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geodesic.c
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/**
* \file geodesic.c
* \brief Implementation of the geodesic routines in C
*
* For the full documentation see geodesic.h.
**********************************************************************/
/** @cond SKIP */
/*
* This is a C implementation of the geodesic algorithms described in
*
* C. F. F. Karney,
* Algorithms for geodesics,
* J. Geodesy <b>87</b>, 43--55 (2013);
* https://doi.org/10.1007/s00190-012-0578-z
* Addenda: https://geographiclib.sourceforge.io/geod-addenda.html
*
* See the comments in geodesic.h for documentation.
*
* Copyright (c) Charles Karney (2012-2022) <[email protected]> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
*/
#include "geodesic.h"
#include <math.h>
#include <float.h>
#if !defined(__cplusplus)
#define nullptr 0
#endif
#define GEOGRAPHICLIB_GEODESIC_ORDER 6
#define nA1 GEOGRAPHICLIB_GEODESIC_ORDER
#define nC1 GEOGRAPHICLIB_GEODESIC_ORDER
#define nC1p GEOGRAPHICLIB_GEODESIC_ORDER
#define nA2 GEOGRAPHICLIB_GEODESIC_ORDER
#define nC2 GEOGRAPHICLIB_GEODESIC_ORDER
#define nA3 GEOGRAPHICLIB_GEODESIC_ORDER
#define nA3x nA3
#define nC3 GEOGRAPHICLIB_GEODESIC_ORDER
#define nC3x ((nC3 * (nC3 - 1)) / 2)
#define nC4 GEOGRAPHICLIB_GEODESIC_ORDER
#define nC4x ((nC4 * (nC4 + 1)) / 2)
#define nC (GEOGRAPHICLIB_GEODESIC_ORDER + 1)
typedef int boolx;
enum booly { FALSE = 0, TRUE = 1 };
/* qd = quarter turn / degree
* hd = half turn / degree
* td = full turn / degree */
enum dms { qd = 90, hd = 2 * qd, td = 2 * hd };
static unsigned init = 0;
static unsigned digits, maxit1, maxit2;
static double epsilon, realmin, pi, degree, NaN,
tiny, tol0, tol1, tol2, tolb, xthresh;
static void Init(void) {
if (!init) {
digits = DBL_MANT_DIG;
epsilon = DBL_EPSILON;
realmin = DBL_MIN;
#if defined(M_PI)
pi = M_PI;
#else
pi = atan2(0.0, -1.0);
#endif
maxit1 = 20;
maxit2 = maxit1 + digits + 10;
tiny = sqrt(realmin);
tol0 = 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) */
tol1 = 200 * tol0;
tol2 = sqrt(tol0);
/* Check on bisection interval */
tolb = tol0 * tol2;
xthresh = 1000 * tol2;
degree = pi/hd;
NaN = nan("0");
init = 1;
}
}
enum captype {
CAP_NONE = 0U,
CAP_C1 = 1U<<0,
CAP_C1p = 1U<<1,
CAP_C2 = 1U<<2,
CAP_C3 = 1U<<3,
CAP_C4 = 1U<<4,
CAP_ALL = 0x1FU,
OUT_ALL = 0x7F80U
};
static double sq(double x) { return x * x; }
static double sumx(double u, double v, double* t) {
volatile double s = u + v;
volatile double up = s - v;
volatile double vpp = s - up;
up -= u;
vpp -= v;
if (t) *t = s != 0 ? 0 - (up + vpp) : s;
/* error-free sum:
* u + v = s + t
* = round(u + v) + t */
return s;
}
static double polyval(int N, const double p[], double x) {
double y = N < 0 ? 0 : *p++;
while (--N >= 0) y = y * x + *p++;
return y;
}
static void swapx(double* x, double* y)
{ double t = *x; *x = *y; *y = t; }
static void norm2(double* sinx, double* cosx) {
#if defined(_MSC_VER) && defined(_M_IX86)
/* hypot for Visual Studio (A=win32) fails monotonicity, e.g., with
* x = 0.6102683302836215
* y1 = 0.7906090004346522
* y2 = y1 + 1e-16
* the test
* hypot(x, y2) >= hypot(x, y1)
* fails. See also
* https://bugs.python.org/issue43088 */
double r = sqrt(*sinx * *sinx + *cosx * *cosx);
#else
double r = hypot(*sinx, *cosx);
#endif
*sinx /= r;
*cosx /= r;
}
static double AngNormalize(double x) {
double y = remainder(x, (double)td);
return fabs(y) == hd ? copysign((double)hd, x) : y;
}
static double LatFix(double x)
{ return fabs(x) > qd ? NaN : x; }
