net.sf.geographiclib.GeoMath Maven / Gradle / Ivy
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/**
* Implementation of the net.sf.geographiclib.GeoMath class
*
* Copyright (c) Charles Karney (2013-2020) and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
package net.sf.geographiclib;
/**
* Mathematical functions needed by GeographicLib.
*
* Define mathematical functions and constants so that any version of Java
* can be used.
**********************************************************************/
public class GeoMath {
/**
* The number of binary digits in the fraction of a double precision
* number (equivalent to C++'s {@code numeric_limits::digits}).
**********************************************************************/
public static final int digits = 53;
/**
* Square a number.
*
* @param x the argument.
* @return x2.
**********************************************************************/
public static double sq(double x) { return x * x; }
/**
* The inverse hyperbolic tangent function. This is defined in terms of
* Math.log1p(x) in order to maintain accuracy near x = 0.
* In addition, the odd parity of the function is enforced.
*
* @param x the argument.
* @return atanh(x).
**********************************************************************/
public static double atanh(double x) {
double y = Math.abs(x); // Enforce odd parity
y = Math.log1p(2 * y/(1 - y))/2;
return x > 0 ? y : (x < 0 ? -y : x);
}
/**
* Normalize a sine cosine pair.
*
* @param p return parameter for normalized quantities with sinx2
* + cosx2 = 1.
* @param sinx the sine.
* @param cosx the cosine.
**********************************************************************/
public static void norm(Pair p, double sinx, double cosx) {
double r = Math.hypot(sinx, cosx);
p.first = sinx/r; p.second = cosx/r;
}
/**
* The error-free sum of two numbers.
*
* @param u the first number in the sum.
* @param v the second number in the sum.
* @param p output Pair(s, t) with s = round(u +
* v) and t = u + v - s.
*
* See D. E. Knuth, TAOCP, Vol 2, 4.2.2, Theorem B.
**********************************************************************/
public static void sum(Pair p, double u, double v) {
double s = u + v;
double up = s - v;
double vpp = s - up;
up -= u;
vpp -= v;
double t = s != 0 ? 0.0 - (up + vpp) : s;
// u + v = s + t
// = round(u + v) + t
p.first = s; p.second = t;
}
/**
* Evaluate a polynomial.
*
* @param N the order of the polynomial.
* @param p the coefficient array (of size N + s + 1 or more).
* @param s starting index for the array.
* @param x the variable.
* @return the value of the polynomial.
*
* Evaluate y = ∑n=0..N
* ps+n
* xN−n. Return 0 if N < 0.
* Return ps, if N = 0 (even if x is
* infinite or a nan). The evaluation uses Horner's method.
**********************************************************************/
public static double polyval(int N, double p[], int s, double x) {
double y = N < 0 ? 0 : p[s++];
while (--N >= 0) y = y * x + p[s++];
return y;
}
/**
* Coarsen a value close to zero.
*
* @param x the argument
* @return the coarsened value.
*
* This makes the smallest gap in x = 1/16 − nextafter(1/16, 0)
* = 1/257 for reals = 0.7 pm on the earth if x is an angle
* in degrees. (This is about 1000 times more resolution than we get with
* angles around 90 degrees.) We use this to avoid having to deal with near
* singular cases when x is non-zero but tiny (e.g.,
* 10−200). This converts −0 to +0; however tiny
* negative numbers get converted to −0.
**********************************************************************/
public static double AngRound(double x) {
final double z = 1/16.0;
double y = Math.abs(x);
// The compiler mustn't "simplify" z - (z - y) to y
y = y < z ? z - (z - y) : y;
return Math.copySign(y, x);
}
/**
* Normalize an angle.
*
* @param x the angle in degrees.
* @return the angle reduced to the range [−180°, 180°).
*
* The range of x is unrestricted.
**********************************************************************/
public static double AngNormalize(double x) {
double y = Math.IEEEremainder(x, 360.0);
return Math.abs(y) == 180 ? Math.copySign(180.0, x) : y;
}
/**
* Normalize a latitude.
*
* @param x the angle in degrees.
* @return x if it is in the range [−90°, 90°], otherwise
* return NaN.
**********************************************************************/
public static double LatFix(double x) {
return Math.abs(x) > 90 ? Double.NaN : x;
}
/**
* The exact difference of two angles reduced to [−180°, 180°].
*
* @param x the first angle in degrees.
* @param y the second angle in degrees.
* @param p output Pair(d, e) with d being the rounded
* difference and e being the error.
*
* This computes z = y − x exactly, reduced to
* [−180°, 180°]; and then sets z = d + e
* where d is the nearest representable number to z and
* e is the truncation error. If z = ±0° or
* ±180°, then the sign of d is given by the sign of
* y − x. The maximum absolute value of e is
* 2−26 (for doubles).
