there be a need to exceed this range (which is not necessary for application in Czechia and Slovakia) then for the calculation of D use:
sin(D1) = cos(U) * sin(V) / cos(T)
cos(D1) = {[cos(alphaC)*sin(T) ? sin(U)] / [sin(alphaC)*cos(T)]}
D = atan2(sin(D1) , cos(D1)) (see implementation notes in preface for atan2 convention)
The reverse formulas to derive the latitude and longitude of a point from its Southing and Westing values are:
Xp' = Southing ? FN
Yp' = Westing ? FE
r' = [(Yp')^2 + (Xp')^2]^(1/2)
theta' = atan2[Yp' , Xp'] (see GN7-2 implementation notes in preface for atan2 convention)
D' = theta' / sin(latp)
T' = 2{atan[((ro / r')^(1/n)).tan(pi/4 + latp/2)] - pi/4}
U' = asin[cos(alphaC).sin(T') - sin(alphaC).cos(T').cos(D')]
V' = asin(cos(T').sin(D') / cos(U'))
Then latitude lat is found by iteration using U' as the value for lat(j-1) in the first iteration:
lat(j) = 2*(atan{to^(-1/B) tan^(1/B).(U'/2 + pi/4).[(1 + e sin(lat(j-1)) / (1 - e sin(lat(j-1))]^(e/2)} - pi/4)
Then
lon = lonO - V' / B where lon is referenced to the same prime meridian as lonO.","For Geographic CRS S-JTSK and Projected CRS S-JTSK (Ferro) / Krovak
Parameters:
Ellipsoid Bessel 1841 a = 6377397.155m 1/f = 299.15281
then e = 0.081696831 e^2 = 0.006674372
Latitude of projection centre = 49?30'00""N = 0.863937979 rad
Longitude of origin = 42?30'00""E of Ferro = 0.741764932 rad
Co-latitude of cone axis = 30?17'17.30311"" = 0.528627763 rad
Latitude of pseudo standard parallel = 78?30'00""N = 1.370083463 rad
Scale factor on pseudo Standard Parallel (ko) = 0.9999
False Easting = 0.00 m
False Northing = 0.00 m
Calculated projection constants:
A=6380703.611
B=1.000597498
gammao=0.863239103
to=1.003419164
n= 0.979924705
ro=1298039.005
Forward calculation for:
Latitude = 50?12'32.442""N = 0.876312568 rad
Longitude = 16?50'59.179""E of Greenwich
Firstly, because the projection definition includes longitudes referenced to the Ferro meridian, the longitude of the point needs to be transformed to be referenced to the Ferro meridian using the Longitude Rotation method (EPSG method code 9601).
Longitude = 16?50'59.1790""E of Greenwich
Longitude of Ferro = 17?40'00"" west of Greenwich
and then
Longitude = 34?30'59.1790""E of Ferro = 0.602425500 rad
Then the forward calculation first gives
U = 0.875596949
V = 0.139422687
T = 1.386275049
D = 0.506554623
theta = 0.496385389
r = 1194731.014
Xp = 1050538.643
Yp = 568990.997
Then Southing (X) = 1050538.64 m
Westing (Y) = 568991.00 m.
Reverse calculation for the same Southing and Westing gives
Xp' = 1050538.643
Yp' = 568990.997
r' = 1194731.014
theta' = 0.496385389
D' = 0.506554623
T' = 1.386275049
U' = 0.875596949
V' = 0.139422687
lat(iteration 1) = 0.876310601
lat(iteration 2) = 0.876312560
lat(iteration 3) = 0.876312566
Latitude = 0.876312566 rad = 50?12'32.442""N.
Longitude of point = 0.602425500 rad = 34?30'59.179""E of Ferro.
Then using the Longitude Rotation method (EPSG method code 9601):
Longitude of Ferro = 17?40'00"" west of Greenwich
and
Longitude of point = 34?30'59.179""E of Greenwich.",,"Research Institute for Geodesy Topography and Cartography (VUGTK); Prague, and Geodesy, Cartography and Cadastre Authority of Slovak Republic (UGKK SR).","IOGP",5/17/2019 0:00:00,"2002.950 2006.180 2007.040 2009.017 2010.071 2010.100 2017.037 2017.024 2019.030",0
9820,"Lambert Azimuthal Equal Area",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
Oblique aspect
To derive the projected coordinates of a point, geodetic latitude (lat) is converted to authalic latitude (?). The formulae to convert geodetic latitude and longitude (lat,lon) to Easting and Northing are:
Easting, E = FE + {(B . D) . [cos ? . sin(lon ? lonO)]}
Northing, N = FN + (B / D) . {(cos ?O . sin ?) ? [sin ?O . cos ? . cos(lon ? lonO)]}
where
B = Rq . (2 / {1 + sin ?O . sin ? + [cos ?O . cos ? . cos(lon ? lonO)]})^0.5
D = a . [cos latO / (1 ? e2 sin2 latO)^0.5] / (Rq . cos ?O)
Rq = a . (qP / 2)^0.5
? = asin (q / qP)
?O = asin (qO / qP)
q = (1 ? e^2) . ([sin(lat) / (1 ? e^2 sin^2(lat))] ? {[1/(2e)] . ln [(1 ? e sin(lat)) / (1 + e sin(lat))]})
qO = (1 ? e^2) . ([sin(latO) / (1 ? e^2 sin^2(latO))] ? {[1/(2e)] . ln [(1 ? e sin(latO)) / (1 + e sin(latO))]})
qP = (1 ? e^2) . ([sin(latP) / (1 ? e^2 sin^2(latP))] ? {[1/(2e)] . ln [(1 ? e sin(latP)) / (1 + e sin(latP))]})
where *P = p/2 radians, thus
qP = (1 ? e^2) . ([1 / (1 ? e^2)] ? {[1/(2e)] . ln [(1 ? e) / (1 + e)]})
The reverse formulas to derive the geodetic latitude and longitude of a point from its Easting and Northing values are:
lat = ?' + [(e^2/3 + 31e^4/180 + 517e^6/5040) . sin 2?'] + [(23e^4/360 + 251e^6/3780) . sin 4?'] + [(761e^6/45360) . sin 6?']
lon = lonO + atan2 {(E-FE) . sin C , [D. rho . cos ?O . cos C ? D^2. (N-FN) . sin ?O . sin C]} (see implementation notes in GN7-2 preface for atan2 convention)
where
?' = asin{(cosC . sin ?O) + [(D . (N-FN) . sinC . cos ?O) / rho]}
C = 2 . asin(rho / 2 . Rq)
rho = {[(E-FE)/D]^2 + [D . (N ?FN)]^2}^0.5
and D, Rq, and ?O are as in the forward equations.
