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/**
* Implementation of the net.sf.geographiclib.Geodesic class
*
* Copyright (c) Charles Karney (2013-2017) and licensed
* under the MIT/X11 License. For more information, see
* http://geographiclib.sourceforge.net/
**********************************************************************/
package net.sf.geographiclib;
/**
* Geodesic calculations.
*
* The shortest path between two points on a ellipsoid at (lat1,
* lon1) and (lat2, lon2) is called the geodesic. Its
* length is s12 and the geodesic from point 1 to point 2 has azimuths
* azi1 and azi2 at the two end points. (The azimuth is the
* heading measured clockwise from north. azi2 is the "forward"
* azimuth, i.e., the heading that takes you beyond point 2 not back to point
* 1.)
*
* Given lat1, lon1, azi1, and s12, we can
* determine lat2, lon2, and azi2. This is the
* direct geodesic problem and its solution is given by the function
* {@link #Direct Direct}. (If s12 is sufficiently large that the
* geodesic wraps more than halfway around the earth, there will be another
* geodesic between the points with a smaller s12.)
*
* Given lat1, lon1, lat2, and lon2, we can
* determine azi1, azi2, and s12. This is the
* inverse geodesic problem, whose solution is given by {@link #Inverse
* Inverse}. Usually, the solution to the inverse problem is unique. In cases
* where there are multiple solutions (all with the same s12, of
* course), all the solutions can be easily generated once a particular
* solution is provided.
*
* The standard way of specifying the direct problem is the specify the
* distance s12 to the second point. However it is sometimes useful
* instead to specify the arc length a12 (in degrees) on the auxiliary
* sphere. This is a mathematical construct used in solving the geodesic
* problems. The solution of the direct problem in this form is provided by
* {@link #ArcDirect ArcDirect}. An arc length in excess of 180° indicates
* that the geodesic is not a shortest path. In addition, the arc length
* between an equatorial crossing and the next extremum of latitude for a
* geodesic is 90°.
*
* This class can also calculate several other quantities related to
* geodesics. These are:
*
* -
* reduced length. If we fix the first point and increase
* azi1 by dazi1 (radians), the second point is displaced
* m12 dazi1 in the direction azi2 + 90°. The
* quantity m12 is called the "reduced length" and is symmetric under
* interchange of the two points. On a curved surface the reduced length
* obeys a symmetry relation, m12 + m21 = 0. On a flat
* surface, we have m12 = s12. The ratio s12/m12
* gives the azimuthal scale for an azimuthal equidistant projection.
*
-
* geodesic scale. Consider a reference geodesic and a second
* geodesic parallel to this one at point 1 and separated by a small distance
* dt. The separation of the two geodesics at point 2 is M12
* dt where M12 is called the "geodesic scale". M21 is
* defined similarly (with the geodesics being parallel at point 2). On a
* flat surface, we have M12 = M21 = 1. The quantity
* 1/M12 gives the scale of the Cassini-Soldner projection.
*
-
* area. The area between the geodesic from point 1 to point 2 and
* the equation is represented by S12; it is the area, measured
* counter-clockwise, of the geodesic quadrilateral with corners
* (lat1,lon1), (0,lon1), (0,lon2), and
* (lat2,lon2). It can be used to compute the area of any
* simple geodesic polygon.
*
*
* The quantities m12, M12, M21 which all specify the
* behavior of nearby geodesics obey addition rules. If points 1, 2, and 3 all
* lie on a single geodesic, then the following rules hold:
*
* -
* s13 = s12 + s23
*
-
* a13 = a12 + a23
*
-
* S13 = S12 + S23
*
-
* m13 = m12 M23 + m23 M21
*
-
* M13 = M12 M23 − (1 − M12
* M21) m23 / m12
*
-
* M31 = M32 M21 − (1 − M23
* M32) m12 / m23
*
*
* The results of the geodesic calculations are bundled up into a {@link
* GeodesicData} object which includes the input parameters and all the
* computed results, i.e., lat1, lon1, azi1, lat2,
* lon2, azi2, s12, a12, m12, M12,
* M21, S12.
*
* The functions {@link #Direct(double, double, double, double, int) Direct},
* {@link #ArcDirect(double, double, double, double, int) ArcDirect}, and
* {@link #Inverse(double, double, double, double, int) Inverse} include an
* optional final argument outmask which allows you specify which
* results should be computed and returned. If you omit outmask, then
* the "standard" geodesic results are computed (latitudes, longitudes,
* azimuths, and distance). outmask is bitor'ed combination of {@link
* GeodesicMask} values. For example, if you wish just to compute the distance
* between two points you would call, e.g.,
*
* {@code
* GeodesicData g = Geodesic.WGS84.Inverse(lat1, lon1, lat2, lon2,
* GeodesicMask.DISTANCE); }
*
* Additional functionality is provided by the {@link GeodesicLine} class,
* which allows a sequence of points along a geodesic to be computed.
*
* The shortest distance returned by the solution of the inverse problem is
* (obviously) uniquely defined. However, in a few special cases there are
* multiple azimuths which yield the same shortest distance. Here is a
* catalog of those cases:
*
* -
* lat1 = −lat2 (with neither point at a pole). If
* azi1 = azi2, the geodesic is unique. Otherwise there are
* two geodesics and the second one is obtained by setting [azi1,
* azi2] → [azi2, azi1], [M12, M21]
* → [M21, M12], S12 → −S12.
* (This occurs when the longitude difference is near ±180° for
* oblate ellipsoids.)
*
-
* lon2 = lon1 ± 180° (with neither point at a
* pole). If azi1 = 0° or ±180°, the geodesic is
* unique. Otherwise there are two geodesics and the second one is obtained
* by setting [ azi1, azi2] → [−azi1,
* −azi2], S12 → − S12. (This occurs
* when lat2 is near −lat1 for prolate ellipsoids.)
*
-
* Points 1 and 2 at opposite poles. There are infinitely many geodesics
* which can be generated by setting [azi1, azi2] →
* [azi1, azi2] + [d, −d], for arbitrary
* d. (For spheres, this prescription applies when points 1 and 2 are
* antipodal.)
*
-
* s12 = 0 (coincident points). There are infinitely many geodesics which
* can be generated by setting [azi1, azi2] →
* [azi1, azi2] + [d, d], for arbitrary d.
*
*
* The calculations are accurate to better than 15 nm (15 nanometers) for the
* WGS84 ellipsoid. See Sec. 9 of
* arXiv:1102.1215v1 for
* details. The algorithms used by this class are based on series expansions
* using the flattening f as a small parameter. These are only accurate
* for |f| < 0.02; however reasonably accurate results will be
* obtained for |f| < 0.2. Here is a table of the approximate
* maximum error (expressed as a distance) for an ellipsoid with the same
* equatorial radius as the WGS84 ellipsoid and different values of the
* flattening.
* |f| error
* 0.01 25 nm
* 0.02 30 nm
* 0.05 10 um
* 0.1 1.5 mm
* 0.2 300 mm
*
* The algorithms are described in
*
* - C. F. F. Karney,
*
* Algorithms for geodesics,
* J. Geodesy 87, 43–55 (2013)
* (addenda).
*
*
* Example of use:
*
* {@code
* // Solve the direct geodesic problem.
*
* // This program reads in lines with lat1, lon1, azi1, s12 and prints
* // out lines with lat2, lon2, azi2 (for the WGS84 ellipsoid).
*
* import java.util.*;
* import net.sf.geographiclib.*;
* public class Direct {
* public static void main(String[] args) {
* try {
* Scanner in = new Scanner(System.in);
* double lat1, lon1, azi1, s12;
* while (true) {
* lat1 = in.nextDouble(); lon1 = in.nextDouble();
* azi1 = in.nextDouble(); s12 = in.nextDouble();
* GeodesicData g = Geodesic.WGS84.Direct(lat1, lon1, azi1, s12);
* System.out.println(g.lat2 + " " + g.lon2 + " " + g.azi2);
* }
* }
* catch (Exception e) {}
* }
* }}
**********************************************************************/
public class Geodesic {
/**
* The order of the expansions used by Geodesic.
