diff --git a/README.md b/README.md index c0efb2d..3ae17d2 100644 --- a/README.md +++ b/README.md @@ -3,7 +3,7 @@ [![Swift Version](https://img.shields.io/badge/swift-5.7-blue.svg)](https://swift.org) ![Platform](https://img.shields.io/badge/platform-macOS|linux--64-lightgray.svg) ![Build](https://github.com/dastrobu/geodesic/actions/workflows/ci.yaml/badge.svg) -[![GeographicLib Version](https://img.shields.io/badge/GeographicLib-2.0-blue.svg)](https://github.com/geographiclib/geographiclib-c/releases/tag/v2.0) +[![GeographicLib Version](https://img.shields.io/badge/GeographicLib-2.1-blue.svg)](https://github.com/geographiclib/geographiclib-c/releases/tag/v2.1) Solver for the inverse geodesic problem in Swift. @@ -24,10 +24,11 @@ and that's it. + ## Table of Contents - [Installation](#installation) - - [Swift Package Manager](#swift-package-manager) + - [Swift Package Manager](#swift-package-manager) - [Implementation Details](#implementation-details) - [Convergence and Tolerance](#convergence-and-tolerance) - [WGS 84 and other Ellipsoids](#wgs-84-and-other-ellipsoids) @@ -52,11 +53,10 @@ let package = Package( ## Implementation Details This Swift package is a wrapper for the -[C library for Geodesics](https://geographiclib.sourceforge.io/html/C/). The author of this library is Charles Karney ( -charles@karney.com). The goal of this Swift package is to make some algorithms from -[GeographicLib](https://github.com/geographiclib/geographiclib-c) available to the Swift world. Alternatively one can employ the -package -[vincenty](https://github.com/dastrobu/vincenty) +[C implementation of the geodesic routines in GeographicLib](https://github.com/geographiclib/geographiclib-c). +The goal of this Swift package is to make some algorithms from +GeographicLib available to the Swift world. Alternatively one can employ the +package [vincenty](https://github.com/dastrobu/vincenty) which is a much simpler solver for the inverse geodesic problem, completely written in Swift. Vincenty's formulae does, however, have some convergence problems in rare cases and may not give the same accuracy as Karney's algorithm. diff --git a/Sources/geographiclib/geodesic.c b/Sources/geographiclib/geodesic.c index d86e508..7f00ad1 100644 --- a/Sources/geographiclib/geodesic.c +++ b/Sources/geographiclib/geodesic.c @@ -55,66 +55,66 @@ 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; + tiny, tol0, tol1, tol2, tolb, xthresh; static void Init(void) { - if (!init) { - digits = DBL_MANT_DIG; - epsilon = DBL_EPSILON; - realmin = DBL_MIN; + if (!init) { + digits = DBL_MANT_DIG; + epsilon = DBL_EPSILON; + realmin = DBL_MIN; #if defined(M_PI) - pi = M_PI; + pi = M_PI; #else - pi = atan2(0.0, -1.0); + 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; - } + 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; + 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 + 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; + 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 double polyvalx(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) @@ -122,7 +122,7 @@ static void swapx(double* x, double* y) 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 + /* hypot for Visual Studio (A=win32) fails monotonicity, e.g., with * x = 0.6102683302836215 * y1 = 0.7906090004346522 * y2 = y1 + 1e-16 @@ -132,148 +132,148 @@ static void norm2(double* sinx, double* cosx) { * https://bugs.python.org/issue43088 */ double r = sqrt(*sinx * *sinx + *cosx * *cosx); #else - double r = hypot(*sinx, *cosx); + double r = hypot(*sinx, *cosx); #endif - *sinx /= r; - *cosx /= r; + *sinx /= r; + *cosx /= r; } static double AngNormalize(double x) { - double y = remainder(x, (double)td); - return fabs(y) == hd ? copysign((double)hd, x) : y; + 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; + /* 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); + /* 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); + /* 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); + /* 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; + /* 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); + 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[]); + 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[]); + 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[]); + 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[]); @@ -291,1732 +291,1738 @@ 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); + int crossings, boolx reverse, boolx sign); static double areareduceB(double area, double area0, - int crossings, boolx reverse, boolx sign); + 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); + 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; + 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); + 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); + 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); + 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 */ + 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)) + + /* 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_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; + 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; } - 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; + 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); + 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); + 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); + 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* 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); + 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); + 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 + 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 * - * echo 20.001 0 20.001 0 | GeodSolve -i + * 0 <= lon12 <= 180 + * -90 <= lat1 <= -0 + * lat1 <= lat2 <= -lat1 * - * 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); + * 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; + } - /* 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, + 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) { + /* 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; - } + if (tripb || + /* Reversed test to allow escape with NaNs */ + !