Polar aspect
For the polar aspect of the Lambert Azimuthal Equal Area projection, some of the above equations are indeterminate. Instead, for the forward case from latitude and longitude (lat, lon) to Easting (E) and Northing (N):
For the north polar case:
Easting, E = FE + [rho sin(lon ? lonO)]
Northing, N = FN ? [rho cos(lon ? lonO)]
where
rho = a (qP ? q)^0.5
and qP and q are found as for the general case above.
For the south polar case:
Easting, E = FE + [rho . sin(lon ? lonO)]
Northing, N = FN + [rho . cos(lon ? lonO)]
where
rho = a (qP + q)^0.5
and qP and q are found as for the general case above.
For the reverse formulas to derive the geodetic latitude and longitude of a point from its Easting and Northing:
lat = ?' + [(e^2/3 + 31e^4/180 + 517e^6/5040) sin 2?'] + [(23e^4/360 + 251e^6/3780) sin 4?'] + [(761e^6/45360) sin 6?']
as for the oblique case, but where
?' = ?asin [1? rho^2 / (a^2{1? [(1? e^2)/2e)) ln[(1-e)/(1+ e)]})], taking the sign of latO
and rho = {[(E ?FE)]^2 + [(N ? FN)]^2}^0.5
Then
lon = lonO + atan2 [(E ? FE)] , (N ? FN)] for the south pole case
and
lon = lonO + atan2 [(E ? FE)] , ?(N ? FN)] for the north pole case.
(see implementation notes in GN7-2 preface for atan2 convention)","For Projected Coordinate Reference System: ETRS89 / ETRS-LAEA
Parameters:
Ellipsoid:GRS 1980 a = 6378137.0 metres 1/f = 298.2572221
then e = 0.081819191
Latitude of natural origin (latO): 52?00'00.000""N = 0.907571211 rad
Longitude of natural origin (lonO): 10?00'00.000""E = 0.174532925 rad
False easting (FE): 4321000.00 metres
False northing (FN) 3210000.00 metres
Forward calculation for:
Latitude (lat) = 50?00'00.000""N = 0.872664626 rad
Longitude(lon) = 5?00'00.000""E = 0.087266463 rad
First gives
qP = 1.995531087
qO = 1.569825704
q = 1.525832247
Rq = 6371007.181
betaO = 0.905397517
beta = 0.870458708
D = 1.000425395
B = 6374393.455
whence
E = 3962799.45 m
N = 2999718.85 m
Reverse calculation for the same Easting and Northing (3962799.45 E, 2999718.85 N) first gives:
rho = 415276.208
C = 0.065193736
beta' = 0.870458708
Then Latitude = 50?00'00.000""N
Longitude = 5?00'00.000""E","This is the ellipsoidal form of the projection.","USGS Professional Paper 1395, ""Map Projections - A Working Manual"" by John P. Snyder.","IOGP",8/29/2018 0:00:00,"2003.350 2004.430 2005.075 2005.290 2006.200 2007.005 2017.018 2017.024",0
9821,"Lambert Azimuthal Equal Area (Spherical)",1,"See information source.","See information source.","This is the spherical form of the projection. See coordinate operation method Lambert Azimuthal Equal Area (code 9820) for ellipsoidal form. Differences of several tens of metres result from comparison of the two methods.","USGS Professional Paper 1395, ""Map Projections - A Working Manual"" by John P. Snyder.","OGP",6/5/2001 0:00:00,"2009.007",1
9822,"Albers Equal Area",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
To derive the projected coordinates of a point, geodetic latitude (lat) is converted to authalic latitude (?). The formulas to convert geodetic latitude and longitude (lat, lon) to Easting (E) and Northing (N) are:
Easting (E) = EF + (rho . sin(theta))
Northing (N) = NF + rhoO ? (rho . cos(theta))
where
theta = n . (lon - lonO)
rho = [a . (C ? n.alpha)^0.5] / n
rhoO = [a . (C ? n.alphaO)^0.5] / n
and
C = m1^2 + (n . alpha1)
n = (m1^2 ? m2^2) / (alpha2 - alpha1)
m1 = cos lat1 / (1 ? e^2 sin^2(lat1))^0.5
m2 = cos lat2 / (1 ? e^2 sin^2(lat2))^0.5
alpha = (1 ? e^2) . {[sin(lat) / (1 ? e^2 sin^2(lat))] ? [1/(2e)] . ln [(1 ? e sin(lat)) / (1 + e sin(lat))]}
alphaO = (1 ? e^2) . {[sin(latO) / (1 ? e^2 sin^2(latO))] ? [1/(2e)] . ln [(1 ? e sin(latO)) / (1 + e sin(latO))]}
alpha1 = (1 ? e^2) . {[sin(lat1) / (1 ? e^2 sin^2(lat1))] ? [1/(2e)] . ln [(1 ? e sin(lat1)) / (1 + e sin(lat1))]}
alpha2 = (1 ? e^2) . {[sin(lat2) / (1 ? e^2 sin^2(lat2))] ? [1/(2e)] . ln [(1 ? e sin(lat2)) / (1 + e sin(lat2))]}
The reverse formulas to derive the geodetic latitude and longitude of a point from its Easting and Northing values are:
lat = ?' + [(e^2/3 + 31e^4/180 + 517e^6/5040) . sin 2?'] + [(23e^4/360 + 251e^6/3780) . sin 4?'] + [(761e^6/45360) . sin 6?']
lon = lonO + (theta / n)
where
?' = asin(alpha' / {1 ? [(1 ? e^2) / 2e] . ln [(1 ? e) / (1 + e)]})
alpha' = [C ? (rho^2 . N^2 / a^2)] / n
rho = {(E ? EF)^2 + [rhoO ? (N ? NF)]^2 }^0.5
theta = atan2 {(E ? EF) , [rhoO ? (N ? NF)]}
and C, n and rhoO are as in the forward equations.