**********************************************************************/
protected static final int GEOGRAPHICLIB_GEODESIC_ORDER = 6;
protected static final int nA1_ = GEOGRAPHICLIB_GEODESIC_ORDER;
protected static final int nC1_ = GEOGRAPHICLIB_GEODESIC_ORDER;
protected static final int nC1p_ = GEOGRAPHICLIB_GEODESIC_ORDER;
protected static final int nA2_ = GEOGRAPHICLIB_GEODESIC_ORDER;
protected static final int nC2_ = GEOGRAPHICLIB_GEODESIC_ORDER;
protected static final int nA3_ = GEOGRAPHICLIB_GEODESIC_ORDER;
protected static final int nA3x_ = nA3_;
protected static final int nC3_ = GEOGRAPHICLIB_GEODESIC_ORDER;
protected static final int nC3x_ = (nC3_ * (nC3_ - 1)) / 2;
protected static final int nC4_ = GEOGRAPHICLIB_GEODESIC_ORDER;
protected static final int nC4x_ = (nC4_ * (nC4_ + 1)) / 2;
private static final int maxit1_ = 20;
private static final int maxit2_ = maxit1_ + GeoMath.digits + 10;
// Underflow guard. We require
// tiny_ * epsilon() > 0
// tiny_ + epsilon() == epsilon()
protected static final double tiny_ = Math.sqrt(GeoMath.min);
private static final double tol0_ = GeoMath.epsilon;
// Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse case
// 52.784459512564 0 -52.784459512563990912 179.634407464943777557
// which otherwise failed for Visual Studio 10 (Release and Debug)
private static final double tol1_ = 200 * tol0_;
private static final double tol2_ = Math.sqrt(tol0_);
// Check on bisection interval
private static final double tolb_ = tol0_ * tol2_;
private static final double xthresh_ = 1000 * tol2_;
protected double _a, _f, _f1, _e2, _ep2, _b, _c2;
private double _n, _etol2;
private double _A3x[], _C3x[], _C4x[];
/**
* Constructor for a ellipsoid with
*
* @param a equatorial radius (meters).
* @param f flattening of ellipsoid. Setting f = 0 gives a sphere.
* Negative f gives a prolate ellipsoid.
* @exception GeographicErr if a or (1 − f ) a is
* not positive.
**********************************************************************/
public Geodesic(double a, double f) {
_a = a;
_f = f;
_f1 = 1 - _f;
_e2 = _f * (2 - _f);
_ep2 = _e2 / GeoMath.sq(_f1); // e2 / (1 - e2)
_n = _f / ( 2 - _f);
_b = _a * _f1;
_c2 = (GeoMath.sq(_a) + GeoMath.sq(_b) *
(_e2 == 0 ? 1 :
(_e2 > 0 ? GeoMath.atanh(Math.sqrt(_e2)) :
Math.atan(Math.sqrt(-_e2))) /
Math.sqrt(Math.abs(_e2))))/2; // authalic radius squared
// The sig12 threshold for "really short". Using the auxiliary sphere
// solution with dnm computed at (bet1 + bet2) / 2, the relative error in
// the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
// (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. For a
// given f and sig12, the max error occurs for lines near the pole. If
// the old rule for computing dnm = (dn1 + dn2)/2 is used, then the error
// increases by a factor of 2.) Setting this equal to epsilon gives
// sig12 = etol2. Here 0.1 is a safety factor (error decreased by 100)
// and max(0.001, abs(f)) stops etol2 getting too large in the nearly
// spherical case.
_etol2 = 0.1 * tol2_ /
Math.sqrt( Math.max(0.001, Math.abs(_f)) *
Math.min(1.0, 1 - _f/2) / 2 );
if (!(GeoMath.isfinite(_a) && _a > 0))
throw new GeographicErr("Equatorial radius is not positive");
if (!(GeoMath.isfinite(_b) && _b > 0))
throw new GeographicErr("Polar semi-axis is not positive");
_A3x = new double[nA3x_];
_C3x = new double[nC3x_];
_C4x = new double[nC4x_];
A3coeff();
C3coeff();
C4coeff();
}
/**
* Solve the direct geodesic problem where the length of the geodesic
* is specified in terms of distance.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param s12 distance between point 1 and point 2 (meters); it can be
* negative.
* @return a {@link GeodesicData} object with the following fields:
* lat1, lon1, azi1, lat2, lon2,
* azi2, s12, a12.
*
* lat1 should be in the range [−90°, 90°]. The values
* of lon2 and azi2 returned are in the range [−180°,
* 180°).
*
* If either point is at a pole, the azimuth is defined by keeping the
* longitude fixed, writing lat = ±(90° − ε),
* and taking the limit ε → 0+. An arc length greater that
* 180° signifies a geodesic which is not a shortest path. (For a
* prolate ellipsoid, an additional condition is necessary for a shortest
* path: the longitudinal extent must not exceed of 180°.)
**********************************************************************/
public GeodesicData Direct(double lat1, double lon1,
double azi1, double s12) {
return Direct(lat1, lon1, azi1, false, s12, GeodesicMask.STANDARD);
}
/**
* Solve the direct geodesic problem where the length of the geodesic is
* specified in terms of distance and with a subset of the geodesic results
* returned.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param s12 distance between point 1 and point 2 (meters); it can be
* negative.
* @param outmask a bitor'ed combination of {@link GeodesicMask} values
* specifying which results should be returned.
* @return a {@link GeodesicData} object with the fields specified by
* outmask computed.
*
* lat1, lon1, azi1, s12, and a12 are
* always included in the returned result. The value of lon2 returned
* is in the range [−180°, 180°), unless the outmask
* includes the {@link GeodesicMask#LONG_UNROLL} flag.
**********************************************************************/
public GeodesicData Direct(double lat1, double lon1,
double azi1, double s12, int outmask) {
return Direct(lat1, lon1, azi1, false, s12, outmask);
}
/**
* Solve the direct geodesic problem where the length of the geodesic
* is specified in terms of arc length.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param a12 arc length between point 1 and point 2 (degrees); it can
* be negative.
* @return a {@link GeodesicData} object with the following fields:
* lat1, lon1, azi1, lat2, lon2,
* azi2, s12, a12.
*
* lat1 should be in the range [−90°, 90°]. The values
* of lon2 and azi2 returned are in the range [−180°,
* 180°).
*
* If either point is at a pole, the azimuth is defined by keeping the
* longitude fixed, writing lat = ±(90° − ε),
* and taking the limit ε → 0+. An arc length greater that
* 180° signifies a geodesic which is not a shortest path. (For a
* prolate ellipsoid, an additional condition is necessary for a shortest
* path: the longitudinal extent must not exceed of 180°.)
**********************************************************************/
public GeodesicData ArcDirect(double lat1, double lon1,
double azi1, double a12) {
return Direct(lat1, lon1, azi1, true, a12, GeodesicMask.STANDARD);
}
/**
* Solve the direct geodesic problem where the length of the geodesic is
* specified in terms of arc length and with a subset of the geodesic results
* returned.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param a12 arc length between point 1 and point 2 (degrees); it can
* be negative.
* @param outmask a bitor'ed combination of {@link GeodesicMask} values
* specifying which results should be returned.
* @return a {@link GeodesicData} object with the fields specified by
* outmask computed.
*
* lat1, lon1, azi1, and a12 are always included
* in the returned result. The value of lon2 returned is in the range
* [−180°, 180°), unless the outmask includes the {@link
* GeodesicMask#LONG_UNROLL} flag.
**********************************************************************/
public GeodesicData ArcDirect(double lat1, double lon1,
double azi1, double a12, int outmask) {
return Direct(lat1, lon1, azi1, true, a12, outmask);
}
/**
* The general direct geodesic problem. {@link #Direct Direct} and
* {@link #ArcDirect ArcDirect} are defined in terms of this function.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param arcmode boolean flag determining the meaning of the
* s12_a12.
* @param s12_a12 if arcmode is false, this is the distance between
* point 1 and point 2 (meters); otherwise it is the arc length between
* point 1 and point 2 (degrees); it can be negative.
* @param outmask a bitor'ed combination of {@link GeodesicMask} values
* specifying which results should be returned.
* @return a {@link GeodesicData} object with the fields specified by
* outmask computed.
*
* The {@link GeodesicMask} values possible for outmask are
*
* -
* outmask |= {@link GeodesicMask#LATITUDE} for the latitude
* lat2;
*
-
* outmask |= {@link GeodesicMask#LONGITUDE} for the latitude
* lon2;
*
-
* outmask |= {@link GeodesicMask#AZIMUTH} for the latitude
* azi2;
*
-
* outmask |= {@link GeodesicMask#DISTANCE} for the distance
* s12;
*
-
* outmask |= {@link GeodesicMask#REDUCEDLENGTH} for the reduced
* length m12;
*
-
* outmask |= {@link GeodesicMask#GEODESICSCALE} for the geodesic
* scales M12 and M21;
*
-
* outmask |= {@link GeodesicMask#AREA} for the area S12;
*
-
* outmask |= {@link GeodesicMask#ALL} for all of the above;
*
-
* outmask |= {@link GeodesicMask#LONG_UNROLL}, if set then
* lon1 is unchanged and lon2 − lon1 indicates
* how many times and in what sense the geodesic encircles the ellipsoid.
* Otherwise lon1 and lon2 are both reduced to the range
* [−180°, 180°).
*
*
* The function value a12 is always computed and returned and this
* equals s12_a12 is arcmode is true. If outmask
* includes {@link GeodesicMask#DISTANCE} and arcmode is false, then
* s12 = s12_a12. It is not necessary to include {@link
* GeodesicMask#DISTANCE_IN} in outmask; this is automatically
* included is arcmode is false.
**********************************************************************/
public GeodesicData Direct(double lat1, double lon1, double azi1,
boolean arcmode, double s12_a12, int outmask) {
// Automatically supply DISTANCE_IN if necessary
if (!arcmode) outmask |= GeodesicMask.DISTANCE_IN;
return new GeodesicLine(this, lat1, lon1, azi1, outmask)
. // Note the dot!