static double AngDiff(double x, double y, double* e) {
/* Use remainder instead of AngNormalize, since we treat boundary cases
* later taking account of the error */
double t, d = sumx(remainder(-x, (double)td), remainder( y, (double)td), &t);
/* This second sum can only change d if abs(d) < 128, so don't need to
* apply remainder yet again. */
d = sumx(remainder(d, (double)td), t, &t);
/* Fix the sign if d = -180, 0, 180. */
if (d == 0 || fabs(d) == hd)
/* If t == 0, take sign from y - x
* else (t != 0, implies d = +/-180), d and t must have opposite signs */
d = copysign(d, t == 0 ? y - x : -t);
if (e) *e = t;
return d;
}
static double AngRound(double x) {
/* False positive in cppcheck requires "1.0" instead of "1" */
const double z = 1.0/16.0;
volatile double y = fabs(x);
volatile double w = z - y;
/* The compiler mustn't "simplify" z - (z - y) to y */
y = w > 0 ? z - w : y;
return copysign(y, x);
}
static void sincosdx(double x, double* sinx, double* cosx) {
/* 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, s, c; int q = 0;
r = remquo(x, (double)qd, &q);
/* now abs(r) <= 45 */
r *= degree;
/* Possibly could call the gnu extension sincos */
s = sin(r); c = cos(r);
switch ((unsigned)q & 3U) {
case 0U: *sinx = s; *cosx = c; break;
case 1U: *sinx = c; *cosx = -s; break;
case 2U: *sinx = -s; *cosx = -c; break;
default: *sinx = -c; *cosx = s; break; /* case 3U */
}
/* http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf */
*cosx += 0; /* special values from F.10.1.12 */
/* special values from F.10.1.13 */
if (*sinx == 0) *sinx = copysign(*sinx, x);
}
static void sincosde(double x, double t, double* sinx, double* cosx) {
/* 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, s, c; int q = 0;
r = AngRound(remquo(x, (double)qd, &q) + t);
/* now abs(r) <= 45 */
r *= degree;
/* Possibly could call the gnu extension sincos */
s = sin(r); c = cos(r);
switch ((unsigned)q & 3U) {
case 0U: *sinx = s; *cosx = c; break;
case 1U: *sinx = c; *cosx = -s; break;
case 2U: *sinx = -s; *cosx = -c; break;
default: *sinx = -c; *cosx = s; break; /* case 3U */
}
/* http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf */
*cosx += 0; /* special values from F.10.1.12 */
/* special values from F.10.1.13 */
if (*sinx == 0) *sinx = copysign(*sinx, x);
}
static double atan2dx(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; double ang;
if (fabs(y) > fabs(x)) { swapx(&x, &y); q = 2; }
if (signbit(x)) { x = -x; ++q; }
/* here x >= 0 and x >= abs(y), so angle is in [-pi/4, pi/4] */
ang = atan2(y, x) / degree;
switch (q) {
case 1: ang = copysign((double)hd, y) - ang; break;
case 2: ang = qd - ang; break;
case 3: ang = -qd + ang; break;
default: break;
}
return ang;
}
static void A3coeff(struct geod_geodesic* g);
static void C3coeff(struct geod_geodesic* g);
static void C4coeff(struct geod_geodesic* g);
static double SinCosSeries(boolx sinp,
double sinx, double cosx,
const double c[], int n);
static void Lengths(const struct geod_geodesic* g,
double eps, double sig12,
double ssig1, double csig1, double dn1,
double ssig2, double csig2, double dn2,
double cbet1, double cbet2,
double* ps12b, double* pm12b, double* pm0,
double* pM12, double* pM21,
/* Scratch area of the right size */
double Ca[]);
static double Astroid(double x, double y);
static double InverseStart(const struct geod_geodesic* g,
double sbet1, double cbet1, double dn1,
double sbet2, double cbet2, double dn2,
double lam12, double slam12, double clam12,
double* psalp1, double* pcalp1,
/* Only updated if return val >= 0 */