(see implementation notes in GN7-2 preface for atan2 convention).","See Information Source.",,"USGS Professional Paper 1395, ""Map Projections - A Working Manual"" by John P. Snyder.","IOGP",8/29/2018 0:00:00,"2006.200 2007.049 2015.022 2017.024",0
9823,"Equidistant Cylindrical (Spherical)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
This method has one of the simplest formulas available. If the latitude of natural origin (latO) is at the equator the method is also known as Plate Carr?e. It is not used for rigorous topographic mapping because its distortion characteristics are unsuitable. Formulas are included to distinguish this map projection method from an approach sometimes mistakenly called by the same name and used for simple computer display of geographic coordinates ? see Pseudo Plate Carr?e (coordinate operation method code 9825).
For the forward calculation:
X = R . (lon - lonO) . cos(latO)
Y = R . lat
where R = ((a^2 * (1 ? e^2)) / (1 ? e^2 sin^2 latO)^2)^0.5
and latO, lonO, lat and lon are expressed in radians.
For the Equidistant Cylindrical method on a sphere (not ellipsoid), e = 0 and R = a.
For the reverse calculation:
lat = Y / R
lon = lonO + (X / R cos(latO))
where R is as for the forward method.","See information source.","See method code 9842 for ellipsoidal development. If the latitude of natural origin is at the equator, also known as Plate Carr?e. See also Pseudo Plate Carree, method code 9825.","US Geological Survey Professional Paper 1395; ""Map Projections - A Working Manual""; J. Snyder.","OGP",3/14/2008 0:00:00,"2002.920 2008.003 2009.023",1
9824,"Transverse Mercator Zoned Grid System",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The standard Transverse Mercator formulas (coordinate operation method 9807) are modified as follows:
Zone number, Z, = [(int((Long - LongI) / W) mod(360 / W)) + 1] with Long, LongI and W in degrees.
where (LongI) is the Initial Longitude of the zoned grid system (the western limit of zone 1)
and W is the width of each zone of the zoned grid system.
In the formula above, mod = (a - floor(a / b) * b) where 'floor' is defined as rounding toward negative infinity. Z is in the range 1 = Z = (360/W) and for UTM 1 = Z = 60.
Then,
Long0 = LongI + (Z * W) ? (W/2)
For the forward calculation,
Easting, E = Z*10^6 + FE + k0.nu[A + (1 - T + C)A^3/6 + (5 - 18T + T^2 + 72C - 58e'^2)A^5/120]
and in the reverse calculation for longitude,
D = (E ? [FE + Z*10^6])/(nu1.k0)","(none)","If locations fall outwith the fixed zones the general Transverse Mercator method (code 9807) must be used for each zone.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",7/14/2017 0:00:00,"2017.018 2017.011",0
9825,"Pseudo Plate Carree",1,"X = Lon
Y = Lat
Lat = Y
Lon = X","(none)","Used only for depiction of graticule (latitude/longitude) coordinates on a computer display. The axes units are decimal degrees and of variable scale. The origin is at Lat = 0, Long = 0. See Equidistant Cylindrical, code 1029, for proper Plate Carr?e.","EPSG guidance note #7-2, http://www.epsg.org","IOGP",9/17/2019 0:00:00,"2019.059",0
9826,"Lambert Conic Conformal (West Orientated)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
In older mapping of Denmark and Greenland the Lambert Conic Conformal is used with axes positive north and west. To derive the projected Westing and Northing coordinates of a point with geographical coordinates (Lat, Lon) the formulas are as for the standard Lambert Conic Conformal (1SP) case (coordinate operation method code 9801) except for:
W = FE ? r.sin(theta)
In this formula the term FE retains its definition, i.e. in the Lambert Conic Conformal (West Orientated) method it increases the Westing value at the natural origin. In this method it is effectively false westing (FW).
The reverse formulas to derive the latitude and longitude of a point from its Westing and Northing values are as for the standard Lambert Conic Conformal (1SP) case except for:
theta' = atan2[(FE ? W),{r0 ? (N ? FN)}] (see implementation notes in GN7-2 preface for atan2 convention)
r' = +/-[(FE ? W)^2 + {r0 ? (N ? FN)}^2]^0.5, taking the sign of n.","See Lambert Conic Conformal (1SP), code 9801, for general methodology.",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"2017.018 2017.024",0
9827,"Bonne",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The formulas to convert geodetic latitude and longitude (lat, lon) to Easting and Northing are:
E = (rho . sin T) + FE
N = (a . mO / sin(latO) ? rho . cos T) + FN
where
m = cos(lat) / (1 ? e^2sin^2(lat))^0.5
with lat in radians and mO for latO, the latitude of the origin, derived in the same way.
M = a[(1 ? e^2/4 ? 3e^4/64 ? 5e^6/256 ?....)lat ? (3e^2/8 + 3e^4/32 + 45e^6/1024+....)sin(2 lat) + (15e^4/256 + 45e^6/1024 +.....)sin(4 lat) ? (35e^6/3072 + ....)sin(6 lat) + .....]
with lat in radians and MO for latO, the latitude of the origin, derived in the same way.
rho = a . mO / sin(latO) + MO ? M
T = a . m (lon ? lonO) / rho with lon and lonO in radians
For the reverse calculation:
X = E ? FE
Y = N ? FN
rho = ? [X^2 + (a . mO / sin(latO) ? Y)^2]^0.5 taking the sign of latO
M = a . mO / sin(latO) + MO ? rho
mu = M / [a (1 ? e^2/4 ? 3e^4/64 ? 5e^6/256 ? ?)]
e1 = [1 ? (1 ? e^2)^0.5] / [1 + (1 ? e^2)^0.5]
lat = mu + ((3 e1 / 2) ? (27 e1^3 / 32) +.....)sin(2 mu) + ((21 e1^2 / 16) ? (55 e1^4 / 32) + ....)sin(4 mu)
+ ((151 e1^3 / 96) +.....)sin(6 mu) + ((1097 e1^4 / 512) ? ....)sin(8 mu) + ......
m = cos(lat) / (1 ? e^2 sin^2(lat))^0.5
If latO is not negative
lon = lonO + rho {atan2[X / (a . mO , sin(latO) ? Y)]} / a . m
but if lonO is negative
lon = lonO + rho {atan2[?X , ?(a . mO , sin(latO) ? Y)]} / a . m
In either case, if lat = ?90?, m = 0 and the equation for lon is indeterminate, so use lon = lonO. See implementation notes in GN7-2 preface for atan2 convention.","See information source.",,"US Geological Survey Professional Paper 1395, ""Map Projections - A Working Manual"" by John P Snyder.","IOGP",8/29/2018 0:00:00,"2017.024",0
9828,"Bonne (South Orientated)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The formulas to convert geodetic latitude and longitude (lat, lon) to Easting and Northing are:
W = FE ? (rho . sin T)
S = FN ? (a . mO / sin(latO) ? rho . cos T)
where
m = cos(lat) / (1 ? e^2sin^2(lat))^0.5
with lat in radians and mO for latO, the latitude of the origin, derived in the same way.