Position(arcmode, s12_a12, outmask);
}
/**
* Define a {@link GeodesicLine} in terms of the direct geodesic problem
* specified in terms of distance with all capabilities included.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param s12 distance between point 1 and point 2 (meters); it can be
* negative.
* @return a {@link GeodesicLine} object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the direct geodesic problem.
*
* lat1 should be in the range [−90°, 90°].
**********************************************************************/
public GeodesicLine DirectLine(double lat1, double lon1, double azi1,
double s12) {
return DirectLine(lat1, lon1, azi1, s12, GeodesicMask.ALL);
}
/**
* Define a {@link GeodesicLine} in terms of the direct geodesic problem
* specified in terms of distance with a subset of the capabilities included.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param s12 distance between point 1 and point 2 (meters); it can be
* negative.
* @param caps bitor'ed combination of {@link GeodesicMask} values
* specifying the capabilities the GeodesicLine object should possess,
* i.e., which quantities can be returned in calls to
* {@link GeodesicLine#Position GeodesicLine.Position}.
* @return a {@link GeodesicLine} object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the direct geodesic problem.
*
* lat1 should be in the range [−90°, 90°].
**********************************************************************/
public GeodesicLine DirectLine(double lat1, double lon1, double azi1,
double s12, int caps) {
return GenDirectLine(lat1, lon1, azi1, false, s12, caps);
}
/**
* Define a {@link GeodesicLine} in terms of the direct geodesic problem
* specified in terms of arc length with all capabilities included.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param a12 arc length between point 1 and point 2 (degrees); it can
* be negative.
* @return a {@link GeodesicLine} object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the direct geodesic problem.
*
* lat1 should be in the range [−90°, 90°].
**********************************************************************/
public GeodesicLine ArcDirectLine(double lat1, double lon1, double azi1,
double a12) {
return ArcDirectLine(lat1, lon1, azi1, a12, GeodesicMask.ALL);
}
/**
* Define a {@link GeodesicLine} in terms of the direct geodesic problem
* specified in terms of arc length with a subset of the capabilities
* included.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param a12 arc length between point 1 and point 2 (degrees); it can
* be negative.
* @param caps bitor'ed combination of {@link GeodesicMask} values
* specifying the capabilities the GeodesicLine object should possess,
* i.e., which quantities can be returned in calls to
* {@link GeodesicLine#Position GeodesicLine.Position}.
* @return a {@link GeodesicLine} object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the direct geodesic problem.
*
* lat1 should be in the range [−90°, 90°].
**********************************************************************/
public GeodesicLine ArcDirectLine(double lat1, double lon1, double azi1,
double a12, int caps) {
return GenDirectLine(lat1, lon1, azi1, true, a12, caps);
}
/**
* Define a {@link GeodesicLine} in terms of the direct geodesic problem
* specified in terms of either distance or arc length with a subset of the
* capabilities included.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param arcmode boolean flag determining the meaning of the s12_a12.
* @param s12_a12 if arcmode is false, this is the distance between
* point 1 and point 2 (meters); otherwise it is the arc length between
* point 1 and point 2 (degrees); it can be negative.
* @param caps bitor'ed combination of {@link GeodesicMask} values
* specifying the capabilities the GeodesicLine object should possess,
* i.e., which quantities can be returned in calls to
* {@link GeodesicLine#Position GeodesicLine.Position}.
* @return a {@link GeodesicLine} object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the direct geodesic problem.
*
* lat1 should be in the range [−90°, 90°].
**********************************************************************/
public GeodesicLine GenDirectLine(double lat1, double lon1, double azi1,
boolean arcmode, double s12_a12, int caps) {
azi1 = GeoMath.AngNormalize(azi1);
double salp1, calp1;
// Guard against underflow in salp0. Also -0 is converted to +0.
{ Pair p = GeoMath.sincosd(GeoMath.AngRound(azi1));
salp1 = p.first; calp1 = p.second; }
// Automatically supply DISTANCE_IN if necessary
if (!arcmode) caps |= GeodesicMask.DISTANCE_IN;
return new GeodesicLine(this, lat1, lon1, azi1, salp1, calp1,
caps, arcmode, s12_a12);
}
/**
* Solve the inverse geodesic problem.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param lat2 latitude of point 2 (degrees).
* @param lon2 longitude of point 2 (degrees).
* @return a {@link GeodesicData} object with the following fields:
* lat1, lon1, azi1, lat2, lon2,
* azi2, s12, a12.
*
* lat1 and lat2 should be in the range [−90°,
* 90°]. The values of azi1 and azi2 returned are in the
* range [−180°, 180°).
*
* If either point is at a pole, the azimuth is defined by keeping the
* longitude fixed, writing lat = ±(90° − ε),
* taking the limit ε → 0+.
*
* The solution to the inverse problem is found using Newton's method. If
* this fails to converge (this is very unlikely in geodetic applications
* but does occur for very eccentric ellipsoids), then the bisection method
* is used to refine the solution.
**********************************************************************/
public GeodesicData Inverse(double lat1, double lon1,
double lat2, double lon2) {
return Inverse(lat1, lon1, lat2, lon2, GeodesicMask.STANDARD);
}
private class InverseData {
private GeodesicData g;
private double salp1, calp1, salp2, calp2;
private InverseData() {
g = new GeodesicData();
salp1 = calp1 = salp2 = calp2 = Double.NaN;
}
}
private InverseData InverseInt(double lat1, double lon1,
double lat2, double lon2, int outmask) {
InverseData result = new InverseData();
GeodesicData r = result.g;
// Compute longitude difference (AngDiff does this carefully). Result is
// in [-180, 180] but -180 is only for west-going geodesics. 180 is for
// east-going and meridional geodesics.
r.lat1 = lat1 = GeoMath.LatFix(lat1); r.lat2 = lat2 = GeoMath.LatFix(lat2);
// If really close to the equator, treat as on equator.
lat1 = GeoMath.AngRound(lat1);
lat2 = GeoMath.AngRound(lat2);
double lon12, lon12s;
{
Pair p = GeoMath.AngDiff(lon1, lon2);
lon12 = p.first; lon12s = p.second;
}
if ((outmask & GeodesicMask.LONG_UNROLL) != 0) {
r.lon1 = lon1; r.lon2 = (lon1 + lon12) + lon12s;
} else {
r.lon1 = GeoMath.AngNormalize(lon1); r.lon2 = GeoMath.AngNormalize(lon2);
}
// Make longitude difference positive.
int lonsign = lon12 >= 0 ? 1 : -1;
// If very close to being on the same half-meridian, then make it so.
lon12 = lonsign * GeoMath.AngRound(lon12);
lon12s = GeoMath.AngRound((180 - lon12) - lonsign * lon12s);
double
lam12 = Math.toRadians(lon12), slam12, clam12;
{ Pair p = GeoMath.sincosd(lon12 > 90 ? lon12s : lon12);
slam12 = p.first; clam12 = (lon12 > 90 ? -1 : 1) * p.second; }
// Swap points so that point with higher (abs) latitude is point 1
// If one latitude is a nan, then it becomes lat1.
int swapp = Math.abs(lat1) < Math.abs(lat2) ? -1 : 1;
if (swapp < 0) {
lonsign *= -1;
{ double t = lat1; lat1 = lat2; lat2 = t; }
}
// Make lat1 <= 0
int latsign = lat1 < 0 ? 1 : -1;
lat1 *= latsign;
lat2 *= latsign;
// Now we have
//
// 0 <= lon12 <= 180
// -90 <= lat1 <= 0
// lat1 <= lat2 <= -lat1
//
// longsign, swapp, latsign register the transformation to bring the
// coordinates to this canonical form. In all cases, 1 means no change was
// made. We make these transformations so that there are few cases to
// check, e.g., on verifying quadrants in atan2. In addition, this
// enforces some symmetries in the results returned.
double sbet1, cbet1, sbet2, cbet2, s12x, m12x;
s12x = m12x = Double.NaN;
{ Pair p = GeoMath.sincosd(lat1); sbet1 = _f1 * p.first; cbet1 = p.second; }
// Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12
// will be <= 2*tiny for two points at the same pole.
{ Pair p = GeoMath.norm(sbet1, cbet1); sbet1 = p.first; cbet1 = p.second; }
cbet1 = Math.max(tiny_, cbet1);
{ Pair p = GeoMath.sincosd(lat2); sbet2 = _f1 * p.first; cbet2 = p.second; }
// Ensure cbet2 = +epsilon at poles
{ Pair p = GeoMath.norm(sbet2, cbet2); sbet2 = p.first; cbet2 = p.second; }
cbet2 = Math.max(tiny_, cbet2);
// If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
// |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
// a better measure. This logic is used in assigning calp2 in Lambda12.