(fabs(v) >= (tripn ? 8 : 1) * tol0) || + /* Enough bisections to get accurate result */ + numit == maxit2) + 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; + if (fabs(dalp1) < pi) { + double + sdalp1 = sin(dalp1), cdalp1 = cos(dalp1), + nsalp1 = salp1 * cdalp1 + calp1 * sdalp1; + if (nsalp1 > 0) { + 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; + } } - /* 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_DISTANCE) + s12 = 0 + s12x; /* Convert -0 to 0 */ - if (outmask & GEOD_REDUCEDLENGTH) - m12 = 0 + m12x; /* 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 (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. */ + if (salp12 == 0 && calp12 < 0) { + salp12 = tiny * calp1; + calp12 = -1; + } + alp12 = atan2(salp12, calp12); + } + S12 += g->c2 * alp12; + S12 *= swapp * lonsign * latsign; + /* Convert -0 to 0 */ + S12 += 0; } - 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. */ - if (salp12 == 0 && calp12 < 0) { - salp12 = tiny * calp1; - calp12 = -1; - } - alp12 = atan2(salp12, calp12); + /* Convert calp, salp to azimuth accounting for lonsign, swapp, latsign. */ + if (swapp < 0) { + swapx(&salp1, &salp2); + swapx(&calp1, &calp2); + if (outmask & GEOD_GEODESICSCALE) + swapx(&M12, &M21); } - S12 += g->c2 * alp12; - S12 *= swapp * lonsign * latsign; - /* Convert -0 to 0 */ - S12 += 0; - } - - /* Convert calp, salp to azimuth accounting for lonsign, swapp, latsign. */ - if (swapp < 0) { - swapx(&salp1, &salp2); - swapx(&calp1, &calp2); - if (outmask & GEOD_GEODESICSCALE) - swapx(&M12, &M21); - } - - salp1 *= swapp * lonsign; calp1 *= swapp * latsign; - salp2 *= swapp * lonsign; calp2 *= swapp * latsign; - - if (psalp1) *psalp1 = salp1; - if (pcalp1) *pcalp1 = calp1; - if (psalp2) *psalp2 = salp2; - if (pcalp2) *pcalp2 = calp2; - - if (outmask & GEOD_DISTANCE) - *ps12 = s12; - if (outmask & GEOD_REDUCEDLENGTH) - *pm12 = m12; - if (outmask & GEOD_GEODESICSCALE) { - if (pM12) *pM12 = M12; - if (pM21) *pM21 = M21; - } - if (outmask & GEOD_AREA) - *pS12 = S12; - - /* Returned value in [0, 180] */ - return a12; + + salp1 *= swapp * lonsign; calp1 *= swapp * latsign; + salp2 *= swapp * lonsign; calp2 *= swapp * latsign; + + if (psalp1) *psalp1 = salp1; + if (pcalp1) *pcalp1 = calp1; + if (psalp2) *psalp2 = salp2; + if (pcalp2) *pcalp2 = calp2; + + if (outmask & GEOD_DISTANCE) + *ps12 = s12; + if (outmask & GEOD_REDUCEDLENGTH) + *pm12 = m12; + if (outmask & GEOD_GEODESICSCALE) { + if (pM12) *pM12 = M12; + if (pM21) *pM21 = M21; + } + if (outmask & GEOD_AREA) + *pS12 = S12; + + /* Returned value in [0, 180] */ + return a12; } double geod_geninverse(const struct geod_geodesic* g, - double lat1, double lon1, double lat2, double lon2, - double* ps12, double* pazi1, double* pazi2, - double* pm12, double* pM12, double* pM21, - double* pS12) { - double salp1, calp1, salp2, calp2, - a12 = geod_geninverse_int(g, lat1, lon1, lat2, lon2, ps12, - &salp1, &calp1, &salp2, &calp2, - pm12, pM12, pM21, pS12); - if (pazi1) *pazi1 = atan2dx(salp1, calp1); - if (pazi2) *pazi2 = atan2dx(salp2, calp2); - return a12; + double lat1, double lon1, double lat2, double lon2, + double* ps12, double* pazi1, double* pazi2, + double* pm12, double* pM12, double* pM21, + double* pS12) { + double salp1, calp1, salp2, calp2, + a12 = geod_geninverse_int(g, lat1, lon1, lat2, lon2, ps12, + &salp1, &calp1, &salp2, &calp2, + pm12, pM12, pM21, pS12); + if (pazi1) *pazi1 = atan2dx(salp1, calp1); + if (pazi2) *pazi2 = atan2dx(salp2, calp2); + return a12; } void geod_inverseline(struct geod_geodesicline* l, - const struct geod_geodesic* g, - double lat1, double lon1, double lat2, double lon2, - unsigned caps) { - double salp1, calp1, - a12 = geod_geninverse_int(g, lat1, lon1, lat2, lon2, nullptr, - &salp1, &calp1, nullptr, nullptr, - nullptr, nullptr, nullptr, nullptr), - azi1 = atan2dx(salp1, calp1); - caps = caps ? caps : GEOD_DISTANCE_IN | GEOD_LONGITUDE; - /* Ensure that a12 can be converted to a distance */ - if (caps & (OUT_ALL & GEOD_DISTANCE_IN)) caps |= GEOD_DISTANCE; - geod_lineinit_int(l, g, lat1, lon1, azi1, salp1, calp1, caps); - geod_setarc(l, a12); + const struct geod_geodesic* g, + double lat1, double lon1, double lat2, double lon2, + unsigned caps) { + double salp1, calp1, + a12 = geod_geninverse_int(g, lat1, lon1, lat2, lon2, nullptr, + &salp1, &calp1, nullptr, nullptr, + nullptr, nullptr, nullptr, nullptr), + azi1 = atan2dx(salp1, calp1); + caps = caps ? caps : GEOD_DISTANCE_IN | GEOD_LONGITUDE; + /* Ensure that a12 can be converted to a distance */ + if (caps & (OUT_ALL & GEOD_DISTANCE_IN)) caps |= GEOD_DISTANCE; + geod_lineinit_int(l, g, lat1, lon1, azi1, salp1, calp1, caps); + geod_setarc(l, a12); } void geod_inverse(const struct geod_geodesic* g, - double lat1, double lon1, double lat2, double lon2, - double* ps12, double* pazi1, double* pazi2) { - geod_geninverse(g, lat1, lon1, lat2, lon2, ps12, pazi1, pazi2, - nullptr, nullptr, nullptr, nullptr); + double lat1, double lon1, double lat2, double lon2, + double* ps12, double* pazi1, double* pazi2) { + geod_geninverse(g, lat1, lon1, lat2, lon2, ps12, pazi1, pazi2, + nullptr, nullptr, nullptr, nullptr); } double SinCosSeries(boolx sinp, double sinx, double cosx, - const double c[], int n) { - /* 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 */ - double ar, y0, y1; - c += (n + sinp); /* Point to one beyond last element */ - ar = 2 * (cosx - sinx) * (cosx + sinx); /* 2 * cos(2 * x) */ - y0 = (n & 1) ? *--c : 0; y1 = 0; /* accumulators for sum */ - /* Now n is even */ - n /= 2; - while (n--) { - /* Unroll loop x 2, so accumulators return to their original role */ - y1 = ar * y0 - y1 + *--c; - y0 = ar * y1 - y0 + *--c; - } - return sinp - ? 