M = a[(1 ? e^2/4 ? 3e^4/64 ? 5e^6/256 ?....)lat ? (3e^2/8 + 3e^4/32 + 45e^6/1024+....)sin(2 lat) + (15e^4/256 + 45e^6/1024 +.....)sin(4 lat) ? (35e^6/3072 + ....)sin(6 lat) + .....]
with lat in radians and MO for latO, the latitude of the origin, derived in the same way.
rho = a . mO / sin(latO) + MO ? M
T = a . m (lon ? lonO) / rho with lon and lonO in radians
For the reverse calculation:
X = FE ? W
Y = FN ? S
rho = ? [X^2 + (a . mO / sin(latO) ? Y)^2]^0.5 taking the sign of latO
M = a . mO / sin(latO) + MO ? rho
mu = M / [a (1 ? e^2/4 ? 3e^4/64 ? 5e^6/256 ? ?)]
e1 = [1 ? (1 ? e^2)^0.5] / [1 + (1 ? e^2)^0.5]
lat = mu + ((3 e1 / 2) ? (27 e1^3 / 32) +.....)sin(2 mu) + ((21 e1^2 / 16) ? (55 e1^4 / 32) + ....)sin(4 mu)
+ ((151 e1^3 / 96) +.....)sin(6 mu) + ((1097 e1^4 / 512) ? ....)sin(8 mu) + ......
m = cos(lat) / (1 ? e^2 sin^2(lat))^0.5
If latO is not negative
lon = lonO + rho {atan2[X , (a . mO / sin(latO) ? Y)]} / a . m
but if lonO is negative
lon = lonO + rho {atan2[?X , ?(a . mO , sin(latO) ? Y)]} / a . m
In either case, if lat = ?90?, m = 0 and the equation for lon is indeterminate, so use lon = lonO. See implementation notes in GN7-2 preface for atan2 convention.
In these formulas the terms FE and FN retain their definition, i.e. in the Bonne (South Orientated) method they increase the Westing and Southing value at the natural origin. In this method they are effectively false westing (FW) and false southing (FS) respectively.","See information source of Bonne, code 9827, for general methodology.",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"2017.018 2017.024",0
9829,"Polar Stereographic (variant B)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
First calculate the scale factor at natural origin:
for the south pole case
tF = tan (pi/4 + latF/2) / {[(1 + e sin(latF)) / (1 ? e sin(latF))]^(e/2)}
but for the north pole case
tF = tan (pi/4 - latF/2) * {[(1 + e sin(latF)) / (1 ? e sin(latF))]^(e/2)}
then for both cases
mF = cos(latF) / (1 ? e^2 sin^2(latF))^0.5
ko = mF {[(1+e)^(1+e) (1?e)^(1?e)]0.5} / (2 tF)
The forward and reverse conversions then follow the formulae for the
Polar Stereographic (variant A) method:
For the forward conversion from latitude and longitude, for the south pole case
E = FE + rho * sin(lon ? lonO)
N = FN + rho * cos(lon ? lonO)
where
t = tan(pi/4 + lat/2) / {[(1 + e sin(lat)) / (1 ? e sin(lat))]^(e/2)}
rho = 2*a*ko*t / {[(1+e)^(1+e) (1?e)^(1?e)]^0.5}
For the north pole case,
rho and E are found as for the south pole case but
t = tan(pi/4 ? lat/2) * {[(1 + e sin(lat)) / (1 ? e sin(lat))]^(e/2)}
N = FN ? rho * cos(lon ? lonO)
For the reverse conversion from easting and northing to latitude and longitude,
lat = chi + (e^2/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2 chi)
+ (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4 chi)
+ (7e^6/120 + 81e^8/1120) sin(6 chi) + (4279e^8/161280) sin(8 chi)
where rho' = [(E-FE)^2 + (N ? FN)^2]^0.5
t' =rho' {[(1+e)^(1+e) * (1? e)^(1-e)]^0.5} / (2 a ko)
and for the south pole case
chi = 2 atan(t' ) ? pi/2
but for the north pole case
chi = pi/2 - 2 atan t'
Then for for both north and south cases if E = FE, lon = lonO
else for the south pole case
lon = lonO + atan2[(E ? FE),(N ? FN)]
and for the north pole case
lon = lonO + atan2[(E ? FE),(FN ? N)]
(see implementation notes in preface for atan2 convention)","For Projected Coordinate Reference System: WGS 84 / Australian Antarctic Polar Stereographic
Parameters:
Ellipsoid: WGS 84
a = 6378137.0 metres 1/f = 298.2572236
then e = 0.081819191
Latitude of standard parallel (latF): 71?00'00.000""S = -1.239183769 rad
Longitude of origin (lonO): 70?00'00.000""E = 1.221730476 rad
False easting (FE): 6000000.00 metres
False northing (FN): 6000000.00 metres
Forward calculation for:
Latitude (lat) = 75?00'00.000""S = -1.308996939 rad
Longitude(lon) = 120?00'00.000""E = 2.094395102 rad
tF = 0.168407325
mF = 0.326546781
ko = 0.97276901
t = 0.132508348
pho = 1638783.238
whence
E = 7255380.79 m
N = 7053389.56 m
Reverse calculation for the same Easting and Northing (7255380.79 E, 7053389.56 N) first gives:
tF = 0.168407325 mF = 0.326546781 and ko = 0.97276901
then rho' = 1638783.236 t' = 0.132508347 chi = -1.3073146
Then Latitude (lat) = 75?00'00.000""S
Longitude (lon) = 120?00'00.000""E",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"2004.430 2010.054 2017.024",0