// Sometimes these quantities vanish and in that case we force bet2 = +/-
// bet1 exactly. An example where is is necessary is the inverse problem
// 48.522876735459 0 -48.52287673545898293 179.599720456223079643
// which failed with Visual Studio 10 (Release and Debug)
if (cbet1 < -sbet1) {
if (cbet2 == cbet1)
sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
} else {
if (Math.abs(sbet2) == -sbet1)
cbet2 = cbet1;
}
double
dn1 = Math.sqrt(1 + _ep2 * GeoMath.sq(sbet1)),
dn2 = Math.sqrt(1 + _ep2 * GeoMath.sq(sbet2));
double a12, sig12, calp1, salp1, calp2, salp2;
a12 = sig12 = calp1 = salp1 = calp2 = salp2 = Double.NaN;
// index zero elements of these arrays are unused
double C1a[] = new double[nC1_ + 1];
double C2a[] = new double[nC2_ + 1];
double C3a[] = new double[nC3_];
boolean meridian = lat1 == -90 || slam12 == 0;
if (meridian) {
// Endpoints are on a single full meridian, so the geodesic might lie on
// a meridian.
calp1 = clam12; salp1 = slam12; // Head to the target longitude
calp2 = 1; salp2 = 0; // At the target we're heading north
double
// tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1, csig1 = calp1 * cbet1,
ssig2 = sbet2, csig2 = calp2 * cbet2;
// sig12 = sig2 - sig1
sig12 = Math.atan2(Math.max(0.0, csig1 * ssig2 - ssig1 * csig2),
csig1 * csig2 + ssig1 * ssig2);
{
LengthsV v =
Lengths(_n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
outmask | GeodesicMask.DISTANCE | GeodesicMask.REDUCEDLENGTH,
C1a, C2a);
s12x = v.s12b; m12x = v.m12b;
if ((outmask & GeodesicMask.GEODESICSCALE) != 0) {
r.M12 = v.M12; r.M21 = v.M21;
}
}
// Add the check for sig12 since zero length geodesics might yield m12 <
// 0. Test case was
//
// echo 20.001 0 20.001 0 | GeodSolve -i
//
// In fact, we will have sig12 > pi/2 for meridional geodesic which is
// not a shortest path.
if (sig12 < 1 || m12x >= 0) {
// Need at least 2, to handle 90 0 90 180
if (sig12 < 3 * tiny_)
sig12 = m12x = s12x = 0;
m12x *= _b;
s12x *= _b;
a12 = Math.toDegrees(sig12);
} else
// m12 < 0, i.e., prolate and too close to anti-podal
meridian = false;
}
double omg12 = Double.NaN, somg12 = 2, comg12 = Double.NaN;
if (!meridian &&
sbet1 == 0 && // and sbet2 == 0
// Mimic the way Lambda12 works with calp1 = 0
(_f <= 0 || lon12s >= _f * 180)) {
// Geodesic runs along equator
calp1 = calp2 = 0; salp1 = salp2 = 1;
s12x = _a * lam12;
sig12 = omg12 = lam12 / _f1;
m12x = _b * Math.sin(sig12);
if ((outmask & GeodesicMask.GEODESICSCALE) != 0)
r.M12 = r.M21 = Math.cos(sig12);
a12 = lon12 / _f1;
} else if (!meridian) {
// Now point1 and point2 belong within a hemisphere bounded by a
// meridian and geodesic is neither meridional or equatorial.
// Figure a starting point for Newton's method
double dnm;
{
InverseStartV v =
InverseStart(sbet1, cbet1, dn1, sbet2, cbet2, dn2,
lam12, slam12, clam12,
C1a, C2a);
sig12 = v.sig12;
salp1 = v.salp1; calp1 = v.calp1;
salp2 = v.salp2; calp2 = v.calp2;
dnm = v.dnm;
}
if (sig12 >= 0) {
// Short lines (InverseStart sets salp2, calp2, dnm)
s12x = sig12 * _b * dnm;
m12x = GeoMath.sq(dnm) * _b * Math.sin(sig12 / dnm);
if ((outmask & GeodesicMask.GEODESICSCALE) != 0)
r.M12 = r.M21 = Math.cos(sig12 / dnm);
a12 = Math.toDegrees(sig12);
omg12 = lam12 / (_f1 * dnm);
} else {
// Newton's method. This is a straightforward solution of f(alp1) =
// lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
// root in the interval (0, pi) and its derivative is positive at the
// root. Thus f(alp) is positive for alp > alp1 and negative for alp <
// alp1. During the course of the iteration, a range (alp1a, alp1b) is
// maintained which brackets the root and with each evaluation of
// f(alp) the range is shrunk, if possible. Newton's method is
// restarted whenever the derivative of f is negative (because the new
// value of alp1 is then further from the solution) or if the new
// estimate of alp1 lies outside (0,pi); in this case, the new starting
// guess is taken to be (alp1a + alp1b) / 2.
double ssig1, csig1, ssig2, csig2, eps, domg12;
ssig1 = csig1 = ssig2 = csig2 = eps = domg12 = Double.NaN;
int numit = 0;
// Bracketing range
double salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
for (boolean tripn = false, tripb = false; numit < maxit2_; ++numit) {
// the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
// WGS84 and random input: mean = 2.85, sd = 0.60
double v, dv;
{
Lambda12V w =
Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
slam12, clam12, numit < maxit1_, C1a, C2a, C3a);
v = w.lam12;
salp2 = w.salp2; calp2 = w.calp2;
sig12 = w.sig12;
ssig1 = w.ssig1; csig1 = w.csig1;
ssig2 = w.ssig2; csig2 = w.csig2;
eps = w.eps; domg12 = w.domg12;
dv = w.dlam12;
}
// 2 * tol0 is approximately 1 ulp for a number in [0, pi].
// Reversed test to allow escape with NaNs
if (tripb || !(Math.abs(v) >= (tripn ? 8 : 1) * tol0_)) break;
// Update bracketing values
if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
{ salp1b = salp1; calp1b = calp1; }
else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
{ salp1a = salp1; calp1a = calp1; }
if (numit < maxit1_ && dv > 0) {
double
dalp1 = -v/dv;
double
sdalp1 = Math.sin(dalp1), cdalp1 = Math.cos(dalp1),
nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
if (nsalp1 > 0 && Math.abs(dalp1) < Math.PI) {
calp1 = calp1 * cdalp1 - salp1 * sdalp1;
salp1 = nsalp1;
{ Pair p = GeoMath.norm(salp1, calp1);
salp1 = p.first; calp1 = p.second; }
// In some regimes we don't get quadratic convergence because
// slope -> 0. So use convergence conditions based on epsilon
// instead of sqrt(epsilon).
tripn = Math.abs(v) <= 16 * tol0_;
continue;
}
}
// Either dv was not postive or updated value was outside legal
// range. Use the midpoint of the bracket as the next estimate.
// This mechanism is not needed for the WGS84 ellipsoid, but it does
// catch problems with more eccentric ellipsoids. Its efficacy is
// such for the WGS84 test set with the starting guess set to alp1 =
// 90deg:
// the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
// WGS84 and random input: mean = 4.74, sd = 0.99
salp1 = (salp1a + salp1b)/2;
calp1 = (calp1a + calp1b)/2;
{ Pair p = GeoMath.norm(salp1, calp1);
salp1 = p.first; calp1 = p.second; }
tripn = false;
tripb = (Math.abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
Math.abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
}
{
// Ensure that the reduced length and geodesic scale are computed in
// a "canonical" way, with the I2 integral.
int lengthmask = outmask |
((outmask &
(GeodesicMask.REDUCEDLENGTH | GeodesicMask.GEODESICSCALE)) != 0 ?
GeodesicMask.DISTANCE : GeodesicMask.NONE);
LengthsV v =
Lengths(eps, sig12,
ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
lengthmask, C1a, C2a);
s12x = v.s12b; m12x = v.m12b;
if ((outmask & GeodesicMask.GEODESICSCALE) != 0) {
r.M12 = v.M12; r.M21 = v.M21;
}
}
m12x *= _b;
s12x *= _b;
a12 = Math.toDegrees(sig12);
if ((outmask & GeodesicMask.AREA) != 0) {
// omg12 = lam12 - domg12
double sdomg12 = Math.sin(domg12), cdomg12 = Math.cos(domg12);
somg12 = slam12 * cdomg12 - clam12 * sdomg12;
comg12 = clam12 * cdomg12 + slam12 * sdomg12;
}
}
}
if ((outmask & GeodesicMask.DISTANCE) != 0)
r.s12 = 0 + s12x; // Convert -0 to 0
if ((outmask & GeodesicMask.REDUCEDLENGTH) != 0)
r.m12 = 0 + m12x; // Convert -0 to 0
if ((outmask & GeodesicMask.AREA) != 0) {
double
// From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1,
calp0 = GeoMath.hypot(calp1, salp1 * sbet1); // calp0 > 0
double alp12;
if (calp0 != 0 && salp0 != 0) {
double
// From Lambda12: tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1, csig1 = calp1 * cbet1,
ssig2 = sbet2, csig2 = calp2 * cbet2,
k2 = GeoMath.sq(calp0) * _ep2,
eps = k2 / (2 * (1 + Math.sqrt(1 + k2)) + k2),
// Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
A4 = GeoMath.sq(_a) * calp0 * salp0 * _e2;
{ Pair p = GeoMath.norm(ssig1, csig1);
ssig1 = p.first; csig1 = p.second; }
{ Pair p = GeoMath.norm(ssig2, csig2);
ssig2 = p.first; csig2 = p.second; }
double C4a[] = new double[nC4_];
C4f(eps, C4a);
double
B41 = SinCosSeries(false, ssig1, csig1, C4a),
B42 = SinCosSeries(false, ssig2, csig2, C4a);
r.S12 = A4 * (B42 - B41);
} else
// Avoid problems with indeterminate sig1, sig2 on equator
r.S12 = 0;
if (!meridian && somg12 > 1) {
somg12 = Math.sin(omg12); comg12 = Math.cos(omg12);
}
if (!meridian &&
comg12 > -0.7071 && // Long difference not too big
sbet2 - sbet1 < 1.75) { // Lat difference not too big
// Use tan(Gamma/2) = tan(omg12/2)
// * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
// with tan(x/2) = sin(x)/(1+cos(x))
double
domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
alp12 = 2 * Math.atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
} else {
// alp12 = alp2 - alp1, used in atan2 so no need to normalize
double
salp12 = salp2 * calp1 - calp2 * salp1,
calp12 = calp2 * calp1 + salp2 * salp1;
// The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
// salp12 = -0 and alp12 = -180. However this depends on the sign
// being attached to 0 correctly. The following ensures the correct
// behavior.