2 * sinx * cosx * y0 /* sin(2 * x) * y0 */ - : cosx * (y0 - y1); /* cos(x) * (y0 - y1) */ + const double c[], int n) { + /* 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 */ + double ar, y0, y1; + c += (n + sinp); /* Point to one beyond last element */ + ar = 2 * (cosx - sinx) * (cosx + sinx); /* 2 * cos(2 * x) */ + y0 = (n & 1) ? *--c : 0; y1 = 0; /* accumulators for sum */ + /* Now n is even */ + n /= 2; + while (n--) { + /* Unroll loop x 2, so accumulators return to their original role */ + y1 = ar * y0 - y1 + *--c; + y0 = ar * y1 - y0 + *--c; + } + return sinp + ? 2 * sinx * cosx * y0 /* sin(2 * x) * y0 */ + : cosx * (y0 - y1); /* cos(x) * (y0 - y1) */ } 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[]) { - double m0 = 0, J12 = 0, A1 = 0, A2 = 0; - double Cb[nC]; - - /* Return m12b = (reduced length)/b; also calculate s12b = distance/b, - * and m0 = coefficient of secular term in expression for reduced length. */ - boolx redlp = pm12b || pm0 || pM12 || pM21; - if (ps12b || redlp) { - A1 = A1m1f(eps); - C1f(eps, Ca); - if (redlp) { - A2 = A2m1f(eps); - C2f(eps, Cb); - m0 = A1 - A2; - A2 = 1 + A2; + 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[]) { + double m0 = 0, J12 = 0, A1 = 0, A2 = 0; + double Cb[nC]; + + /* Return m12b = (reduced length)/b; also calculate s12b = distance/b, + * and m0 = coefficient of secular term in expression for reduced length. */ + boolx redlp = pm12b || pm0 || pM12 || pM21; + if (ps12b || redlp) { + A1 = A1m1f(eps); + C1f(eps, Ca); + if (redlp) { + A2 = A2m1f(eps); + C2f(eps, Cb); + m0 = A1 - A2; + A2 = 1 + A2; + } + A1 = 1 + A1; + } + if (ps12b) { + double B1 = SinCosSeries(TRUE, ssig2, csig2, Ca, nC1) - + SinCosSeries(TRUE, ssig1, csig1, Ca, nC1); + /* Missing a factor of b */ + *ps12b = A1 * (sig12 + B1); + if (redlp) { + double B2 = SinCosSeries(TRUE, ssig2, csig2, Cb, nC2) - + SinCosSeries(TRUE, ssig1, csig1, Cb, nC2); + J12 = m0 * sig12 + (A1 * B1 - A2 * B2); + } + } else if (redlp) { + /* Assume here that nC1 >= nC2 */ + int l; + for (l = 1; l <= nC2; ++l) + Cb[l] = A1 * Ca[l] - A2 * Cb[l]; + J12 = m0 * sig12 + (SinCosSeries(TRUE, ssig2, csig2, Cb, nC2) - + SinCosSeries(TRUE, ssig1, csig1, Cb, nC2)); } - A1 = 1 + A1; - } - if (ps12b) { - double B1 = SinCosSeries(TRUE, ssig2, csig2, Ca, nC1) - - SinCosSeries(TRUE, ssig1, csig1, Ca, nC1); - /* Missing a factor of b */ - *ps12b = A1 * (sig12 + B1); - if (redlp) { - double B2 = SinCosSeries(TRUE, ssig2, csig2, Cb, nC2) - - SinCosSeries(TRUE, ssig1, csig1, Cb, nC2); - J12 = m0 * sig12 + (A1 * B1 - A2 * B2); + if (pm0) *pm0 = m0; + if (pm12b) + /* Missing a factor of b. + * Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure + * accurate cancellation in the case of coincident points. */ + *pm12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - + csig1 * csig2 * J12; + if (pM12 || pM21) { + double csig12 = csig1 * csig2 + ssig1 * ssig2; + double t = g->ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2); + if (pM12) + *pM12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1; + if (pM21) + *pM21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2; } - } else if (redlp) { - /* Assume here that nC1 >= nC2 */ - int l; - for (l = 1; l <= nC2; ++l) - Cb[l] = A1 * Ca[l] - A2 * Cb[l]; - J12 = m0 * sig12 + (SinCosSeries(TRUE, ssig2, csig2, Cb, nC2) - - SinCosSeries(TRUE, ssig1, csig1, Cb, nC2)); - } - if (pm0) *pm0 = m0; - if (pm12b) - /* Missing a factor of b. - * Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure - * accurate cancellation in the case of coincident points. */ - *pm12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - - csig1 * csig2 * J12; - if (pM12 || pM21) { - double csig12 = csig1 * csig2 + ssig1 * ssig2; - double t = g->ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2); - if (pM12) - *pM12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1; - if (pM21) - *pM21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2; - } } 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 = sq(x), - q = sq(y), - r = (p + q - 1) / 6; - if ( !(q == 0 && r <= 0) ) { + /* 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 - /* 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 = 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; - double v, uv, w; - if (disc >= 0) { - double T3 = S + r3, T; - /* 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 ? -sqrt(disc) : sqrt(disc); /* T3 = (r * t)^3 */ - /* N.B. cbrt always returns the double root. cbrt(-8) = -2. */ - T = 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 = atan2(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 * cos(ang / 3); + p = sq(x), + q = 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 = 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; + double v, uv, w; + if (disc >= 0) { + double T3 = S + r3, T; + /* 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 ? -sqrt(disc) : sqrt(disc); /* T3 = (r * t)^3 */ + /* N.B. cbrt always returns the double root. cbrt(-8) = -2. */ + T = 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 = atan2(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 * cos(ang / 3); + } + v = sqrt(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 / (sqrt(uv + 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; } - v = sqrt(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 / (sqrt(uv + 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; + return k; } 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[]) { - double salp1 = 0, calp1 = 0, salp2 = 0, calp2 = 0, dnm = 0; - - /* 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. */ - double - sig12 = -1, /* Return value */ + 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[]) { + double salp1 = 0, calp1 = 0, salp2 = 0, calp2 = 0, dnm = 0; + + /* 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. */ + double + sig12 = -1, /* Return value */ /* bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0] */ sbet12 = sbet2 * cbet1 - cbet2 * sbet1, - cbet12 = cbet2 * cbet1 + sbet2 * sbet1; - double sbet12a; - boolx shortline = cbet12 >= 0 && sbet12 < 0.5 && cbet2 * lam12 < 0.5; - double somg12, comg12, ssig12, csig12; - sbet12a = sbet2 * cbet1 + cbet2 * sbet1; - if (shortline) { - double sbetm2 = sq(sbet1 + sbet2), omg12; - /* sin((bet1+bet2)/2)^2 - * = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2) */ - sbetm2 /= sbetm2 + sq(cbet1 + cbet2); - dnm = sqrt(1 + g->ep2 * sbetm2); - omg12 = lam12 / (g->f1 * dnm); - somg12 = sin(omg12); comg12 = cos(omg12); - } else { - somg12 = slam12; comg12 = clam12; - } - - salp1 = cbet2 * somg12; - calp1 = comg12 >= 0 ? - sbet12 + cbet2 * sbet1 * sq(somg12) / (1 + comg12) : - sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12); - - ssig12 = hypot(salp1, calp1); - csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12; - - if (shortline && ssig12 < g->etol2) { - /* really short lines */ - salp2 = cbet1 * somg12; - calp2 = sbet12 - cbet1 * sbet2 * - (comg12 >= 0 ? sq(somg12) / (1 + comg12) : 1 - comg12); - norm2(&salp2, &calp2); - /* Set return value */ - sig12 = atan2(ssig12, csig12); - } else if (fabs(g->n) > 0.1 || /* No astroid calc if too eccentric */ - csig12 >= 0 || - ssig12 >= 6 * fabs(g->n) * pi * 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 x, y, lamscale, betscale; - double lam12x = atan2(-slam12, -clam12); /* lam12 - pi */ - if (g->f >= 0) { /* In fact f == 0 does not get here */ - /* x = dlong, y = dlat */ - { - double - k2 = sq(sbet1) * g->ep2, - eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2); - lamscale = g->f * cbet1 * A3f(g, eps) * pi; - } - betscale = lamscale * cbet1; - - x = lam12x / lamscale; - y = sbet12a / betscale; - } else { /* f < 0 */ - /* x = dlat, y = dlong */ - double - cbet12a = cbet2 * cbet1 - sbet2 * sbet1, - bet12a = atan2(sbet12a, cbet12a); - double m12b, m0; - /* In the case of lon12 = 180, this repeats a calculation made in - * Inverse. */ - Lengths(g, g->n, pi + bet12a, - sbet1, -cbet1, dn1, sbet2, cbet2, dn2, - cbet1, cbet2, nullptr, &m12b, &m0, nullptr, nullptr, Ca); - x = -1 + m12b / (cbet1 * cbet2 * m0 * pi); - betscale = x < -0.01 ? sbet12a / x : - -g->f * sq(cbet1) * pi; - lamscale = betscale / cbet1; - y = lam12x / lamscale; + cbet12 = cbet2 * cbet1 + sbet2 * sbet1; + double sbet12a; + boolx shortline = cbet12 >= 0 && sbet12 < 0.5 && cbet2 * lam12 < 0.5; + double somg12, comg12, ssig12, csig12; + sbet12a = sbet2 * cbet1 + cbet2 * sbet1; + if (shortline) { + double sbetm2 = sq(sbet1 + sbet2), omg12; + /* sin((bet1+bet2)/2)^2 + * = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2) */ + sbetm2 /= sbetm2 + sq(cbet1 + cbet2); + dnm = sqrt(1 + g->ep2 * sbetm2); + omg12 = lam12 / (g->f1 * dnm); + somg12 = sin(omg12); comg12 = cos(omg12); + } else { + somg12 = slam12; comg12 = clam12; } - if (y > -tol1 && x > -1 - xthresh) { - /* strip near cut */ - if (g->f >= 0) { - salp1 = fmin(1.0, -x); calp1 = - sqrt(1 - sq(salp1)); - } else { - calp1 = fmax(x > -tol1 ? 0.0 : -1.0, x); - salp1 = sqrt(1 - sq(calp1)); - } + salp1 = cbet2 * somg12; + calp1 = comg12 >= 0 ? + sbet12 + cbet2 * sbet1 * sq(somg12) / (1 + comg12) : + sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12); + + ssig12 = hypot(salp1, calp1); + csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12; + + if (shortline && ssig12 < g->etol2) { + /* really short lines */ + salp2 = cbet1 * somg12; + calp2 = sbet12 - cbet1 * sbet2 * + (comg12 >= 0 ? sq(somg12) / (1 + comg12) : 1 - comg12); + norm2(&salp2, &calp2); + /* Set return value */ + sig12 = atan2(ssig12, csig12); + } else if (fabs(g->n) > 0.1 || /* No astroid calc if too eccentric */ + csig12 >= 0 || + ssig12 >= 6 * fabs(g->n) * pi * sq(cbet1)) { + /* Nothing to do, zeroth order spherical approximation is OK */ } 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 * ( g->f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k ); - somg12 = sin(omg12a); comg12 = -cos(omg12a); - /* Update spherical estimate of alp1 using omg12 instead of lam12 */ - salp1 = cbet2 * somg12; - calp1 = sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12); + /* 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 x, y, lamscale, betscale; + double lam12x = atan2(-slam12, -clam12); /* lam12 - pi */ + if (g->f >= 0) { /* In fact f == 0 does not get here */ + /* x = dlong, y = dlat */ + { + double + k2 = sq(sbet1) * g->ep2, + eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2); + lamscale = g->f * cbet1 * A3f(g, eps) * pi; + } + betscale = lamscale * cbet1; + + x = lam12x / lamscale; + y = sbet12a / betscale; + } else { /* f < 0 */ + /* x = dlat, y = dlong */ + double + cbet12a = cbet2 * cbet1 - sbet2 * sbet1, + bet12a = atan2(sbet12a, cbet12a); + double m12b, m0; + /* In the case of lon12 = 180, this repeats a calculation made in + * Inverse. */ + Lengths(g, g->n, pi + bet12a, + sbet1, -cbet1, dn1, sbet2, cbet2, dn2, + cbet1, cbet2, nullptr, &m12b, &m0, nullptr, nullptr, Ca); + x = -1 + m12b / (cbet1 * cbet2 * m0 * pi); + betscale = x < -0.01 ? sbet12a / x : + -g->f * sq(cbet1) * pi; + lamscale = betscale / cbet1; + y = lam12x / lamscale; + } + + if (y > -tol1 && x > -1 - xthresh) { + /* strip near cut */ + if (g->f >= 0) { + salp1 = fmin(1.0, -x); calp1 = - sqrt(1 - sq(salp1)); + } else { + calp1 = fmax(x > -tol1 ? 0.0 : -1.0, x); + salp1 = sqrt(1 - sq(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 * ( g->f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k ); + somg12 = sin(omg12a); comg12 = -cos(omg12a); + /* Update spherical estimate of alp1 using omg12 instead of lam12 */ + salp1 = cbet2 * somg12; + calp1 = sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12); + } } - } - /* Sanity check on starting guess. Backwards check allows NaN through. */ - if (!(salp1 <= 0)) - norm2(&salp1, &calp1); - else { - salp1 = 1; calp1 = 0; - } - - *psalp1 = salp1; - *pcalp1 = calp1; - if (shortline) - *pdnm = dnm; - if (sig12 >= 0) { - *psalp2 = salp2; - *pcalp2 = calp2; - } - return sig12; + /* Sanity check on starting guess. Backwards check allows NaN through. */ + if (!