double* psalp2, double* pcalp2,
/* Only updated for short lines */
double* pdnm,
/* Scratch area of the right size */
double Ca[]);
static double Lambda12(const struct geod_geodesic* g,
double sbet1, double cbet1, double dn1,
double sbet2, double cbet2, double dn2,
double salp1, double calp1,
double slam120, double clam120,
double* psalp2, double* pcalp2,
double* psig12,
double* pssig1, double* pcsig1,
double* pssig2, double* pcsig2,
double* peps,
double* pdomg12,
boolx diffp, double* pdlam12,
/* Scratch area of the right size */
double Ca[]);
static double A3f(const struct geod_geodesic* g, double eps);
static void C3f(const struct geod_geodesic* g, double eps, double c[]);
static void C4f(const struct geod_geodesic* g, double eps, double c[]);
static double A1m1f(double eps);
static void C1f(double eps, double c[]);
static void C1pf(double eps, double c[]);
static double A2m1f(double eps);
static void C2f(double eps, double c[]);
static int transit(double lon1, double lon2);
static int transitdirect(double lon1, double lon2);
static void accini(double s[]);
static void acccopy(const double s[], double t[]);
static void accadd(double s[], double y);
static double accsum(const double s[], double y);
static void accneg(double s[]);
static void accrem(double s[], double y);
static double areareduceA(double area[], double area0,
int crossings, boolx reverse, boolx sign);
static double areareduceB(double area, double area0,
int crossings, boolx reverse, boolx sign);
void geod_init(struct geod_geodesic* g, double a, double f) {
if (!init) Init();
g->a = a;
g->f = f;
g->f1 = 1 - g->f;
g->e2 = g->f * (2 - g->f);
g->ep2 = g->e2 / sq(g->f1); /* e2 / (1 - e2) */
g->n = g->f / ( 2 - g->f);
g->b = g->a * g->f1;
g->c2 = (sq(g->a) + sq(g->b) *
(g->e2 == 0 ? 1 :
(g->e2 > 0 ? atanh(sqrt(g->e2)) : atan(sqrt(-g->e2))) /
sqrt(fabs(g->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. */
g->etol2 = 0.1 * tol2 /
sqrt( fmax(0.001, fabs(g->f)) * fmin(1.0, 1 - g->f/2) / 2 );
A3coeff(g);
C3coeff(g);
C4coeff(g);
}
static void geod_lineinit_int(struct geod_geodesicline* l,
const struct geod_geodesic* g,
double lat1, double lon1,
double azi1, double salp1, double calp1,
unsigned caps) {
double cbet1, sbet1, eps;
l->a = g->a;
l->f = g->f;
l->b = g->b;
l->c2 = g->c2;
l->f1 = g->f1;
/* If caps is 0 assume the standard direct calculation */
l->caps = (caps ? caps : GEOD_DISTANCE_IN | GEOD_LONGITUDE) |
/* always allow latitude and azimuth and unrolling of longitude */
GEOD_LATITUDE | GEOD_AZIMUTH | GEOD_LONG_UNROLL;
l->lat1 = LatFix(lat1);
l->lon1 = lon1;
l->azi1 = azi1;
l->salp1 = salp1;
l->calp1 = calp1;
sincosdx(AngRound(l->lat1), &sbet1, &cbet1); sbet1 *= l->f1;
/* Ensure cbet1 = +epsilon at poles */
norm2(&sbet1, &cbet1); cbet1 = fmax(tiny, cbet1);
l->dn1 = sqrt(1 + g->ep2 * sq(sbet1));
/* Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0), */
l->salp0 = l->salp1 * cbet1; /* alp0 in [0, pi/2 - |bet1|] */
/* Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
* is slightly better (consider the case salp1 = 0). */
l->calp0 = hypot(l->calp1, l->salp1 * sbet1);
/* Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
* sig = 0 is nearest northward crossing of equator.
* With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
* With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
* With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
* Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
* With alp0 in (0, pi/2], quadrants for sig and omg coincide.