9830,"Polar Stereographic (variant C)",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
For the forward conversion from latitude and longitude, for the south pole case
E = EF + rho * sin (lon ? lonO)
N = NF ? rhoF + rho * cos (lon ? lonO)
where
mF = cos latF / (1 ? e^2 sin^2(latF))^0.5
tF = tan (p/4 + latF/2) / {[(1 + e sin(latF)) / (1 ? e sin(latF))]^(e/2)}
t = tan (p/4 + lat/2) / {[(1 + e sin(lat)) / (1 ? e sin(lat))]^(e/2)}
rhoF = a mF
rho = rhoF * t / tF
For the north pole case, mF, *F, * and E are found as for the south pole case but
tF = tan (p/4 ? latF/2) * {[(1 + e sin(latF)) / (1 ? e sin(latF))]^(e/2)}
t = tan (p/4 ? lat/2) * {[(1 + e sin(lat)) / (1 ? e sin(lat))]^(e/2)}
N = NF + rhoF ? [rho * cos (lon ? lonO)]
For the reverse conversion from easting and northing to latitude and longitude,
lat = chi + (e^2/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2 chi)
+ (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4 chi)
+ (7e^6/120 + 81e^8/1120) sin(6 chi) + (4279e^8/161280) sin(8 chi)
where for the south pole case
rho' = [(E-EF)^2 + (N ? NF + rhoF)^2] ^0.5
t' = rho' * tF / rhoF
chi = 2 atan(t') ? pi/2
and where mF and tF are as for the forward conversion
For reverse conversion north pole case, mF, tF and rhoF are found as for the north pole case of the forward conversion, and
rho' = [(E-EF)^2 + (N ? NF ? rhoF)^2]^0.5
t' is found as for the south pole case of the reverse conversion = rho' * tF / rhoF
chi = pi/2 - 2 atan(t')
Then for for both north and south pole cases
if E = EF, lon = lonO
else for the south pole case
lon = lonO + atan2[(E ? EF),(N ? NF + rhoF)]
and for the north pole case
lon = lonO + atan2[(E ? EF),(NF + rhoF ? N)]
(see implementation notes in GN7-2 preface for atan2 convention)","For Projected Coordinate Reference System: Petrels 1972 / Terre Adelie Polar Stereographic
Parameters:
Ellipsoid:International 1924
a = 6378388.0 metres 1/f = 297.0
then e = 0.081991890
Latitude of false origin (latF): 67?00'00.000""S = -1.169370599 rad
Longitude of origin (lonO): 140?00'00.000""E = 2.443460953 rad
Easting at false origin (EF): 300000.00 metres
Northing at false origin (NF): 200000.00 metres
Forward calculation for:
Latitude (lat) = 66?36'18.820""S = -1.162480524 rad
Longitude (lon) = 140?04'17.040""E = 2.444707118 rad
mF = 0.391848769
rhoF = 2499363.488
tF = 0.204717630
t = 0.208326304
rho = 2543421.183
whence
E = 303169.52 m
N = 244055.72 m
Reverse calculation for the same Easting and Northing (303169.522 E, 244055.721 N) first gives:
mF = 0.391848769
rhoF = 2499363.488
tF = 0.204717630
then
rho' = 2543421.183
t' = 0.208326304
chi = -1.1600190
Then Latitude (lat) = 66?36'18.820""S
Longitude (lon) =140?04'17.040""E",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"2010.054 2017.024",0
9831,"Guam Projection",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
the forward conversion from latitude and longitude is given by:
x = (lon ? lonO) cos(lat) / [(1 ? e^2 sin^2(lat))^0.5]
E = FE + x
N = FN + M ? MO + ^2 tan(lat) [(1 ? e^2 sin^2(lat))^0.5] / (2a)}
where
M = a[(1 ? ^2/4 ? 3e^4/64 ? 5e^6/256 ?....)lat ? (3e^2/8 + 3e^4/32 + 45e^6/1024+....)sin(2 lat)
+ (^4/256 + 45e^6/1024 +.....)sin(4 lat) ? (35e^6/3072 + ....)sin(6 lat) + .....]
with lat in radians and MO for latO, the latitude of the natural origin, derived in the same way.
The reverse conversion from easting and northing to latitude and longitude requires iteration of three equations. The Guam projection uses three iterations, which is satisfactory over the small area of application. First MO for the latitude of the origin latO is derived as for the forward conversion. Then:
e' = [1 ? (1 ?^2)^0.5] / [1 + (1 ? e^2)^0.5]
and
M' = MO + (N ? FN) ? {(E ? FE)^2 tan(latO) [(1 ? e^2 sin^2(latO)^0.5] / (2a)}
mu' = M' / a(1 ? e^2/4 ? 3e^4/64 ? 5e^6/256 ?....)
lat' = mu' + (3e'/2 ? 27e'^3/32)sin(2mu') + (21e'^2/16 ? 55e'^4/32)sin(4mu') + (151e'^3/96)sin(6mu')
+ (1097e'^4/512)sin(8mu')
M"" = MO + (N ? FN) ? {(E FE)^2 tan(lat') [(1 ? e^2 sin^2(lat'))^0] / (2a)}
mu"" = M"" / a(1 ? e^2/4 ? 3e^4/64 ? 5e^6/256 ?....)
lat"" = mu"" + (3e'/2 ? 27e'^3/32)sin(2mu"") + (21e'^2/16 ? 55e'^4/32)sin(4mu"") + (151e'^3/96)sin(6mu"")
(1097 e'^4/512)sin(8mu"")
M''' = MO + (N ? FN) ? {(E ? F)^2 ta(lat"") [(1 ? e^2 sin^2(lat"")^0.5] / (2a)}
mu''' = M''' / a(1 ? e^2/4 ? 3e^4/64 ? 5e^6/256 ?....)