if (salp12 == 0 && calp12 < 0) {
salp12 = tiny_ * calp1;
calp12 = -1;
}
alp12 = Math.atan2(salp12, calp12);
}
r.S12 += _c2 * alp12;
r.S12 *= swapp * lonsign * latsign;
// Convert -0 to 0
r.S12 += 0;
}
// Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
if (swapp < 0) {
{ double t = salp1; salp1 = salp2; salp2 = t; }
{ double t = calp1; calp1 = calp2; calp2 = t; }
if ((outmask & GeodesicMask.GEODESICSCALE) != 0)
{ double t = r.M12; r.M12 = r.M21; r.M21 = t; }
}
salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
// Returned value in [0, 180]
r.a12 = a12;
result.salp1 = salp1; result.calp1 = calp1;
result.salp2 = salp2; result.calp2 = calp2;
return result;
}
/**
* Solve the inverse geodesic problem with a subset of the geodesic results
* returned.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param lat2 latitude of point 2 (degrees).
* @param lon2 longitude of point 2 (degrees).
* @param outmask a bitor'ed combination of {@link GeodesicMask} values
* specifying which results should be returned.
* @return a {@link GeodesicData} object with the fields specified by
* outmask computed.
*
* The {@link GeodesicMask} values possible for outmask are
*
* -
* outmask |= {@link GeodesicMask#DISTANCE} for the distance
* s12;
*
-
* outmask |= {@link GeodesicMask#AZIMUTH} for the latitude
* azi2;
*
-
* outmask |= {@link GeodesicMask#REDUCEDLENGTH} for the reduced
* length m12;
*
-
* outmask |= {@link GeodesicMask#GEODESICSCALE} for the geodesic
* scales M12 and M21;
*
-
* outmask |= {@link GeodesicMask#AREA} for the area S12;
*
-
* outmask |= {@link GeodesicMask#ALL} for all of the above.
*
-
* outmask |= {@link GeodesicMask#LONG_UNROLL}, if set then
* lon1 is unchanged and lon2 − lon1 indicates
* whether the geodesic is east going or west going. Otherwise lon1
* and lon2 are both reduced to the range [−180°,
* 180°).
*
*
* lat1, lon1, lat2, lon2, and a12 are
* always included in the returned result.
**********************************************************************/
public GeodesicData Inverse(double lat1, double lon1,
double lat2, double lon2, int outmask) {
outmask &= GeodesicMask.OUT_MASK;
InverseData result = InverseInt(lat1, lon1, lat2, lon2, outmask);
GeodesicData r = result.g;
if ((outmask & GeodesicMask.AZIMUTH) != 0) {
r.azi1 = GeoMath.atan2d(result.salp1, result.calp1);
r.azi2 = GeoMath.atan2d(result.salp2, result.calp2);
}
return r;
}
/**
* Define a {@link GeodesicLine} in terms of the inverse geodesic problem
* with all capabilities included.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param lat2 latitude of point 2 (degrees).
* @param lon2 longitude of point 2 (degrees).
* @return a {@link GeodesicLine} object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the inverse geodesic problem.
*
* lat1 and lat2 should be in the range [−90°,
* 90°].
**********************************************************************/
public GeodesicLine InverseLine(double lat1, double lon1,
double lat2, double lon2) {
return InverseLine(lat1, lon1, lat2, lon2, GeodesicMask.ALL);
}
/**
* Define a {@link GeodesicLine} in terms of the inverse geodesic problem
* with a subset of the capabilities included.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param lat2 latitude of point 2 (degrees).
* @param lon2 longitude of point 2 (degrees).
* @param caps bitor'ed combination of {@link GeodesicMask} values specifying
* the capabilities the GeodesicLine object should possess, i.e., which
* quantities can be returned in calls to
* {@link GeodesicLine#Position GeodesicLine.Position}.
* @return a {@link GeodesicLine} object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the inverse geodesic problem.
*
* lat1 and lat2 should be in the range [−90°,
* 90°].
**********************************************************************/
public GeodesicLine InverseLine(double lat1, double lon1,
double lat2, double lon2, int caps) {
InverseData result = InverseInt(lat1, lon1, lat2, lon2, 0);
double salp1 = result.salp1, calp1 = result.calp1,
azi1 = GeoMath.atan2d(salp1, calp1), a12 = result.g.a12;
// Ensure that a12 can be converted to a distance
if ((caps & (GeodesicMask.OUT_MASK & GeodesicMask.DISTANCE_IN)) != 0)
caps |= GeodesicMask.DISTANCE;
return new GeodesicLine(this, lat1, lon1, azi1, salp1, calp1, caps,
true, a12);
}
/**
* Set up to compute several points on a single geodesic with all
* capabilities included.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @return a {@link GeodesicLine} object.
*
* lat1 should be in the range [−90°, 90°]. The full
* set of capabilities is included.
*
* If the point is at a pole, the azimuth is defined by keeping the
* lon1 fixed, writing lat1 = ±(90 − ε),
* taking the limit ε → 0+.
**********************************************************************/
public GeodesicLine Line(double lat1, double lon1, double azi1) {
return Line(lat1, lon1, azi1, GeodesicMask.ALL);
}
/**
* Set up to compute several points on a single geodesic with a subset of the
* capabilities included.
*
* @param lat1 latitude of point 1 (degrees).
* @param lon1 longitude of point 1 (degrees).
* @param azi1 azimuth at point 1 (degrees).
* @param caps bitor'ed combination of {@link GeodesicMask} values specifying
* the capabilities the {@link GeodesicLine} object should possess, i.e.,
* which quantities can be returned in calls to {@link
* GeodesicLine#Position GeodesicLine.Position}.
* @return a {@link GeodesicLine} object.
*
* The {@link GeodesicMask} values are
*
* -
* caps |= {@link GeodesicMask#LATITUDE} for the latitude
* lat2; this is added automatically;
*
-
* caps |= {@link GeodesicMask#LONGITUDE} for the latitude
* lon2;
*
-
* caps |= {@link GeodesicMask#AZIMUTH} for the azimuth azi2;
* this is added automatically;
*
-
* caps |= {@link GeodesicMask#DISTANCE} for the distance
* s12;
*
-
* caps |= {@link GeodesicMask#REDUCEDLENGTH} for the reduced length
* m12;
*
-
* caps |= {@link GeodesicMask#GEODESICSCALE} for the geodesic
* scales M12 and M21;
*
-
* caps |= {@link GeodesicMask#AREA} for the area S12;
*
-
* caps |= {@link GeodesicMask#DISTANCE_IN} permits the length of
* the geodesic to be given in terms of s12; without this capability
* the length can only be specified in terms of arc length;
*
-
* caps |= {@link GeodesicMask#ALL} for all of the above.
*
*
* If the point is at a pole, the azimuth is defined by keeping lon1
* fixed, writing lat1 = ±(90 − ε), and taking
* the limit ε → 0+.
**********************************************************************/
public GeodesicLine Line(double lat1, double lon1, double azi1, int caps) {
return new GeodesicLine(this, lat1, lon1, azi1, caps);
}
/**
* @return a the equatorial radius of the ellipsoid (meters). This is
* the value used in the constructor.
**********************************************************************/
public double MajorRadius() { return _a; }
/**
* @return f the flattening of the ellipsoid. This is the
* value used in the constructor.
**********************************************************************/
public double Flattening() { return _f; }
/**
* @return total area of ellipsoid in meters2. The area of a
* polygon encircling a pole can be found by adding EllipsoidArea()/2 to
* the sum of S12 for each side of the polygon.
**********************************************************************/
public double EllipsoidArea() { return 4 * Math.PI * _c2; }
/**
* A global instantiation of Geodesic with the parameters for the WGS84
* ellipsoid.