* No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
* With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi. */
l->ssig1 = sbet1; l->somg1 = l->salp0 * sbet1;
l->csig1 = l->comg1 = sbet1 != 0 || l->calp1 != 0 ? cbet1 * l->calp1 : 1;
norm2(&l->ssig1, &l->csig1); /* sig1 in (-pi, pi] */
/* norm2(somg1, comg1); -- don't need to normalize! */
l->k2 = sq(l->calp0) * g->ep2;
eps = l->k2 / (2 * (1 + sqrt(1 + l->k2)) + l->k2);
if (l->caps & CAP_C1) {
double s, c;
l->A1m1 = A1m1f(eps);
C1f(eps, l->C1a);
l->B11 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C1a, nC1);
s = sin(l->B11); c = cos(l->B11);
/* tau1 = sig1 + B11 */
l->stau1 = l->ssig1 * c + l->csig1 * s;
l->ctau1 = l->csig1 * c - l->ssig1 * s;
/* Not necessary because C1pa reverts C1a
* B11 = -SinCosSeries(TRUE, stau1, ctau1, C1pa, nC1p); */
}
if (l->caps & CAP_C1p)
C1pf(eps, l->C1pa);
if (l->caps & CAP_C2) {
l->A2m1 = A2m1f(eps);
C2f(eps, l->C2a);
l->B21 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C2a, nC2);
}
if (l->caps & CAP_C3) {
C3f(g, eps, l->C3a);
l->A3c = -l->f * l->salp0 * A3f(g, eps);
l->B31 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C3a, nC3-1);
}
if (l->caps & CAP_C4) {
C4f(g, eps, l->C4a);
/* Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) */
l->A4 = sq(l->a) * l->calp0 * l->salp0 * g->e2;
l->B41 = SinCosSeries(FALSE, l->ssig1, l->csig1, l->C4a, nC4);
}
l->a13 = l->s13 = NaN;
}
void geod_lineinit(struct geod_geodesicline* l,
const struct geod_geodesic* g,
double lat1, double lon1, double azi1, unsigned caps) {
double salp1, calp1;
azi1 = AngNormalize(azi1);
/* Guard against underflow in salp0 */
sincosdx(AngRound(azi1), &salp1, &calp1);
geod_lineinit_int(l, g, lat1, lon1, azi1, salp1, calp1, caps);
}
void geod_gendirectline(struct geod_geodesicline* l,
const struct geod_geodesic* g,
double lat1, double lon1, double azi1,
unsigned flags, double s12_a12,
unsigned caps) {
geod_lineinit(l, g, lat1, lon1, azi1, caps);
geod_gensetdistance(l, flags, s12_a12);
}
void geod_directline(struct geod_geodesicline* l,
const struct geod_geodesic* g,
double lat1, double lon1, double azi1,
double s12, unsigned caps) {
geod_gendirectline(l, g, lat1, lon1, azi1, GEOD_NOFLAGS, s12, caps);
}
double geod_genposition(const struct geod_geodesicline* l,
unsigned flags, double s12_a12,
double* plat2, double* plon2, double* pazi2,
double* ps12, double* pm12,
double* pM12, double* pM21,
double* pS12) {
double lat2 = 0, lon2 = 0, azi2 = 0, s12 = 0,
m12 = 0, M12 = 0, M21 = 0, S12 = 0;
/* Avoid warning about uninitialized B12. */
double sig12, ssig12, csig12, B12 = 0, AB1 = 0;
double omg12, lam12, lon12;
double ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2, dn2;
unsigned outmask =
(plat2 ? GEOD_LATITUDE : GEOD_NONE) |
(plon2 ? GEOD_LONGITUDE : GEOD_NONE) |
(pazi2 ? GEOD_AZIMUTH : GEOD_NONE) |
(ps12 ? GEOD_DISTANCE : GEOD_NONE) |
(pm12 ? GEOD_REDUCEDLENGTH : GEOD_NONE) |
(pM12 || pM21 ? GEOD_GEODESICSCALE : GEOD_NONE) |
(pS12 ? GEOD_AREA : GEOD_NONE);
outmask &= l->caps & OUT_ALL;
if (!( (flags & GEOD_ARCMODE || (l->caps & (GEOD_DISTANCE_IN & OUT_ALL))) ))
/* Impossible distance calculation requested */
return NaN;
if (flags & GEOD_ARCMODE) {
/* Interpret s12_a12 as spherical arc length */
sig12 = s12_a12 * degree;
sincosdx(s12_a12, &ssig12, &csig12);
} else {
/* Interpret s12_a12 as distance */
double
tau12 = s12_a12 / (l->b * (1 + l->A1m1)),
s = sin(tau12),
c = cos(tau12);
/* tau2 = tau1 + tau12 */
B12 = - SinCosSeries(TRUE,
l->stau1 * c + l->ctau1 * s,
l->ctau1 * c - l->stau1 * s,
l->C1pa, nC1p);
sig12 = tau12 - (B12 - l->B11);
ssig12 = sin(sig12); csig12 = cos(sig12);
if (fabs(l->f) > 0.01) {
/* Reverted distance series is inaccurate for |f| > 1/100, so correct
* sig12 with 1 Newton iteration. The following table shows the
* approximate maximum error for a = WGS_a() and various f relative to
* GeodesicExact.