lat''' = mu''' + (3e'/2 ? 27e'^3/32)sin(2mu''') + (21e'^2/16 ? 55e'^4/32)sin(4mu''') + (151e'^3/96)sin(6mu''')
+ (1097e'^4/512)sin(8mu''')
Then
lon = lonO + {(E ? FE) . [(1 ? e^2 sin^2 lat''')^0.5] / (a cos lat''')}","See information source or EPSG Guidance Note 7.","Simplified form of Oblique Azimuthal Equidistant projection method.","US Geological Survey Professional Paper 1395; ""Map Projections - A Working Manual""; J. Snyder.","IOGP",6/13/2017 0:00:00,"2017.018",0
9832,"Modified Azimuthal Equidistant",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
First calculate a constant for the projection:
nu_O = a /(1 ? e^2 sin^2(latO))^0.5
Then the forward conversion from latitude and longitude is given by:
nu = a /(1 ? e^2 sin^2(lat))^0.5
psi = atan [(1 ? e^2) tan(lat) + e^2 * nu_O * sin(latO) / (nu * cos(lat))]
alpha = atan2{sin (lon ? lonO) , [cos(latO) * tan(psi) ? sin(latO) * cos (lon ? lonO)]} (see implementation notes in GN7-2 preface for atan2 convention)
G = e sin(latO) / (1 ? e^2)^0.5
H = e cos(latO) * cos(alpha) / (1 ? e^2)^0.5
Then
if sin(alpha)) = 0, s = asin (cos(latO) * sin(psi) ? sin(latO) * cos(psi)) * SIGN(cos(alpha))
else s = asin [sin (lon ? lonO) * cos(psi) / sin(alpha))
and in either case
c = nu_O * s {[1 ? s^2 * H^2 (1 ? H^2) /6] + [(s^3/8)GH(1-2H^2)] + (s^4/120)[H^2(4-7H^2) ? 3G^2(1-7H^2)] ? [(s^5/48)GH]}
Then
E = FE + * sin(alpha)
N = FN + * cos(alpha)
For the reverse conversion from easting and northing to latitude and longitude:
c' = [(E FE)^2 + (N ? FN)^2]^0.5
alpha' = atan2 [(E ? FE) , (N ? FN)]
A = e^2 * cos^2(latO) * cos^2(alpha') / (1 ? e^2)
B 3e^2 * (1-A) * sin(latO) * cos(latO) * cos(alpha') / (1 ? e^2)
D = c'nu_O
J = D ? [A (1 + AD^3 / 6] ? [B (1 + 3A) D^4 / 24]
K = 1 ? (* J^2 / 2) ? (B *J^3 / 6)
psi' = asin (sin(latO) cos(J) + cos(latO) sin(J) cos(alpha'))
Then
lat = atan [(1 ? e^2 * K sin(latO) / sin(psi')) * tan(psi') / (1 ? e^2)]
lon = lonO + asin (sin(alpha') * sin(J) / cos(psi'))","See information source or EPSG Guidance Note 7.","Modified form of Oblique Azimuthal Equidistant projection method developed for Polynesian islands. For the distances over which these projections are used (under 800km) this modification introduces no significant error.","US Geological Survey Professional Paper 1395; ""Map Projections - A Working Manual""; J. Snyder.","IOGP",8/29/2018 0:00:00,"2006.200 2017.018 2017.024",0
9833,"Hyperbolic Cassini-Soldner",1,"See information source.","See information source.",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9834,"Lambert Cylindrical Equal Area (Spherical)",1,"For the forward calculation for the normal aspect of the projection in which lat1 is the latitude of the standard parallel:
E = FE R (lon – lonO) cos(lat1)
N = FN R sin(lat) / cos(lat1)
where lat1, lat and lon are expressed in radians
R is the radius of the sphere and will normally be one of the CRS parameters. If the figure of the earth used is an ellipsoid rather than a sphere then R should be calculated as the radius of the authalic sphere using the formula for RA given in EPSG Guidance Note 7-2, section 1.2, table 3.
For the reverse calculation:
lat = asin{[(N – FN) / R] cos(lat1)}
lon = lonO {[E – FE] / [R cos(lat1)]}
where R is as for the forward method.
See information source for formulas for oblique and polar aspects and examples.","See information source.","This is the spherical form of the projection. See coordinate operation method Lambert Cylindrical Equal Area (code 9835) for ellipsoidal form. Differences of several tens of metres result from comparison of the two methods.","USGS Professional Paper 1395, ""Map Projections - A Working Manual"" by John P. Snyder.","OGP",6/7/2009 0:00:00,"2007.078 2009.023",0
9835,"Lambert Cylindrical Equal Area",1,"See information source.","See information source.","This is the ellipsoidal form of the projection.","USGS Professional Paper 1395, ""Map Projections - A Working Manual"" by John P. Snyder.","OGP",11/1/2007 0:00:00,,0
9836,"Geocentric/topocentric conversions",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
First it is necessary to derive ellipsoidal values Po, Lo of the topocentric origin from their geocentric values Xo, Yo, Zo through the reverse formulas given in method 9602.