**********************************************************************/
public static void AngDiff(Pair p, double x, double y) {
sum(p, Math.IEEEremainder(-x, 360.0), Math.IEEEremainder(y, 360.0));
sum(p, Math.IEEEremainder(p.first, 360.0), p.second);
if (p.first == 0 || Math.abs(p.first) == 180)
// p = [d, e]...
// If e == 0, take sign from y - x
// else (e != 0, implies d = +/-180), d and e must have opposite signs
p.first = Math.copySign(p.first, p.second == 0 ? y - x : -p.second);
}
/**
* Evaluate the sine and cosine function with the argument in degrees
*
* @param p return Pair(s, t) with s = sin(x) and
* c = cos(x).
* @param x in degrees.
*
* The results obey exactly the elementary properties of the trigonometric
* functions, e.g., sin 9° = cos 81° = − sin 123456789°.
**********************************************************************/
public static void sincosd(Pair p, double x) {
// In order to minimize round-off errors, this function exactly reduces
// the argument to the range [-45, 45] before converting it to radians.
double r; int q;
r = x % 360.0;
q = (int)Math.round(r / 90); // If r is NaN this returns 0
r -= 90 * q;
// now abs(r) <= 45
r = Math.toRadians(r);
// Possibly could call the gnu extension sincos
double s = Math.sin(r), c = Math.cos(r);
double sinx, cosx;
switch (q & 3) {
case 0: sinx = s; cosx = c; break;
case 1: sinx = c; cosx = -s; break;
case 2: sinx = -s; cosx = -c; break;
default: sinx = -c; cosx = s; break; // case 3
}
if (sinx == 0) sinx = Math.copySign(sinx, x);
p.first = sinx; p.second = 0.0 + cosx;
}
/**
* Evaluate the sine and cosine function with reduced argument plus correction
*
* @param p return Pair(s, t) with s =
* sin(x + t) and c = cos(x + t).
* @param x reduced angle in degrees.
* @param t correction in degrees.
*
* This is a variant of GeoMath.sincosd allowing a correction to the angle to
* be supplied. x x must be in [−180°, 180°] and
* t is assumed to be a small correction. GeoMath.AngRound is
* applied to the reduced angle to prevent problems with x + t
* being extremely close but not exactly equal to one of the four cardinal
* directions.
**********************************************************************/
public static void sincosde(Pair p, double x, double t) {
// In order to minimize round-off errors, this function exactly reduces
// the argument to the range [-45, 45] before converting it to radians.
double r; int q;
q = (int)Math.round(x / 90); // If r is NaN this returns 0
r = x - 90 * q;
// now abs(r) <= 45
r = Math.toRadians(GeoMath.AngRound(r + t));
// Possibly could call the gnu extension sincos
double s = Math.sin(r), c = Math.cos(r);
double sinx, cosx;
switch (q & 3) {
case 0: sinx = s; cosx = c; break;
case 1: sinx = c; cosx = -s; break;
case 2: sinx = -s; cosx = -c; break;
default: sinx = -c; cosx = s; break; // case 3
}
if (sinx == 0) sinx = Math.copySign(sinx, x);
p.first = sinx; p.second = 0.0 + cosx;
}
/**
* Evaluate the atan2 function with the result in degrees
*
* @param y the sine of the angle
* @param x the cosine of the angle
* @return atan2(y, x) in degrees.
*
* The result is in the range [−180° 180°]. N.B.,
* atan2d(±0, −1) = ±180°.
**********************************************************************/
public static double atan2d(double y, double x) {
// In order to minimize round-off errors, this function rearranges the
// arguments so that result of atan2 is in the range [-pi/4, pi/4] before
// converting it to degrees and mapping the result to the correct
// quadrant.
int q = 0;
if (Math.abs(y) > Math.abs(x)) { double t; t = x; x = y; y = t; q = 2; }
if (x < 0) { x = -x; ++q; }
// here x >= 0 and x >= abs(y), so angle is in [-pi/4, pi/4]
double ang = Math.toDegrees(Math.atan2(y, x));
switch (q) {
// Note that atan2d(-0.0, 1.0) will return -0. However, we expect that
// atan2d will not be called with y = -0. If need be, include
//
// case 0: ang = 0 + ang; break;
//
// and handle mpfr as in AngRound.
case 1: ang = Math.copySign(180.0, y) - ang; break;
case 2: ang = 90 - ang; break;
case 3: ang = -90 + ang; break;
default: break;
}
return ang;
}
private GeoMath() {}
}