* erri = the error in the inverse solution (nm)
* errd = the error in the direct solution (series only) (nm)
* errda = the error in the direct solution (series + 1 Newton) (nm)
*
* f erri errd errda
* -1/5 12e6 1.2e9 69e6
* -1/10 123e3 12e6 765e3
* -1/20 1110 108e3 7155
* -1/50 18.63 200.9 27.12
* -1/100 18.63 23.78 23.37
* -1/150 18.63 21.05 20.26
* 1/150 22.35 24.73 25.83
* 1/100 22.35 25.03 25.31
* 1/50 29.80 231.9 30.44
* 1/20 5376 146e3 10e3
* 1/10 829e3 22e6 1.5e6
* 1/5 157e6 3.8e9 280e6 */
double serr;
ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12;
csig2 = l->csig1 * csig12 - l->ssig1 * ssig12;
B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1);
serr = (1 + l->A1m1) * (sig12 + (B12 - l->B11)) - s12_a12 / l->b;
sig12 = sig12 - serr / sqrt(1 + l->k2 * sq(ssig2));
ssig12 = sin(sig12); csig12 = cos(sig12);
/* Update B12 below */
}
}
/* sig2 = sig1 + sig12 */
ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12;
csig2 = l->csig1 * csig12 - l->ssig1 * ssig12;
dn2 = sqrt(1 + l->k2 * sq(ssig2));
if (outmask & (GEOD_DISTANCE | GEOD_REDUCEDLENGTH | GEOD_GEODESICSCALE)) {
if (flags & GEOD_ARCMODE || fabs(l->f) > 0.01)
B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1);
AB1 = (1 + l->A1m1) * (B12 - l->B11);
}
/* sin(bet2) = cos(alp0) * sin(sig2) */
sbet2 = l->calp0 * ssig2;
/* Alt: cbet2 = hypot(csig2, salp0 * ssig2); */
cbet2 = hypot(l->salp0, l->calp0 * csig2);
if (cbet2 == 0)
/* I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case */
cbet2 = csig2 = tiny;
/* tan(alp0) = cos(sig2)*tan(alp2) */
salp2 = l->salp0; calp2 = l->calp0 * csig2; /* No need to normalize */
if (outmask & GEOD_DISTANCE)
s12 = (flags & GEOD_ARCMODE) ?
l->b * ((1 + l->A1m1) * sig12 + AB1) :
s12_a12;
if (outmask & GEOD_LONGITUDE) {
double E = copysign(1, l->salp0); /* east or west going? */
/* tan(omg2) = sin(alp0) * tan(sig2) */
somg2 = l->salp0 * ssig2; comg2 = csig2; /* No need to normalize */
/* omg12 = omg2 - omg1 */
omg12 = (flags & GEOD_LONG_UNROLL)
? E * (sig12
- (atan2( ssig2, csig2) - atan2( l->ssig1, l->csig1))
+ (atan2(E * somg2, comg2) - atan2(E * l->somg1, l->comg1)))
: atan2(somg2 * l->comg1 - comg2 * l->somg1,
comg2 * l->comg1 + somg2 * l->somg1);
lam12 = omg12 + l->A3c *
( sig12 + (SinCosSeries(TRUE, ssig2, csig2, l->C3a, nC3-1)
- l->B31));
lon12 = lam12 / degree;
lon2 = (flags & GEOD_LONG_UNROLL) ? l->lon1 + lon12 :
AngNormalize(AngNormalize(l->lon1) + AngNormalize(lon12));
}
if (outmask & GEOD_LATITUDE)
lat2 = atan2dx(sbet2, l->f1 * cbet2);
if (outmask & GEOD_AZIMUTH)
azi2 = atan2dx(salp2, calp2);
if (outmask & (GEOD_REDUCEDLENGTH | GEOD_GEODESICSCALE)) {
double
B22 = SinCosSeries(TRUE, ssig2, csig2, l->C2a, nC2),
AB2 = (1 + l->A2m1) * (B22 - l->B21),
J12 = (l->A1m1 - l->A2m1) * sig12 + (AB1 - AB2);
if (outmask & GEOD_REDUCEDLENGTH)
/* Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
* accurate cancellation in the case of coincident points. */
m12 = l->b * ((dn2 * (l->csig1 * ssig2) - l->dn1 * (l->ssig1 * csig2))
- l->csig1 * csig2 * J12);
if (outmask & GEOD_GEODESICSCALE) {
double t = l->k2 * (ssig2 - l->ssig1) * (ssig2 + l->ssig1) /
(l->dn1 + dn2);
M12 = csig12 + (t * ssig2 - csig2 * J12) * l->ssig1 / l->dn1;
M21 = csig12 - (t * l->ssig1 - l->csig1 * J12) * ssig2 / dn2;
}
}
if (outmask & GEOD_AREA) {
double
B42 = SinCosSeries(FALSE, ssig2, csig2, l->C4a, nC4);
double salp12, calp12;