(salp1 <= 0)) + norm2(&salp1, &calp1); + else { + salp1 = 1; calp1 = 0; + } + + *psalp1 = salp1; + *pcalp1 = calp1; + if (shortline) + *pdnm = dnm; + if (sig12 >= 0) { + *psalp2 = salp2; + *pcalp2 = calp2; + } + return sig12; } 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[]) { - double salp2 = 0, calp2 = 0, sig12 = 0, - ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, - domg12 = 0, dlam12 = 0; - double salp0, calp0; - double somg1, comg1, somg2, comg2, somg12, comg12, lam12; - double B312, eta, k2; - - if (sbet1 == 0 && calp1 == 0) - /* Break degeneracy of equatorial line. This case has already been - * handled. */ - calp1 = -tiny; - - /* sin(alp1) * cos(bet1) = sin(alp0) */ - salp0 = salp1 * cbet1; - calp0 = hypot(calp1, salp1 * sbet1); /* calp0 > 0 */ - - /* tan(bet1) = tan(sig1) * cos(alp1) - * tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1) */ - ssig1 = sbet1; somg1 = salp0 * sbet1; - csig1 = comg1 = calp1 * cbet1; - norm2(&ssig1, &csig1); - /* norm2(&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) */ - 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]). */ - calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ? - sqrt(sq(calp1 * cbet1) + - (cbet1 < -sbet1 ? - (cbet2 - cbet1) * (cbet1 + cbet2) : - (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 : - fabs(calp1); - /* tan(bet2) = tan(sig2) * cos(alp2) - * tan(omg2) = sin(alp0) * tan(sig2). */ - ssig2 = sbet2; somg2 = salp0 * sbet2; - csig2 = comg2 = calp2 * cbet2; - norm2(&ssig2, &csig2); - /* norm2(&somg2, &comg2); -- don't need to normalize! */ - - /* sig12 = sig2 - sig1, limit to [0, pi] */ - sig12 = atan2(fmax(0.0, csig1 * ssig2 - ssig1 * csig2) + 0, - csig1 * csig2 + ssig1 * ssig2); - - /* omg12 = omg2 - omg1, limit to [0, pi] */ - somg12 = fmax(0.0, comg1 * somg2 - somg1 * comg2) + 0; - comg12 = comg1 * comg2 + somg1 * somg2; - /* eta = omg12 - lam120 */ - eta = atan2(somg12 * clam120 - comg12 * slam120, - comg12 * clam120 + somg12 * slam120); - k2 = sq(calp0) * g->ep2; - eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2); - C3f(g, eps, Ca); - B312 = (SinCosSeries(TRUE, ssig2, csig2, Ca, nC3-1) - - SinCosSeries(TRUE, ssig1, csig1, Ca, nC3-1)); - domg12 = -g->f * A3f(g, eps) * salp0 * (sig12 + B312); - lam12 = eta + domg12; - - if (diffp) { - if (calp2 == 0) - dlam12 = - 2 * g->f1 * dn1 / sbet1; - else { - Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, - cbet1, cbet2, nullptr, &dlam12, nullptr, nullptr, nullptr, Ca); - dlam12 *= g->f1 / (calp2 * cbet2); + 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[]) { + double salp2 = 0, calp2 = 0, sig12 = 0, + ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, + domg12 = 0, dlam12 = 0; + double salp0, calp0; + double somg1, comg1, somg2, comg2, somg12, comg12, lam12; + double B312, eta, k2; + + if (sbet1 == 0 && calp1 == 0) + /* Break degeneracy of equatorial line. This case has already been + * handled. */ + calp1 = -tiny; + + /* sin(alp1) * cos(bet1) = sin(alp0) */ + salp0 = salp1 * cbet1; + calp0 = hypot(calp1, salp1 * sbet1); /* calp0 > 0 */ + + /* tan(bet1) = tan(sig1) * cos(alp1) + * tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1) */ + ssig1 = sbet1; somg1 = salp0 * sbet1; + csig1 = comg1 = calp1 * cbet1; + norm2(&ssig1, &csig1); + /* norm2(&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) */ + 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]). */ + calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ? + sqrt(sq(calp1 * cbet1) + + (cbet1 < -sbet1 ? + (cbet2 - cbet1) * (cbet1 + cbet2) : + (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 : + fabs(calp1); + /* tan(bet2) = tan(sig2) * cos(alp2) + * tan(omg2) = sin(alp0) * tan(sig2). */ + ssig2 = sbet2; somg2 = salp0 * sbet2; + csig2 = comg2 = calp2 * cbet2; + norm2(&ssig2, &csig2); + /* norm2(&somg2, &comg2); -- don't need to normalize! */ + + /* sig12 = sig2 - sig1, limit to [0, pi] */ + sig12 = atan2(fmax(0.0, csig1 * ssig2 - ssig1 * csig2) + 0, + csig1 * csig2 + ssig1 * ssig2); + + /* omg12 = omg2 - omg1, limit to [0, pi] */ + somg12 = fmax(0.0, comg1 * somg2 - somg1 * comg2) + 0; + comg12 = comg1 * comg2 + somg1 * somg2; + /* eta = omg12 - lam120 */ + eta = atan2(somg12 * clam120 - comg12 * slam120, + comg12 * clam120 + somg12 * slam120); + k2 = sq(calp0) * g->ep2; + eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2); + C3f(g, eps, Ca); + B312 = (SinCosSeries(TRUE, ssig2, csig2, Ca, nC3-1) - + SinCosSeries(TRUE, ssig1, csig1, Ca, nC3-1)); + domg12 = -g->f * A3f(g, eps) * salp0 * (sig12 + B312); + lam12 = eta + domg12; + + if (diffp) { + if (calp2 == 0) + dlam12 = - 2 * g->f1 * dn1 / sbet1; + else { + Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, + cbet1, cbet2, nullptr, &dlam12, nullptr, nullptr, nullptr, Ca); + dlam12 *= g->f1 / (calp2 * cbet2); + } } - } - - *psalp2 = salp2; - *pcalp2 = calp2; - *psig12 = sig12; - *pssig1 = ssig1; - *pcsig1 = csig1; - *pssig2 = ssig2; - *pcsig2 = csig2; - *peps = eps; - *pdomg12 = domg12; - if (diffp) - *pdlam12 = dlam12; - - return lam12; + + *psalp2 = salp2; + *pcalp2 = calp2; + *psig12 = sig12; + *pssig1 = ssig1; + *pcsig1 = csig1; + *pssig2 = ssig2; + *pcsig2 = csig2; + *peps = eps; + *pdomg12 = domg12; + if (diffp) + *pdlam12 = dlam12; + + return lam12; } double A3f(const struct geod_geodesic* g, double eps) { - /* Evaluate A3 */ - return polyval(nA3 - 1, g->A3x, eps); + /* Evaluate A3 */ + return polyvalx(nA3 - 1, g->A3x, eps); } void C3f(const struct geod_geodesic* g, double eps, double c[]) { - /* Evaluate C3 coeffs - * Elements c[1] through c[nC3 - 1] are set */ - double mult = 1; - int o = 0, l; - for (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 * polyval(m, g->C3x + o, eps); - o += m + 1; - } + /* Evaluate C3 coeffs + * Elements c[1] through c[nC3 - 1] are set */ + double mult = 1; + int o = 0, l; + for (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 * polyvalx(m, g->C3x + o, eps); + o += m + 1; + } } void C4f(const struct geod_geodesic* g, double eps, double c[]) { - /* Evaluate C4 coeffs - * Elements c[0] through c[nC4 - 1] are set */ - double mult = 1; - int o = 0, l; - for (l = 0; l < nC4; ++l) { /* l is index of C4[l] */ - int m = nC4 - l - 