**********************************************************************/
public static final Geodesic WGS84 =
new Geodesic(Constants.WGS84_a, Constants.WGS84_f);
// This is a reformulation of the geodesic problem. The notation is as
// follows:
// - at a general point (no suffix or 1 or 2 as suffix)
// - phi = latitude
// - beta = latitude on auxiliary sphere
// - omega = longitude on auxiliary sphere
// - lambda = longitude
// - alpha = azimuth of great circle
// - sigma = arc length along great circle
// - s = distance
// - tau = scaled distance (= sigma at multiples of pi/2)
// - at northwards equator crossing
// - beta = phi = 0
// - omega = lambda = 0
// - alpha = alpha0
// - sigma = s = 0
// - a 12 suffix means a difference, e.g., s12 = s2 - s1.
// - s and c prefixes mean sin and cos
protected static double SinCosSeries(boolean sinp,
double sinx, double cosx,
double c[]) {
// Evaluate
// y = sinp ? sum(c[i] * sin( 2*i * x), i, 1, n) :
// sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
// using Clenshaw summation. N.B. c[0] is unused for sin series
// Approx operation count = (n + 5) mult and (2 * n + 2) add
int
k = c.length, // Point to one beyond last element
n = k - (sinp ? 1 : 0);
double
ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
y0 = (n & 1) != 0 ? c[--k] : 0, y1 = 0; // accumulators for sum
// Now n is even
n /= 2;
while (n-- != 0) {
// Unroll loop x 2, so accumulators return to their original role
y1 = ar * y0 - y1 + c[--k];
y0 = ar * y1 - y0 + c[--k];
}
return sinp
? 2 * sinx * cosx * y0 // sin(2 * x) * y0
: cosx * (y0 - y1); // cos(x) * (y0 - y1)
}
private class LengthsV {
private double s12b, m12b, m0, M12, M21;
private LengthsV() {
s12b = m12b = m0 = M12 = M21 = Double.NaN;
}
}
private LengthsV Lengths(double eps, double sig12,
double ssig1, double csig1, double dn1,
double ssig2, double csig2, double dn2,
double cbet1, double cbet2,
int outmask,
// Scratch areas of the right size
double C1a[], double C2a[]) {
// Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
// and m0 = coefficient of secular term in expression for reduced length.
outmask &= GeodesicMask.OUT_MASK;
LengthsV v = new LengthsV(); // To hold s12b, m12b, m0, M12, M21;
double m0x = 0, J12 = 0, A1 = 0, A2 = 0;
if ((outmask & (GeodesicMask.DISTANCE | GeodesicMask.REDUCEDLENGTH |
GeodesicMask.GEODESICSCALE)) != 0) {
A1 = A1m1f(eps);
C1f(eps, C1a);
if ((outmask & (GeodesicMask.REDUCEDLENGTH |
GeodesicMask.GEODESICSCALE)) != 0) {
A2 = A2m1f(eps);
C2f(eps, C2a);
m0x = A1 - A2;
A2 = 1 + A2;
}
A1 = 1 + A1;
}
if ((outmask & GeodesicMask.DISTANCE) != 0) {
double B1 = SinCosSeries(true, ssig2, csig2, C1a) -
SinCosSeries(true, ssig1, csig1, C1a);
// Missing a factor of _b
v.s12b = A1 * (sig12 + B1);
if ((outmask & (GeodesicMask.REDUCEDLENGTH |
GeodesicMask.GEODESICSCALE)) != 0) {
double B2 = SinCosSeries(true, ssig2, csig2, C2a) -
SinCosSeries(true, ssig1, csig1, C2a);
J12 = m0x * sig12 + (A1 * B1 - A2 * B2);
}
} else if ((outmask & (GeodesicMask.REDUCEDLENGTH |
GeodesicMask.GEODESICSCALE)) != 0) {
// Assume here that nC1_ >= nC2_
for (int l = 1; l <= nC2_; ++l)
C2a[l] = A1 * C1a[l] - A2 * C2a[l];
J12 = m0x * sig12 + (SinCosSeries(true, ssig2, csig2, C2a) -
SinCosSeries(true, ssig1, csig1, C2a));
}
if ((outmask & GeodesicMask.REDUCEDLENGTH) != 0) {
v.m0 = m0x;
// Missing a factor of _b.
// Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
// accurate cancellation in the case of coincident points.
v.m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
csig1 * csig2 * J12;
}
if ((outmask & GeodesicMask.GEODESICSCALE) != 0) {
double csig12 = csig1 * csig2 + ssig1 * ssig2;
double t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
v.M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
v.M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
}
return v;
}
private static double Astroid(double x, double y) {
// Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
// This solution is adapted from Geocentric::Reverse.
double k;
double
p = GeoMath.sq(x),
q = GeoMath.sq(y),
r = (p + q - 1) / 6;
if ( !(q == 0 && r <= 0) ) {
double
// Avoid possible division by zero when r = 0 by multiplying equations
// for s and t by r^3 and r, resp.
S = p * q / 4, // S = r^3 * s
r2 = GeoMath.sq(r),
r3 = r * r2,
// The discriminant of the quadratic equation for T3. This is zero on
// the evolute curve p^(1/3)+q^(1/3) = 1
disc = S * (S + 2 * r3);
double u = r;
if (disc >= 0) {
double T3 = S + r3;
// Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
// of precision due to cancellation. The result is unchanged because
// of the way the T is used in definition of u.
T3 += T3 < 0 ? -Math.sqrt(disc) : Math.sqrt(disc); // T3 = (r * t)^3
// N.B. cbrt always returns the double root. cbrt(-8) = -2.
double T = GeoMath.cbrt(T3); // T = r * t
// T can be zero; but then r2 / T -> 0.
u += T + (T != 0 ? r2 / T : 0);
} else {
// T is complex, but the way u is defined the result is double.
double ang = Math.atan2(Math.sqrt(-disc), -(S + r3));
// There are three possible cube roots. We choose the root which
// avoids cancellation. Note that disc < 0 implies that r < 0.
u += 2 * r * Math.cos(ang / 3);
}
double
v = Math.sqrt(GeoMath.sq(u) + q), // guaranteed positive
// Avoid loss of accuracy when u < 0.
uv = u < 0 ? q / (v - u) : u + v, // u+v, guaranteed positive
w = (uv - q) / (2 * v); // positive?
// Rearrange expression for k to avoid loss of accuracy due to
// subtraction. Division by 0 not possible because uv > 0, w >= 0.
k = uv / (Math.sqrt(uv + GeoMath.sq(w)) + w); // guaranteed positive
} else { // q == 0 && r <= 0
// y = 0 with |x| <= 1. Handle this case directly.
// for y small, positive root is k = abs(y)/sqrt(1-x^2)
k = 0;
}
return k;
}
private class InverseStartV {
private double sig12, salp1, calp1,
// Only updated if return val >= 0
salp2, calp2,
// Only updated for short lines
dnm;
private InverseStartV() {
sig12 = salp1 = calp1 = salp2 = calp2 = dnm = Double.NaN;
}
}
private InverseStartV InverseStart(double sbet1, double cbet1, double dn1,
double sbet2, double cbet2, double dn2,
double lam12, double slam12, double clam12,
// Scratch areas of the right size
double C1a[], double C2a[]) {
// Return a starting point for Newton's method in salp1 and calp1 (function
// value is -1). If Newton's method doesn't need to be used, return also
// salp2 and calp2 and function value is sig12.
// To hold sig12, salp1, calp1, salp2, calp2, dnm.
InverseStartV w = new InverseStartV();
w.sig12 = -1; // Return value
double
// bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
double sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
boolean shortline = cbet12 >= 0 && sbet12 < 0.5 &&
cbet2 * lam12 < 0.5;
double somg12, comg12;
if (shortline) {
double sbetm2 = GeoMath.sq(sbet1 + sbet2);
// sin((bet1+bet2)/2)^2
// = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
sbetm2 /= sbetm2 + GeoMath.sq(cbet1 + cbet2);
w.dnm = Math.sqrt(1 + _ep2 * sbetm2);
double omg12 = lam12 / (_f1 * w.dnm);
somg12 = Math.sin(omg12); comg12 = Math.cos(omg12);
} else {
somg12 = slam12; comg12 = clam12;
}
w.salp1 = cbet2 * somg12;
w.calp1 = comg12 >= 0 ?
sbet12 + cbet2 * sbet1 * GeoMath.sq(somg12) / (1 + comg12) :
sbet12a - cbet2 * sbet1 * GeoMath.sq(somg12) / (1 - comg12);
double
ssig12 = GeoMath.hypot(w.salp1, w.calp1),
csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
if (shortline && ssig12 < _etol2) {
// really short lines
w.salp2 = cbet1 * somg12;
w.calp2 = sbet12 - cbet1 * sbet2 *
(comg12 >= 0 ? GeoMath.sq(somg12) / (1 + comg12) : 1 - comg12);
{ Pair p = GeoMath.norm(w.salp2, w.calp2);
w.salp2 = p.first; w.calp2 = p.second; }
// Set return value
w.sig12 = Math.atan2(ssig12, csig12);
} else if (Math.abs(_n) > 0.1 || // Skip astroid calc if too eccentric
csig12 >= 0 ||
ssig12 >= 6 * Math.abs(_n) * Math.PI * GeoMath.sq(cbet1)) {
// Nothing to do, zeroth order spherical approximation is OK
} else {
// Scale lam12 and bet2 to x, y coordinate system where antipodal point
// is at origin and singular point is at y = 0, x = -1.