if (l->calp0 == 0 || l->salp0 == 0) {
/* alp12 = alp2 - alp1, used in atan2 so no need to normalize */
salp12 = salp2 * l->calp1 - calp2 * l->salp1;
calp12 = calp2 * l->calp1 + salp2 * l->salp1;
} else {
/* tan(alp) = tan(alp0) * sec(sig)
* tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
* = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
* If csig12 > 0, write
* csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
* else
* csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
* No need to normalize */
salp12 = l->calp0 * l->salp0 *
(csig12 <= 0 ? l->csig1 * (1 - csig12) + ssig12 * l->ssig1 :
ssig12 * (l->csig1 * ssig12 / (1 + csig12) + l->ssig1));
calp12 = sq(l->salp0) + sq(l->calp0) * l->csig1 * csig2;
}
S12 = l->c2 * atan2(salp12, calp12) + l->A4 * (B42 - l->B41);
}
/* In the pattern
*
* if ((outmask & GEOD_XX) && pYY)
* *pYY = YY;
*
* the second check "&& pYY" is redundant. It's there to make the CLang
* static analyzer happy.
*/
if ((outmask & GEOD_LATITUDE) && plat2)
*plat2 = lat2;
if ((outmask & GEOD_LONGITUDE) && plon2)
*plon2 = lon2;
if ((outmask & GEOD_AZIMUTH) && pazi2)
*pazi2 = azi2;
if ((outmask & GEOD_DISTANCE) && ps12)
*ps12 = s12;
if ((outmask & GEOD_REDUCEDLENGTH) && pm12)
*pm12 = m12;
if (outmask & GEOD_GEODESICSCALE) {
if (pM12) *pM12 = M12;
if (pM21) *pM21 = M21;
}
if ((outmask & GEOD_AREA) && pS12)
*pS12 = S12;
return (flags & GEOD_ARCMODE) ? s12_a12 : sig12 / degree;
}
void geod_setdistance(struct geod_geodesicline* l, double s13) {
l->s13 = s13;
l->a13 = geod_genposition(l, GEOD_NOFLAGS, l->s13, nullptr, nullptr, nullptr,
nullptr, nullptr, nullptr, nullptr, nullptr);
}
static void geod_setarc(struct geod_geodesicline* l, double a13) {
l->a13 = a13; l->s13 = NaN;
geod_genposition(l, GEOD_ARCMODE, l->a13, nullptr, nullptr, nullptr, &l->s13,
nullptr, nullptr, nullptr, nullptr);
}
void geod_gensetdistance(struct geod_geodesicline* l,
unsigned flags, double s13_a13) {
(flags & GEOD_ARCMODE) ?
geod_setarc(l, s13_a13) :
geod_setdistance(l, s13_a13);
}
void geod_position(const struct geod_geodesicline* l, double s12,
double* plat2, double* plon2, double* pazi2) {
geod_genposition(l, FALSE, s12, plat2, plon2, pazi2,
nullptr, nullptr, nullptr, nullptr, nullptr);
}
double geod_gendirect(const struct geod_geodesic* g,
double lat1, double lon1, double azi1,
unsigned flags, double s12_a12,
double* plat2, double* plon2, double* pazi2,
double* ps12, double* pm12, double* pM12, double* pM21,
double* pS12) {
struct geod_geodesicline l;
unsigned outmask =
(plat2 ? GEOD_LATITUDE : GEOD_NONE) |
(plon2 ? GEOD_LONGITUDE : GEOD_NONE) |
(pazi2 ? GEOD_AZIMUTH : GEOD_NONE) |
(ps12 ? GEOD_DISTANCE : GEOD_NONE) |
(pm12 ? GEOD_REDUCEDLENGTH : GEOD_NONE) |
(pM12 || pM21 ? GEOD_GEODESICSCALE : GEOD_NONE) |
(pS12 ? GEOD_AREA : GEOD_NONE);
geod_lineinit(&l, g, lat1, lon1, azi1,
/* Automatically supply GEOD_DISTANCE_IN if necessary */
outmask |
((flags & GEOD_ARCMODE) ? GEOD_NONE : GEOD_DISTANCE_IN));
return geod_genposition(&l, flags, s12_a12,
plat2, plon2, pazi2, ps12, pm12, pM12, pM21, pS12);
}
void geod_direct(const struct geod_geodesic* g,
double lat1, double lon1, double azi1,
double s12,
double* plat2, double* plon2, double* pazi2) {
geod_gendirect(g, lat1, lon1, azi1, GEOD_NOFLAGS, s12, plat2, plon2, pazi2,
nullptr, nullptr, nullptr, nullptr, nullptr);
}
static double geod_geninverse_int(const struct geod_geodesic* g,