1; /* order of polynomial in eps */ - c[l] = mult * polyval(m, g->C4x + o, eps); - o += m + 1; - mult *= eps; - } + /* Evaluate C4 coeffs + * Elements c[0] through c[nC4 - 1] are set */ + double mult = 1; + int o = 0, l; + for (l = 0; l < nC4; ++l) { /* l is index of C4[l] */ + int m = nC4 - l - 1; /* order of polynomial in eps */ + c[l] = mult * polyvalx(m, g->C4x + o, eps); + o += m + 1; + mult *= eps; + } } /* The scale factor A1-1 = mean value of (d/dsigma)I1 - 1 */ double A1m1f(double eps) { - static const double coeff[] = { - /* (1-eps)*A1-1, polynomial in eps2 of order 3 */ - 1, 4, 64, 0, 256, - }; - int m = nA1/2; - double t = polyval(m, coeff, sq(eps)) / coeff[m + 1]; - return (t + eps) / (1 - eps); + static const double coeff[] = { + /* (1-eps)*A1-1, polynomial in eps2 of order 3 */ + 1, 4, 64, 0, 256, + }; + int m = nA1/2; + double t = polyvalx(m, coeff, sq(eps)) / coeff[m + 1]; + return (t + eps) / (1 - eps); } /* The coefficients C1[l] in the Fourier expansion of B1 */ void C1f(double eps, double c[]) { - static const 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 = sq(eps), - d = eps; - int o = 0, l; - for (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 * polyval(m, coeff + o, eps2) / coeff[o + m + 1]; - o += m + 2; - d *= eps; - } + static const 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 = sq(eps), + d = eps; + int o = 0, l; + for (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 * polyvalx(m, coeff + o, eps2) / coeff[o + m + 1]; + o += m + 2; + d *= eps; + } } /* The coefficients C1p[l] in the Fourier expansion of B1p */ void C1pf(double eps, double c[]) { - static const 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 = sq(eps), - d = eps; - int o = 0, l; - for (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 * polyval(m, coeff + o, eps2) / coeff[o + m + 1]; - o += m + 2; - d *= eps; - } + static const 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 = sq(eps), + d = eps; + int o = 0, l; + for (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 * polyvalx(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 */ double A2m1f(double eps) { - static const double coeff[] = { - /* (eps+1)*A2-1, polynomial in eps2 of order 3 */ - -11, -28, -192, 0, 256, - }; - int m = nA2/2; - double t = polyval(m, coeff, sq(eps)) / coeff[m + 1]; - return (t - eps) / (1 + eps); + static const double coeff[] = { + /* (eps+1)*A2-1, polynomial in eps2 of order 3 */ + -11, -28, -192, 0, 256, + }; + int m = nA2/2; + double t = polyvalx(m, coeff, sq(eps)) / coeff[m + 1]; + return (t - eps) / (1 + eps); } /* The coefficients C2[l] in the Fourier expansion of B2 */ void C2f(double eps, double c[]) { - static const 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 = sq(eps), - d = eps; - int o = 0, l; - for (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 * polyval(m, coeff + o, eps2) / coeff[o + m + 1]; - o += m + 2; - d *= eps; - } + static const 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 = sq(eps), + d = eps; + int o = 0, l; + for (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 * polyvalx(m, coeff + o, eps2) / coeff[o + m + 1]; + o += m + 2; + d *= eps; + } } /* The scale factor A3 = mean value of (d/dsigma)I3 */ void A3coeff(struct geod_geodesic* g) { - static const 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, j; - for (j = nA3 - 1; j >= 0; --j) { /* coeff of eps^j */ - int m = nA3 - j - 1 < j ? nA3 - j - 1 : j; /* order of polynomial in n */ - g->A3x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1]; - o += m + 2; - } + static const 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, j; + for (j = nA3 - 1; j >= 0; --j) { /* coeff of eps^j */ + int m = nA3 - j - 1 < j ? nA3 - j - 1 : j; /* order of polynomial in n */ + g->A3x[k++] = polyvalx(m, coeff + o, g->n) / coeff[o + m + 1]; + o += m + 2; + } } /* The coefficients C3[l] in the Fourier expansion of B3 */ void C3coeff(struct geod_geodesic* g) { - static const 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, l, j; - for (l = 1; l < nC3; ++l) { /* l is index of C3[l] */ - for (j = nC3 - 1; j >= l; --j) { /* coeff of eps^j */ - int m = nC3 - j - 1 < j ? nC3 - j - 1 : j; /* order of polynomial in n */ - g->C3x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1]; - o += m + 2; + static const 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, l, j; + for (l = 1; l < nC3; ++l) { /* l is index of C3[l] */ + for (j = nC3 - 1; j >= l; --j) { /* coeff of eps^j */ + int m = nC3 - j - 1 < j ? nC3 - j - 1 : j; /* order of polynomial in n */ + g->C3x[k++] = polyvalx(m, coeff + o, g->n) / coeff[o + m + 1]; + o += m + 2; + } } - } } /* The coefficients C4[l] in the Fourier expansion of I4 */ void C4coeff(struct geod_geodesic* g) { - static const 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, l, j; - for (l = 0; l < nC4; ++l) { /* l is index of C4[l] */ - for (j = nC4 - 1; j >= l; --j) { /* coeff of eps^j */ - int m = nC4 - j - 1; /* order of polynomial in n */ - g->C4x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1]; - o += m + 2; + static const 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, l, j; + for (l = 0; l < nC4; ++l) { /* l is index of C4[l] */ + for (j = nC4 - 1; j >= l; --j) { /* coeff of eps^j */ + int m = nC4 - j - 1; /* order of polynomial in n */ + g->C4x[k++] = polyvalx(m, coeff + o, g->n) / coeff[o + m + 1]; + o += m + 2; + } } - } } int transit(double lon1, double lon2) { - double lon12; - /* Return 1 or -1 if crossing prime meridian in east or west direction. - * Otherwise return zero. */ - /* Compute lon12 the same way as Geodesic::Inverse. */ - lon12 = AngDiff(lon1, lon2, nullptr); - lon1 = AngNormalize(lon1); - lon2 = AngNormalize(lon2); - return - lon12 > 0 && ((lon1 < 0 && lon2 >= 0) || - (lon1 > 0 && lon2 == 0)) ? 1 : - (lon12 < 0 && lon1 >= 0 && lon2 < 0 ? -1 : 0); + double lon12; + /* Return 1 or -1 if crossing prime meridian in east or west direction. + * Otherwise return zero. */ + /* Compute lon12 the same way as Geodesic::Inverse. */ + lon12 = AngDiff(lon1, lon2, nullptr); + lon1 = AngNormalize(lon1); + lon2 = AngNormalize(lon2); + return + lon12 > 0 && ((lon1 < 0 && lon2 >= 0) || + (lon1 > 0 && lon2 == 0)) ? 