double y, lamscale, betscale;
// In C++ volatile declaration needed to fix inverse case
// 56.320923501171 0 -56.320923501171 179.664747671772880215
// which otherwise fails with g++ 4.4.4 x86 -O3
double x;
double lam12x = Math.atan2(-slam12, -clam12); // lam12 - pi
if (_f >= 0) { // In fact f == 0 does not get here
// x = dlong, y = dlat
{
double
k2 = GeoMath.sq(sbet1) * _ep2,
eps = k2 / (2 * (1 + Math.sqrt(1 + k2)) + k2);
lamscale = _f * cbet1 * A3f(eps) * Math.PI;
}
betscale = lamscale * cbet1;
x = lam12x / lamscale;
y = sbet12a / betscale;
} else { // _f < 0
// x = dlat, y = dlong
double
cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
bet12a = Math.atan2(sbet12a, cbet12a);
double m12b, m0;
// In the case of lon12 = 180, this repeats a calculation made in
// Inverse.
LengthsV v =
Lengths(_n, Math.PI + bet12a,
sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
cbet1, cbet2, GeodesicMask.REDUCEDLENGTH, C1a, C2a);
m12b = v.m12b; m0 = v.m0;
x = -1 + m12b / (cbet1 * cbet2 * m0 * Math.PI);
betscale = x < -0.01 ? sbet12a / x :
-_f * GeoMath.sq(cbet1) * Math.PI;
lamscale = betscale / cbet1;
y = lam12x / lamscale;
}
if (y > -tol1_ && x > -1 - xthresh_) {
// strip near cut
if (_f >= 0) {
w.salp1 = Math.min(1.0, -x);
w.calp1 = - Math.sqrt(1 - GeoMath.sq(w.salp1));
} else {
w.calp1 = Math.max(x > -tol1_ ? 0.0 : -1.0, x);
w.salp1 = Math.sqrt(1 - GeoMath.sq(w.calp1));
}
} else {
// Estimate alp1, by solving the astroid problem.
//
// Could estimate alpha1 = theta + pi/2, directly, i.e.,
// calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
// calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
//
// However, it's better to estimate omg12 from astroid and use
// spherical formula to compute alp1. This reduces the mean number of
// Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
// (min 0 max 5). The changes in the number of iterations are as
// follows:
//
// change percent
// 1 5
// 0 78
// -1 16
// -2 0.6
// -3 0.04
// -4 0.002
//
// The histogram of iterations is (m = number of iterations estimating
// alp1 directly, n = number of iterations estimating via omg12, total
// number of trials = 148605):
//
// iter m n
// 0 148 186
// 1 13046 13845
// 2 93315 102225
// 3 36189 32341
// 4 5396 7
// 5 455 1
// 6 56 0
//
// Because omg12 is near pi, estimate work with omg12a = pi - omg12
double k = Astroid(x, y);
double
omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
somg12 = Math.sin(omg12a); comg12 = -Math.cos(omg12a);
// Update spherical estimate of alp1 using omg12 instead of lam12
w.salp1 = cbet2 * somg12;
w.calp1 = sbet12a - cbet2 * sbet1 * GeoMath.sq(somg12) / (1 - comg12);
}
}
// Sanity check on starting guess. Backwards check allows NaN through.
if (!(w.salp1 <= 0))
{ Pair p = GeoMath.norm(w.salp1, w.calp1);
w.salp1 = p.first; w.calp1 = p.second; }
else {
w.salp1 = 1; w.calp1 = 0;
}
return w;
}
private class Lambda12V {
private double lam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
eps, domg12, dlam12;
private Lambda12V() {
lam12 = salp2 = calp2 = sig12 = ssig1 = csig1 = ssig2 = csig2
= eps = domg12 = dlam12 = Double.NaN;
}
}
private Lambda12V Lambda12(double sbet1, double cbet1, double dn1,
double sbet2, double cbet2, double dn2,
double salp1, double calp1,
double slam120, double clam120,
boolean diffp,
// Scratch areas of the right size
double C1a[], double C2a[], double C3a[]) {
// Object to hold lam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
// eps, domg12, dlam12;
Lambda12V w = new Lambda12V();
if (sbet1 == 0 && calp1 == 0)
// Break degeneracy of equatorial line. This case has already been
// handled.
calp1 = -tiny_;
double
// sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1,
calp0 = GeoMath.hypot(calp1, salp1 * sbet1); // calp0 > 0
double somg1, comg1, somg2, comg2, somg12, comg12;
// tan(bet1) = tan(sig1) * cos(alp1)
// tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
w.ssig1 = sbet1; somg1 = salp0 * sbet1;
w.csig1 = comg1 = calp1 * cbet1;
{ Pair p = GeoMath.norm(w.ssig1, w.csig1);
w.ssig1 = p.first; w.csig1 = p.second; }
// GeoMath.norm(somg1, comg1); -- don't need to normalize!
// Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
// about this case, since this can yield singularities in the Newton
// iteration.
// sin(alp2) * cos(bet2) = sin(alp0)
w.salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
// calp2 = sqrt(1 - sq(salp2))
// = sqrt(sq(calp0) - sq(sbet2)) / cbet2
// and subst for calp0 and rearrange to give (choose positive sqrt
// to give alp2 in [0, pi/2]).
w.calp2 = cbet2 != cbet1 || Math.abs(sbet2) != -sbet1 ?
Math.sqrt(GeoMath.sq(calp1 * cbet1) +
(cbet1 < -sbet1 ?
(cbet2 - cbet1) * (cbet1 + cbet2) :
(sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
Math.abs(calp1);
// tan(bet2) = tan(sig2) * cos(alp2)
// tan(omg2) = sin(alp0) * tan(sig2).
w.ssig2 = sbet2; somg2 = salp0 * sbet2;
w.csig2 = comg2 = w.calp2 * cbet2;
{ Pair p = GeoMath.norm(w.ssig2, w.csig2);
w.ssig2 = p.first; w.csig2 = p.second; }
// GeoMath.norm(somg2, comg2); -- don't need to normalize!