double lat1, double lon1,
double lat2, double lon2,
double* ps12,
double* psalp1, double* pcalp1,
double* psalp2, double* pcalp2,
double* pm12, double* pM12, double* pM21,
double* pS12) {
double s12 = 0, m12 = 0, M12 = 0, M21 = 0, S12 = 0;
double lon12, lon12s;
int latsign, lonsign, swapp;
double sbet1, cbet1, sbet2, cbet2, s12x = 0, m12x = 0;
double dn1, dn2, lam12, slam12, clam12;
double a12 = 0, sig12, calp1 = 0, salp1 = 0, calp2 = 0, salp2 = 0;
double Ca[nC];
boolx meridian;
/* somg12 == 2 marks that it needs to be calculated */
double omg12 = 0, somg12 = 2, comg12 = 0;
unsigned outmask =
(ps12 ? GEOD_DISTANCE : GEOD_NONE) |
(pm12 ? GEOD_REDUCEDLENGTH : GEOD_NONE) |
(pM12 || pM21 ? GEOD_GEODESICSCALE : GEOD_NONE) |
(pS12 ? GEOD_AREA : GEOD_NONE);
outmask &= OUT_ALL;
/* 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. */
lon12 = AngDiff(lon1, lon2, &lon12s);
/* Make longitude difference positive. */
lonsign = signbit(lon12) ? -1 : 1;
lon12 *= lonsign; lon12s *= lonsign;
lam12 = lon12 * degree;
/* Calculate sincos of lon12 + error (this applies AngRound internally). */
sincosde(lon12, lon12s, &slam12, &clam12);
lon12s = (hd - lon12) - lon12s; /* the supplementary longitude difference */
/* If really close to the equator, treat as on equator. */
lat1 = AngRound(LatFix(lat1));
lat2 = AngRound(LatFix(lat2));
/* Swap points so that point with higher (abs) latitude is point 1
* If one latitude is a nan, then it becomes lat1. */
swapp = fabs(lat1) < fabs(lat2) || lat2 != lat2 ? -1 : 1;
if (swapp < 0) {
lonsign *= -1;
swapx(&lat1, &lat2);
}
/* Make lat1 <= -0 */
latsign = signbit(lat1) ? 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. */
sincosdx(lat1, &sbet1, &cbet1); sbet1 *= g->f1;
/* Ensure cbet1 = +epsilon at poles */
norm2(&sbet1, &cbet1); cbet1 = fmax(tiny, cbet1);
sincosdx(lat2, &sbet2, &cbet2); sbet2 *= g->f1;
/* Ensure cbet2 = +epsilon at poles */
norm2(&sbet2, &cbet2); cbet2 = fmax(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 = copysign(sbet1, sbet2);
} else {
if (fabs(sbet2) == -sbet1)
cbet2 = cbet1;
}
dn1 = sqrt(1 + g->ep2 * sq(sbet1));
dn2 = sqrt(1 + g->ep2 * sq(sbet2));
meridian = lat1 == -qd || slam12 == 0;
if (meridian) {
/* Endpoints are on a single full meridian, so the geodesic might lie on
* a meridian. */
double ssig1, csig1, ssig2, csig2;
calp1 = clam12; salp1 = slam12; /* Head to the target longitude */
calp2 = 1; salp2 = 0; /* At the target we're heading north */
/* tan(bet) = tan(sig) * cos(alp) */
ssig1 = sbet1; csig1 = calp1 * cbet1;
ssig2 = sbet2; csig2 = calp2 * cbet2;
/* sig12 = sig2 - sig1 */
sig12 = atan2(fmax(0.0, csig1 * ssig2 - ssig1 * csig2) + 0,
csig1 * csig2 + ssig1 * ssig2);
Lengths(g, g->n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
cbet1, cbet2, &s12x, &m12x, nullptr,
(outmask & GEOD_GEODESICSCALE) ? &M12 : nullptr,
(outmask & GEOD_GEODESICSCALE) ? &M21 : nullptr,
Ca);
/* 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 ||
/* Prevent negative s12 or m12 for short lines */
(sig12 < tol0 && (s12x < 0 || m12x < 0)))
sig12 = m12x = s12x = 0;
m12x *= g->b;
s12x *= g->b;
a12 = sig12 / degree;
} else
/* m12 < 0, i.e., prolate and too close to anti-podal */
meridian = FALSE;
}
if (!meridian &&
sbet1 == 0 && /* and sbet2 == 0 */
/* Mimic the way Lambda12 works with calp1 = 0 */
(g->f <= 0 || lon12s >= g->f * hd)) {
/* Geodesic runs along equator */