1 : + (lon12 < 0 && lon1 >= 0 && lon2 < 0 ? -1 : 0); } int transitdirect(double lon1, double lon2) { - /* Compute exactly the parity of - * int(floor(lon2 / 360)) - int(floor(lon1 / 360)) */ - lon1 = remainder(lon1, 2.0 * td); lon2 = remainder(lon2, 2.0 * td); - return ( (lon2 >= 0 && lon2 < td ? 0 : 1) - - (lon1 >= 0 && lon1 < td ? 0 : 1) ); + /* Compute exactly the parity of + * int(floor(lon2 / 360)) - int(floor(lon1 / 360)) */ + lon1 = remainder(lon1, 2.0 * td); lon2 = remainder(lon2, 2.0 * td); + return ( (lon2 >= 0 && lon2 < td ? 0 : 1) - + (lon1 >= 0 && lon1 < td ? 0 : 1) ); } void accini(double s[]) { - /* Initialize an accumulator; this is an array with two elements. */ - s[0] = s[1] = 0; + /* Initialize an accumulator; this is an array with two elements. */ + s[0] = s[1] = 0; } void acccopy(const double s[], double t[]) { - /* Copy an accumulator; t = s. */ - t[0] = s[0]; t[1] = s[1]; + /* Copy an accumulator; t = s. */ + t[0] = s[0]; t[1] = s[1]; } void accadd(double s[], double y) { - /* Add y to an accumulator. */ - double u, z = sumx(y, s[1], &u); - s[0] = sumx(z, s[0], &s[1]); - if (s[0] == 0) - s[0] = u; - else - s[1] = s[1] + u; + /* Add y to an accumulator. */ + double u, z = sumx(y, s[1], &u); + s[0] = sumx(z, s[0], &s[1]); + if (s[0] == 0) + s[0] = u; + else + s[1] = s[1] + u; } double accsum(const double s[], double y) { - /* Return accumulator + y (but don't add to accumulator). */ - double t[2]; - acccopy(s, t); - accadd(t, y); - return t[0]; + /* Return accumulator + y (but don't add to accumulator). */ + double t[2]; + acccopy(s, t); + accadd(t, y); + return t[0]; } void accneg(double s[]) { - /* Negate an accumulator. */ - s[0] = -s[0]; s[1] = -s[1]; + /* Negate an accumulator. */ + s[0] = -s[0]; s[1] = -s[1]; } void accrem(double s[], double y) { - /* Reduce to [-y/2, y/2]. */ - s[0] = remainder(s[0], y); - accadd(s, 0.0); + /* Reduce to [-y/2, y/2]. */ + s[0] = remainder(s[0], y); + accadd(s, 0.0); } void geod_polygon_init(struct geod_polygon* p, boolx polylinep) { - p->polyline = (polylinep != 0); - geod_polygon_clear(p); + p->polyline = (polylinep != 0); + geod_polygon_clear(p); } void geod_polygon_clear(struct geod_polygon* p) { - p->lat0 = p->lon0 = p->lat = p->lon = NaN; - accini(p->P); - accini(p->A); - p->num = p->crossings = 0; + p->lat0 = p->lon0 = p->lat = p->lon = NaN; + accini(p->P); + accini(p->A); + p->num = p->crossings = 0; } void geod_polygon_addpoint(const struct geod_geodesic* g, - struct geod_polygon* p, - double lat, double lon) { - if (p->num == 0) { - p->lat0 = p->lat = lat; - p->lon0 = p->lon = lon; - } else { - double s12, S12 = 0; /* Initialize S12 to stop Visual Studio warning */ - geod_geninverse(g, p->lat, p->lon, lat, lon, - &s12, nullptr, nullptr, nullptr, nullptr, nullptr, - p->polyline ? nullptr : &S12); - accadd(p->P, s12); - if (!p->polyline) { - accadd(p->A, S12); - p->crossings += transit(p->lon, lon); + struct geod_polygon* p, + double lat, double lon) { + if (p->num == 0) { + p->lat0 = p->lat = lat; + p->lon0 = p->lon = lon; + } else { + double s12, S12 = 0; /* Initialize S12 to stop Visual Studio warning */ + geod_geninverse(g, p->lat, p->lon, lat, lon, + &s12, nullptr, nullptr, nullptr, nullptr, nullptr, + p->polyline ? nullptr : &S12); + accadd(p->P, s12); + if (!p->polyline) { + accadd(p->A, S12); + p->crossings += transit(p->lon, lon); + } + p->lat = lat; p->lon = lon; } - p->lat = lat; p->lon = lon; - } - ++p->num; + ++p->num; } void geod_polygon_addedge(const struct geod_geodesic* g, - struct geod_polygon* p, - double azi, double s) { - if (p->num) { /* Do nothing is num is zero */ - /* Initialize S12 to stop Visual Studio warning. Initialization of lat and - * lon is to make CLang static analyzer happy. */ - double lat = 0, lon = 0, S12 = 0; - geod_gendirect(g, p->lat, p->lon, azi, GEOD_LONG_UNROLL, s, - &lat, &lon, nullptr, - nullptr, nullptr, nullptr, nullptr, - p->polyline ? nullptr : &S12); - accadd(p->P, s); - if (!p->polyline) { - accadd(p->A, S12); - p->crossings += transitdirect(p->lon, lon); + struct geod_polygon* p, + double azi, double s) { + if (p->num) { /* Do nothing is num is zero */ + /* Initialize S12 to stop Visual Studio warning. Initialization of lat and + * lon is to make CLang static analyzer happy. */ + double lat = 0, lon = 0, S12 = 0; + geod_gendirect(g, p->lat, p->lon, azi, GEOD_LONG_UNROLL, s, + &lat, &lon, nullptr, + nullptr, nullptr, nullptr, nullptr, + p->polyline ? nullptr : &S12); + accadd(p->P, s); + if (!p->polyline) { + accadd(p->A, S12); + p->crossings += transitdirect(p->lon, lon); + } + p->lat = lat; p->lon = lon; + ++p->num; } - p->lat = lat; p->lon = lon; - ++p->num; - } } unsigned geod_polygon_compute(const struct geod_geodesic* g, - const struct geod_polygon* p, - boolx reverse, boolx sign, - double* pA, double* pP) { - double s12, S12, t[2]; - if (p->num < 2) { - if (pP) *pP = 0; - if (!p->polyline && pA) *pA = 0; - return p->num; - } - if (p->polyline) { - if (pP) *pP = p->P[0]; + const struct geod_polygon* p, + boolx reverse, boolx sign, + double* pA, double* pP) { + double s12, S12, t[2]; + if (p->num < 2) { + if (pP) *pP = 0; + if (!p->polyline && pA) *pA = 0; + return p->num; + } + if (p->polyline) { + if (pP) *pP = p->P[0]; + return p->num; + } + geod_geninverse(g, p->lat, p->lon, p->lat0, p->lon0, + &s12, nullptr, nullptr, nullptr, nullptr, nullptr, &S12); + if (pP) *pP = accsum(p->P, s12); + acccopy(p->A, t); + accadd(t, S12); + if (pA) *pA = areareduceA(t, 4 * pi * g->c2, + p->crossings + transit(p->lon, p->lon0), + reverse, sign); return p->num; - } - geod_geninverse(g, p->lat, p->lon, p->lat0, p->lon0, - &s12, nullptr, nullptr, nullptr, nullptr, nullptr, &S12); - if (pP) *pP = accsum(p->P, s12); - acccopy(p->A, t); - accadd(t, S12); - if (pA) *pA = areareduceA(t, 4 * pi * g->c2, - p->crossings + transit(p->lon, p->lon0), - reverse, sign); - return p->num; } unsigned geod_polygon_testpoint(const struct geod_geodesic* g, - const struct geod_polygon* p, - double lat, double lon, - boolx reverse, boolx sign, - double* pA, double* pP) { - double perimeter, tempsum; - int crossings, i; - unsigned num = p->num + 1; - if (num == 1) { - if (pP) *pP = 0; - if (!p->polyline && pA) *pA = 0; - return num; - } - perimeter = p->P[0]; - tempsum = p->polyline ? 