Then topocentric coordinates [U, V, W] are computed as follows:
U = ? (X-Xo) sin Lo + (Y-Yo) cos Lo
V = ? (X?Xo) sin Po cos Lo ? (Y?Yo) sin Po sin Lo + (Z?Zo) cos Po
W = (X?Xo) cos Po cos Lo + (Y?Yo) cos Po sin Lo + (Z?Zo) sin Po
The reverse formulas to calculate geocentric coordinates from topocentric coordinates are:
X = Xo ? U sin Lo ? V sin Po cos Lo + W cos Po cos Lo
Y = Yo + U cos Lo ? V sin Po sin Lo + W cos Po sin Lo
Z = Zo + V cos Po + W sin Po","For Geocentric CRS = WGS 84 (EPSG CRS code 4978)
and
Topocentric origin Xo = 3652 755.3058 m
Topocentric origin Yo = 319 574.6799 m
Topocentric origin Zo = 5201 547.3536 m
Ellipsoid parameters: a = 6378137m.0 1/f = 298.2572236
First calculate additional ellipsoid parameters:
e^2 = 0.006694380
eta = 0.006739497
b = 6356752.314
Next, derive Po, Lo from Xo,Yo,Zo by the formulas given in method 9602:
p = 3666708.2376
q = 0.9583523313
Po = 0.9599310885 rad
Lo = 0.0872664625 rad
Forward calculation for point with geocentric coordinates:
X= 3771 793.968
Y= 140 253.342
Z= 5124 304.349
gives topocentric coordinates
U= -189 013.869
V= -128 642.040
W= -4 220.171",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9837,"Geographic/topocentric conversions",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
To convert latitude P, longitude L and ellipsoidal height h into topocentric coordinates U,V,W:
U = (nu + h) cos(P) sin (L ? Lo)
V = (nu + h) [sin(P) cos(Po) ? cos(P) sin(Po) cos(L ? Lo)] + e^2 (nuO sin(Po) ? nu sin(P)) cos(Po)
W = (nu + h) [sin(P) sin(Po) + cos(P) cos(Po) cos(L ? Lo)] + e^2 (nuO sin(Po) ? nu sin(P)) sin(Po) ? (nuO ? ho)
where Po, Lo and ho are the ellipsoidal coordinates of the topocentric origin
and nu is the radius of curvature in the prime vertical at latitude P = a /(1 ? e^2 sin^2 P)^0.5
nuO is the radius of curvature in the prime vertical at latitude Po = a /(1 ? e^2 sin^2 Po)^0.5
e is the eccentricity of the ellipsoid here e^2 = (a^2 ? b^2)/a^2 = 2f ? f^2
The reverse formulae to convert topocentric coordinates (U, V, W) into latitude, longitude and ellipsoidal height (P, L, h) first draws on the reverse case of method 9836 to derive geocentric coordinates X, Y, Z and then on the reverse case of method 9602 to derive latitude, longitude and height.
First,
X = Xo ? U sin(Lo) ? V sin(Po) cos(Lo) + W cos(Po) cos(Lo)
Y = Yo + U cos(Lo) ? V sin(Po) sin(Lo) + W cos(Po) sin(Lo)
Z = Zo + V cos(Po) + W sin(Po)
where,
XO = (nuO + hO) cos(Po) cos(Lo)
YO = (nuO + hO) cos(Po) sin(Lo)
ZO = [(1 ? e2) nuO + hO] sin(Po)
Po, Lo, hO are the ellipsoidal coordinates of the topocentric origin,
nuO is the radius of curvature in the prime vertical at latitude Po = a /(1 ? e^2 sin^2(Po))^0.5, and
e is the eccentricity of the ellipsoid where e^2 = (a^2 ? b^2)/a^2 = 2f ? f^2.
Then,
lat = atan2[(Z + e b sin^3(q)) , (p ? e^2 a cos^3(q))]
lon = atan2 (Y , X)
where
e = e^2 / (1 ? e^2)
b = a(1 ? f)
p = (X^2 + Y^2)^0.5
q = atan2[(Z a) , (p b)]
L is relative to the Greenwich prime meridian.
and
h = (p / cos(P)) ? nu
where
nu is the radius of curvature in the prime vertical at latitude lat = a /(1 ? e^2sin^2(P))^0.5
(see implementation notes in preface for atan2 convention)","For Geographic 3D CRS = WGS 84 (EPSG CRS code 4979)
and
Topocentric origin latitude Po = 55deg N
Topocentric origin longitude Lo = 5 deg E
Topocentric origin ellipsoidal height ho = 200 metres
Ellipsoid parameters: = 6378137.0 metres, 1/f = 298.25722236
First calculate additional ellipsoid parameter e^2 and radius of curvature nuO at the topocentric origin:
e^2=0.006694380
nuO = 0. 6392510.727
Forward calculation for:
Latitude P = 53?48'382""N
Longitude L = 2?07'468""E
Height h = 73.0 metres
nu = 6392088.017
then
U = ?189 013.869 m
V = ?128 642.040 m
W = ? 4 220.171 m
Reverse calculation for:
U = ?189 013.869 m
V = ?128 642.040 m
W = ? 4 220.171 m
First calculate additional ellipsoid parameter e^2 and radius of curvature nuO at the topocentric origin:
e^2 = 0.006694380
nuO = 6392510.727
then the following intermediate terms:
Xo = 3652755.306
Yo = 319574.680
Zo = 5201547.353
X = 3771793.968
Y = 140253.342
Z = 5124304.349
eta = 0.006739496674
b = 6356752.314
p = 3774400.712
q = .937549875
P = 0.9391511015 rad
L = 0.0371676591 rad
nu = 6392088.017
for a final result of:
Latitude P = 53?48'33.820""N
Longitude L = 2?07'46.380""E
Height h = 73.0 metres",,"EPSG guidance note #7-2, http://www.epsg.org","IOGP",8/29/2018 0:00:00,"2017.018 2017.024",0
9838,"Vertical Perspective",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
This general case deals with a viewing point at a finite height hv above the origin. If the viewing point is at infinity, the formulas for the orthographic case (method code 9839) should be used.
The forward equations for the Vertical Perspective to convert geographical 3D coordinates (lat, lon, h) to Easting (E) and Northing (N) begin with the formulas of thje Geographic/topocentric conversions method (code 9836) to convert the geographical coordinates to topocentric coordinates U, V, W. The perspective projection origin is coincident with the topographic origin and has coordinates (latO, lonO, hO).
U = (nu + h) cos P sin (L ? Lo)
V = (nu + h) [sin P cos Po ? cos P sin Po cos (L ? Lo)] + e^2 (nuO sin Po ? nu sin P ) cos Po
W = (nu + h) [sin P sin Po + cos P cos Po cos (L ? Lo)] + e^2 (nuO sin Po ? nu sin P ) sin Po ? (nuO ? ho)
Then, given the height hv of the perspective viewing point above the origin, the perspective coordinates (E, N) are calculated from topocentric coordinates (U, V, W) as:
E = U * hv / (hv ? W)
N = V * hv / (hv ? W)
The reverse calculation from E,N to U,V,W and lat,lon,h is indeterminate.","See EPSG Guidance Note 7-2.","For a viewing point height approaching or at infinity, see the Vertical Perspective (orthographic case) (method code 9839).","EPSG guidance note #7-2, http://www.epsg.org","IOGP",6/13/2017 0:00:00,"2017.018",0
9839,"Vertical Perspective (Orthographic case)",0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The orthographic vertical perspective is a special case of the vertical perspective with the viewing point (hv) at infinity. Therefore, all projection ""rays"" are parallel to one another and all are perpendicular to the tangent plane. Since the rays are parallel, coordinates in the tangent-plane are the same in any other parallel mapping plane, i.e. are consistent for any value of ho, which therefore becomes irrelevant to the forward formulas.