// sig12 = sig2 - sig1, limit to [0, pi]
w.sig12 = Math.atan2(Math.max(0.0, w.csig1 * w.ssig2 - w.ssig1 * w.csig2),
w.csig1 * w.csig2 + w.ssig1 * w.ssig2);
// omg12 = omg2 - omg1, limit to [0, pi]
somg12 = Math.max(0.0, comg1 * somg2 - somg1 * comg2);
comg12 = comg1 * comg2 + somg1 * somg2;
// eta = omg12 - lam120
double eta = Math.atan2(somg12 * clam120 - comg12 * slam120,
comg12 * clam120 + somg12 * slam120);
double B312;
double k2 = GeoMath.sq(calp0) * _ep2;
w.eps = k2 / (2 * (1 + Math.sqrt(1 + k2)) + k2);
C3f(w.eps, C3a);
B312 = (SinCosSeries(true, w.ssig2, w.csig2, C3a) -
SinCosSeries(true, w.ssig1, w.csig1, C3a));
w.domg12 = -_f * A3f(w.eps) * salp0 * (w.sig12 + B312);
w.lam12 = eta + w.domg12;
if (diffp) {
if (w.calp2 == 0)
w.dlam12 = - 2 * _f1 * dn1 / sbet1;
else {
LengthsV v =
Lengths(w.eps, w.sig12, w.ssig1, w.csig1, dn1, w.ssig2, w.csig2, dn2,
cbet1, cbet2, GeodesicMask.REDUCEDLENGTH, C1a, C2a);
w.dlam12 = v.m12b;
w.dlam12 *= _f1 / (w.calp2 * cbet2);
}
}
return w;
}
protected double A3f(double eps) {
// Evaluate A3
return GeoMath.polyval(nA3_ - 1, _A3x, 0, eps);
}
protected void C3f(double eps, double c[]) {
// Evaluate C3 coeffs
// Elements c[1] thru c[nC3_ - 1] are set
double mult = 1;
int o = 0;
for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
int m = nC3_ - l - 1; // order of polynomial in eps
mult *= eps;
c[l] = mult * GeoMath.polyval(m, _C3x, o, eps);
o += m + 1;
}
}
protected void C4f(double eps, double c[]) {
// Evaluate C4 coeffs
// Elements c[0] thru c[nC4_ - 1] are set
double mult = 1;
int o = 0;
for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
int m = nC4_ - l - 1; // order of polynomial in eps
c[l] = mult * GeoMath.polyval(m, _C4x, o, eps);
o += m + 1;
mult *= eps;
}
}
// The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
protected static double A1m1f(double eps) {
final double coeff[] = {
// (1-eps)*A1-1, polynomial in eps2 of order 3
1, 4, 64, 0, 256,
};
int m = nA1_/2;
double t = GeoMath.polyval(m, coeff, 0, GeoMath.sq(eps)) / coeff[m + 1];
return (t + eps) / (1 - eps);
}
// The coefficients C1[l] in the Fourier expansion of B1
protected static void C1f(double eps, double c[]) {
final double coeff[] = {
// C1[1]/eps^1, polynomial in eps2 of order 2
-1, 6, -16, 32,
// C1[2]/eps^2, polynomial in eps2 of order 2
-9, 64, -128, 2048,
// C1[3]/eps^3, polynomial in eps2 of order 1
9, -16, 768,
// C1[4]/eps^4, polynomial in eps2 of order 1
3, -5, 512,
// C1[5]/eps^5, polynomial in eps2 of order 0
-7, 1280,
// C1[6]/eps^6, polynomial in eps2 of order 0
-7, 2048,
};
double
eps2 = GeoMath.sq(eps),
d = eps;
int o = 0;
for (int l = 1; l <= nC1_; ++l) { // l is index of C1p[l]
int m = (nC1_ - l) / 2; // order of polynomial in eps^2
c[l] = d * GeoMath.polyval(m, coeff, o, eps2) / coeff[o + m + 1];
o += m + 2;
d *= eps;
}
}
// The coefficients C1p[l] in the Fourier expansion of B1p
protected static void C1pf(double eps, double c[]) {
final double coeff[] = {
// C1p[1]/eps^1, polynomial in eps2 of order 2
205, -432, 768, 1536,
// C1p[2]/eps^2, polynomial in eps2 of order 2
4005, -4736, 3840, 12288,
// C1p[3]/eps^3, polynomial in eps2 of order 1
-225, 116, 384,
// C1p[4]/eps^4, polynomial in eps2 of order 1
-7173, 2695, 7680,
// C1p[5]/eps^5, polynomial in eps2 of order 0
3467, 7680,
// C1p[6]/eps^6, polynomial in eps2 of order 0
38081, 61440,
};
double
eps2 = GeoMath.sq(eps),
d = eps;
int o = 0;
for (int l = 1; l <= nC1p_; ++l) { // l is index of C1p[l]
int m = (nC1p_ - l) / 2; // order of polynomial in eps^2
c[l] = d * GeoMath.polyval(m, coeff, o, eps2) / coeff[o + m + 1];
o += m + 2;
d *= eps;
}
}
// The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
protected static double A2m1f(double eps) {
final double coeff[] = {
// (eps+1)*A2-1, polynomial in eps2 of order 3
-11, -28, -192, 0, 256,
};
int m = nA2_/2;
double t = GeoMath.polyval(m, coeff, 0, GeoMath.sq(eps)) / coeff[m + 1];
return (t - eps) / (1 + eps);
}
// The coefficients C2[l] in the Fourier expansion of B2
protected static void C2f(double eps, double c[]) {
final double coeff[] = {
// C2[1]/eps^1, polynomial in eps2 of order 2
1, 2, 16, 32,
// C2[2]/eps^2, polynomial in eps2 of order 2
35, 64, 384, 2048,
// C2[3]/eps^3, polynomial in eps2 of order 1
15, 80, 768,
// C2[4]/eps^4, polynomial in eps2 of order 1
7, 35, 512,
// C2[5]/eps^5, polynomial in eps2 of order 0
63, 1280,
// C2[6]/eps^6, polynomial in eps2 of order 0
77, 2048,
};
double
eps2 = GeoMath.sq(eps),
d = eps;
int o = 0;
for (int l = 1; l <= nC2_; ++l) { // l is index of C2[l]
int m = (nC2_ - l) / 2; // order of polynomial in eps^2
c[l] = d * GeoMath.polyval(m, coeff, o, eps2) / coeff[o + m + 1];
o += m + 2;
d *= eps;
}
}
// The scale factor A3 = mean value of (d/dsigma)I3
protected void A3coeff() {
final double coeff[] = {
// A3, coeff of eps^5, polynomial in n of order 0
-3, 128,
// A3, coeff of eps^4, polynomial in n of order 1
-2, -3, 64,
// A3, coeff of eps^3, polynomial in n of order 2
-1, -3, -1, 16,
// A3, coeff of eps^2, polynomial in n of order 2
3, -1, -2, 8,
// A3, coeff of eps^1, polynomial in n of order 1
1, -1, 2,
// A3, coeff of eps^0, polynomial in n of order 0
1, 1,
};
int o = 0, k = 0;
for (int j = nA3_ - 1; j >= 0; --j) { // coeff of eps^j
int m = Math.min(nA3_ - j - 1, j); // order of polynomial in n
_A3x[k++] = GeoMath.polyval(m, coeff, o, _n) / coeff[o + m + 1];
o += m + 2;
}
}
// The coefficients C3[l] in the Fourier expansion of B3
protected void C3coeff() {
final double coeff[] = {
// C3[1], coeff of eps^5, polynomial in n of order 0
3, 128,
// C3[1], coeff of eps^4, polynomial in n of order 1
2, 5, 128,
// C3[1], coeff of eps^3, polynomial in n of order 2
-1, 3, 3, 64,
// C3[1], coeff of eps^2, polynomial in n of order 2
-1, 0, 1, 8,
// C3[1], coeff of eps^1, polynomial in n of order 1
-1, 1, 4,
// C3[2], coeff of eps^5, polynomial in n of order 0
5, 256,
// C3[2], coeff of eps^4, polynomial in n of order 1
1, 3, 128,
// C3[2], coeff of eps^3, polynomial in n of order 2
-3, -2, 3, 64,
// C3[2], coeff of eps^2, polynomial in n of order 2
1, -3, 2, 32,
// C3[3], coeff of eps^5, polynomial in n of order 0
7, 512,
// C3[3], coeff of eps^4, polynomial in n of order 1
-10, 9, 384,
// C3[3], coeff of eps^3, polynomial in n of order 2
5, -9, 5, 192,
// C3[4], coeff of eps^5, polynomial in n of order 0
7, 512,
// C3[4], coeff of eps^4, polynomial in n of order 1
-14, 7, 512,
// C3[5], coeff of eps^5, polynomial in n of order 0
21, 2560,
};
int o = 0, k = 0;
for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
for (int j = nC3_ - 1; j >= l; --j) { // coeff of eps^j
int m = Math.min(nC3_ - j - 1, j); // order of polynomial in n
_C3x[k++] = GeoMath.polyval(m, coeff, o, _n) / coeff[o + m + 1];
o += m + 2;
}
}
}
protected void C4coeff() {
final double coeff[] = {
// C4[0], coeff of eps^5, polynomial in n of order 0
97, 15015,
// C4[0], coeff of eps^4, polynomial in n of order 1
1088, 156, 45045,
// C4[0], coeff of eps^3, polynomial in n of order 2
-224, -4784, 1573, 45045,
// C4[0], coeff of eps^2, polynomial in n of order 3
-10656, 14144, -4576, -858, 45045,
// C4[0], coeff of eps^1, polynomial in n of order 4
64, 624, -4576, 6864, -3003, 15015,
// C4[0], coeff of eps^0, polynomial in n of order 5
100, 208, 572, 3432, -12012, 30030, 45045,
// C4[1], coeff of eps^5, polynomial in n of order 0
1, 9009,
// C4[1], coeff of eps^4, polynomial in n of order 1
-2944, 468, 135135,
// C4[1], coeff of eps^3, polynomial in n of order 2
5792, 1040, -1287, 135135,
// C4[1], coeff of eps^2, polynomial in n of order 3
5952, -11648, 9152, -2574, 135135,
// C4[1], coeff of eps^1, polynomial in n of order 4
-64, -624, 4576, -6864, 3003, 135135,
// C4[2], coeff of eps^5, polynomial in n of order 0
8, 10725,
// C4[2], coeff of eps^4, polynomial in n of order 1
1856, -936, 225225,
// C4[2], coeff of eps^3, polynomial in n of order 2
-8448, 4992, -1144, 225225,
// C4[2], coeff of eps^2, polynomial in n of order 3
-1440, 4160, -4576, 1716, 225225,
// C4[3], coeff of eps^5, polynomial in n of order 0
-136, 63063,
// C4[3], coeff of eps^4, polynomial in n of order 1
1024, -208, 105105,
// C4[3], coeff of eps^3, polynomial in n of order 2
3584, -3328, 1144, 315315,
// C4[4], coeff of eps^5, polynomial in n of order 0
-128, 135135,
// C4[4], coeff of eps^4, polynomial in n of order 1
-2560, 832, 405405,
// C4[5], coeff of eps^5, polynomial in n of order 0
128, 99099,
};
int o = 0, k = 0;
for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
for (int j = nC4_ - 1; j >= l; --j) { // coeff of eps^j
int m = nC4_ - j - 1; // order of polynomial in n
_C4x[k++] = GeoMath.polyval(m, coeff, o, _n) / coeff[o + m + 1];
o += m + 2;
}
}
}
}