calp1 = calp2 = 0; salp1 = salp2 = 1;
s12x = g->a * lam12;
sig12 = omg12 = lam12 / g->f1;
m12x = g->b * sin(sig12);
if (outmask & GEOD_GEODESICSCALE)
M12 = M21 = cos(sig12);
a12 = lon12 / g->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 = 0;
sig12 = InverseStart(g, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
lam12, slam12, clam12,
&salp1, &calp1, &salp2, &calp2, &dnm,
Ca);
if (sig12 >= 0) {
/* Short lines (InverseStart sets salp2, calp2, dnm) */
s12x = sig12 * g->b * dnm;
m12x = sq(dnm) * g->b * sin(sig12 / dnm);
if (outmask & GEOD_GEODESICSCALE)
M12 = M21 = cos(sig12 / dnm);
a12 = sig12 / degree;
omg12 = lam12 / (g->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 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, domg12 = 0;
unsigned numit = 0;
/* Bracketing range */
double salp1a = tiny, calp1a = 1, salp1b = tiny, calp1b = -1;
boolx tripn = FALSE;
boolx tripb = FALSE;
for (; 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 dv = 0,
v = Lambda12(g, sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
slam12, clam12,
&salp2, &calp2, &sig12, &ssig1, &csig1, &ssig2, &csig2,
&eps, &domg12, numit < maxit1, &dv, Ca);
/* Reversed test to allow escape with NaNs */
if (tripb || !(fabs(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 = sin(dalp1), cdalp1 = cos(dalp1),
nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
if (nsalp1 > 0 && fabs(dalp1) < pi) {
calp1 = calp1 * cdalp1 - salp1 * sdalp1;
salp1 = nsalp1;
norm2(&salp1, &calp1);
/* In some regimes we don't get quadratic convergence because
* slope -> 0. So use convergence conditions based on epsilon
* instead of sqrt(epsilon). */
tripn = fabs(v) <= 16 * tol0;
continue;
}
}
/* Either dv was not positive 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;
norm2(&salp1, &calp1);
tripn = FALSE;
tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb ||
fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb);
}
Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
cbet1, cbet2, &s12x, &m12x, nullptr,
(outmask & GEOD_GEODESICSCALE) ? &M12 : nullptr,
(outmask & GEOD_GEODESICSCALE) ? &M21 : nullptr, Ca);
m12x *= g->b;
s12x *= g->b;
a12 = sig12 / degree;
if (outmask & GEOD_AREA) {
/* omg12 = lam12 - domg12 */
double sdomg12 = sin(domg12), cdomg12 = cos(domg12);
somg12 = slam12 * cdomg12 - clam12 * sdomg12;
comg12 = clam12 * cdomg12 + slam12 * sdomg12;
}
}
}
if (outmask & GEOD_DISTANCE)
s12 = 0 + s12x; /* Convert -0 to 0 */
if (outmask & GEOD_REDUCEDLENGTH)
m12 = 0 + m12x; /* Convert -0 to 0 */
if (outmask & GEOD_AREA) {
double
/* From Lambda12: sin(alp1) * cos(bet1) = sin(alp0) */
salp0 = salp1 * cbet1,
calp0 = 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 = sq(calp0) * g->ep2,
eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
/* Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0). */
A4 = sq(g->a) * calp0 * salp0 * g->e2;
double B41, B42;
norm2(&ssig1, &csig1);
norm2(&ssig2, &csig2);
C4f(g, eps, Ca);
B41 = SinCosSeries(FALSE, ssig1, csig1, Ca, nC4);
B42 = SinCosSeries(FALSE, ssig2, csig2, Ca, nC4);
S12 = A4 * (B42 - B41);
} else
/* Avoid problems with indeterminate sig1, sig2 on equator */
S12 = 0;
if (!meridian && somg12 == 2) {
somg12 = sin(omg12); comg12 = cos(omg12);
}
if (!meridian &&
/* omg12 < 3/4 * pi */
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 * 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. */