The orthographic vertical perspective forward conversion from 3D geographic coordinates latitude, longitude and ellipsoidal height (lat, lon, h) to Easting (E) and Northing (N) is given by:
E = U = limit (U hv / (hv ? W), hv -> infinity)
N = V = limit (V hv / (hv ? W), hv -> infinity)
where, as in Sections 2.2.3 and 1.3.17.2:
U = (* + h) cos * sin (* ? *O)
V = (* + h) [sin * cos *O ? cos * sin *O cos (* ? *O)] + e2 (*O sin *O ? * sin * ) cos *O
The reverse calculation from E,N to U,V,W and *,*,h is indeterminate.","See EPSG Guidance Note 7-2.","This is a special case of the general Vertical Perspective (method code 9838) in which the viewing point at infinity.","EPSG guidance note #7-2, http://www.epsg.org","OGP",11/1/2007 0:00:00,,0
9840,"Orthographic",1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The Orthographic Projection forward conversion from 2D geographic coordinates latitude and longitude (lat, lon) and the origin on the ellipsoid (latO, lonO) is given by:
E = FE + nu cos(lat) sin (lon ? lonO)
N = FN + nu [sin(lat) cos(latO) ? cos(lat) sin(latO) cos (lon ? lonO)] + e^2 (nuO sin(latO) ? nu sin(lat)) cos(latO)
where
nu is the prime vertical radius of curvature at latitude lat; nu = a /(1 ? e^2 sin^2(lat))^0.5,
nuO is the prime vertical radius of curvature at latitude of origin latO; nuO = a /(1 ? e^2 sin^2(latO)^0.5,
e is the eccentricity of the ellipsoid and e^2 = (a^2 ? b^2)/a^2 = 2f ? f^2
a and b are the ellipsoidal semi-major and semi-minor axes,
1/f is the inverse flattening, and
the latitude and longitude of the projection origin are latO and lonO.
These formulas are similar to those for the orthographic case of the vertical perspective (method code 9839) except that, for the Orthographic Projection given here, h = 0 and the term (nu + h) reduces to nu. The projection origin is at the topocentric system origin latO, lonO with false origin coordinates FE and FN.
For the reverse formulas for latitude and longitude corresponding to a given Easting (E) and Northing (N), iteration is required as the prime vertical radius (nu) is a function of latitude.
Begin by seeding the iteration with the center of projection (or some better guess):
lat = latO
lon = lonO
Enter the iteration here with the (next) best estimates of lat and lon. Then solve for the radii of curvature in the prime vertical (nu) and meridian (rho):
nu = a / (1 ? e^2 sin^2(lat))^0.5
rho = a (1 ? e^2) / (1 ? e^2 sin^2(lat))^1.5
Compute test values of E and N (E' and N') using the forward equations:
E' = FE + nu cos(lat) sin(lon ? lonO)
N' = FN + nu [sin(lat) cos(latO) ? cos(lat) sin(latO) cos(lon ? lonO)] + e^2 (nuO sin(latO) ? nu sin(lat) ) cos(latO)
Partially differentiate the forward equations to solve for the elements of the Jacobian matrix:
J11 = dE/dlat = ? rho sin(lat) sin(lon ? lonO)
J12 = dE/dlon = nu cos(lat) cos(lon ? lonO)
J21 = dN/dlat = rho [cos(lat) cos(latO) + sin(lat) sin(latO) cos(lon ? lonO)]
J22 = dN/dlon = nu sin(latO) cos(lat) sin (lon ? lonO)
Solve for the determinant of the Jacobian:
D = J11 J22 ? J12 J21
Solve the northerly and easterly differences this iteration:
dE = E ? E'
dN= N ? N'
Adjust the latitude and longitude for the next iteration by inverting the Jacobian and multiplying by the differences:
lat = lat + (J22 dE ? J12 dN) / D
lon = lon + (?J21 dE + J11 dN) / D
Return to the entry point with new estimates of latitude and longitude and iterate until the change in lat and lon is not significant.","See EPSG Guidance Note 7-2.","If the natural origin of the projection is at the topocentric origin, this is a special case of the Vertical Perspective (orthographic case) (method code 9839) in which the ellipsoid height of all mapped points is zero (h = 0).","EPSG guidance note #7-2, http://www.epsg.org","OGP",11/1/2007 0:00:00,,0
9841,"Mercator (1SP) (Spherical)",1,"See information source.","See information source.",,"USGS Professional Paper 1395, ""Map Projections - A Working Manual"" by John P. Snyder.","OGP",3/14/2008 0:00:00,"2008.114",1
9842,"Equidistant Cylindrical",1,"See EPSG Guidance Note #7-2.",,"See method code 9823 for spherical development. See also Pseudo Plate Carree, method code 9825.","ESRI","OGP",3/14/2008 0:00:00,"2009.023",1
9843,"Axis Order Reversal (2D)",1,"The axis order reversal operates on coordinates of a point whose order change is described by the following matrix operation:
[Derived CRS 1st coordinate ] = [ 0 1 ] * [Base CRS 1st coordinate ]
[Derived CRS 2nd coordinate] [ 1 0 ] [Base CRS 2nd coordinate]","(none)","This is a parameter-less conversion to reverse the order of the axes of a 2D CRS.","OGP","IOGP",10/16/2017 0:00:00,"2015.007 2015.018 2014.027",0
9844,"Axis Order Reversal (Geographic3D horizontal)",1,"The axis order change operates on coordinates of a point whose order change is described by the following matrix operation:
[Derived CRS 1st coordinate] [ 0 1 0 ] [Base CRS 1st coordinate]
[Derived CRS 2nd coordinate] = [ 1 0 0 ] * [Base CRS 2nd coordinate]
[Derived CRS 3rd coordinate] [ 0 0 1 ] [Base CRS 3rd coordinate]","(none)","This is a parameter-less conversion to change the order of horizontal coordinates of a geographic 3D CRS.","OGP","IOGP",10/16/2017 0:00:00,"2015.018 2014.027",0
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