0 : p->A[0]; - crossings = p->crossings; - for (i = 0; i < (p->polyline ? 1 : 2); ++i) { - double s12, S12 = 0; /* Initialize S12 to stop Visual Studio warning */ - geod_geninverse(g, - i == 0 ? p->lat : lat, i == 0 ? p->lon : lon, - i != 0 ? p->lat0 : lat, i != 0 ? p->lon0 : lon, - &s12, nullptr, nullptr, nullptr, nullptr, nullptr, - p->polyline ? nullptr : &S12); - perimeter += s12; - if (!p->polyline) { - tempsum += S12; - crossings += transit(i == 0 ? p->lon : lon, - i != 0 ? p->lon0 : lon); + const struct geod_polygon* p, + double lat, double lon, + boolx reverse, boolx sign, + double* pA, double* pP) { + double perimeter, tempsum; + int crossings, i; + unsigned num = p->num + 1; + if (num == 1) { + if (pP) *pP = 0; + if (!p->polyline && pA) *pA = 0; + return num; + } + perimeter = p->P[0]; + tempsum = p->polyline ? 0 : p->A[0]; + crossings = p->crossings; + for (i = 0; i < (p->polyline ? 1 : 2); ++i) { + double s12, S12 = 0; /* Initialize S12 to stop Visual Studio warning */ + geod_geninverse(g, + i == 0 ? p->lat : lat, i == 0 ? p->lon : lon, + i != 0 ? p->lat0 : lat, i != 0 ? p->lon0 : lon, + &s12, nullptr, nullptr, nullptr, nullptr, nullptr, + p->polyline ? nullptr : &S12); + perimeter += s12; + if (!p->polyline) { + tempsum += S12; + crossings += transit(i == 0 ? p->lon : lon, + i != 0 ? p->lon0 : lon); + } } - } - if (pP) *pP = perimeter; - if (p->polyline) - return num; + if (pP) *pP = perimeter; + if (p->polyline) + return num; - if (pA) *pA = areareduceB(tempsum, 4 * pi * g->c2, crossings, reverse, sign); - return num; + if (pA) *pA = areareduceB(tempsum, 4 * pi * g->c2, crossings, reverse, sign); + return num; } unsigned geod_polygon_testedge(const struct geod_geodesic* g, - const struct geod_polygon* p, - double azi, double s, - boolx reverse, boolx sign, - double* pA, double* pP) { - double perimeter, tempsum; - int crossings; - unsigned num = p->num + 1; - if (num == 1) { /* we don't have a starting point! */ - if (pP) *pP = NaN; - if (!p->polyline && pA) *pA = NaN; - return 0; - } - perimeter = p->P[0] + s; - if (p->polyline) { + const struct geod_polygon* p, + double azi, double s, + boolx reverse, boolx sign, + double* pA, double* pP) { + double perimeter, tempsum; + int crossings; + unsigned num = p->num + 1; + if (num == 1) { /* we don't have a starting point! */ + if (pP) *pP = NaN; + if (!p->polyline && pA) *pA = NaN; + return 0; + } + perimeter = p->P[0] + s; + if (p->polyline) { + if (pP) *pP = perimeter; + return num; + } + + tempsum = p->A[0]; + crossings = p->crossings; + { + /* Initialization of lat, lon, and S12 is to make CLang static analyzer + * happy. */ + double lat = 0, lon = 0, s12, S12 = 0; + geod_gendirect(g, p->lat, p->lon, azi, GEOD_LONG_UNROLL, s, + &lat, &lon, nullptr, + nullptr, nullptr, nullptr, nullptr, &S12); + tempsum += S12; + crossings += transitdirect(p->lon, lon); + geod_geninverse(g, lat, lon, p->lat0, p->lon0, + &s12, nullptr, nullptr, nullptr, nullptr, nullptr, &S12); + perimeter += s12; + tempsum += S12; + crossings += transit(lon, p->lon0); + } + if (pP) *pP = perimeter; + if (pA) *pA = areareduceB(tempsum, 4 * pi * g->c2, crossings, reverse, sign); return num; - } - - tempsum = p->A[0]; - crossings = p->crossings; - { - /* Initialization of lat, lon, and S12 is to make CLang static analyzer - * happy. */ - double lat = 0, lon = 0, s12, S12 = 0; - geod_gendirect(g, p->lat, p->lon, azi, GEOD_LONG_UNROLL, s, - &lat, &lon, nullptr, - nullptr, nullptr, nullptr, nullptr, &S12); - tempsum += S12; - crossings += transitdirect(p->lon, lon); - geod_geninverse(g, lat, lon, p->lat0, p->lon0, - &s12, nullptr, nullptr, nullptr, nullptr, nullptr, &S12); - perimeter += s12; - tempsum += S12; - crossings += transit(lon, p->lon0); - } - - if (pP) *pP = perimeter; - if (pA) *pA = areareduceB(tempsum, 4 * pi * g->c2, crossings, reverse, sign); - return num; } void geod_polygonarea(const struct geod_geodesic* g, - double lats[], double lons[], int n, - double* pA, double* pP) { - int i; - struct geod_polygon p; - geod_polygon_init(&p, FALSE); - for (i = 0; i < n; ++i) - geod_polygon_addpoint(g, &p, lats[i], lons[i]); - geod_polygon_compute(g, &p, FALSE, TRUE, pA, pP); + double lats[], double lons[], int n, + double* pA, double* pP) { + int i; + struct geod_polygon p; + geod_polygon_init(&p, FALSE); + for (i = 0; i < n; ++i) + geod_polygon_addpoint(g, &p, lats[i], lons[i]); + geod_polygon_compute(g, &p, FALSE, TRUE, pA, pP); } double areareduceA(double area[], double area0, - int crossings, boolx reverse, boolx sign) { - accrem(area, area0); - if (crossings & 1) - accadd(area, (area[0] < 0 ? 1 : -1) * area0/2); - /* area is with the clockwise sense. If !reverse convert to - * counter-clockwise convention. */ - if (!reverse) - accneg(area); - /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */ - if (sign) { - if (area[0] > area0/2) - accadd(area, -area0); - else if (area[0] <= -area0/2) - accadd(area, +area0); - } else { - if (area[0] >= area0) - accadd(area, -area0); - else if (area[0] < 0) - accadd(area, +area0); - } - return 0 + area[0]; + int crossings, boolx reverse, boolx sign) { + accrem(area, area0); + if (crossings & 1) + accadd(area, (area[0] < 0 ? 1 : -1) * area0/2); + /* area is with the clockwise sense. If !reverse convert to + * counter-clockwise convention. */ + if (!reverse) + accneg(area); + /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */ + if (sign) { + if (area[0] > area0/2) + accadd(area, -area0); + else if (area[0] <= -area0/2) + accadd(area, +area0); + } else { + if (area[0] >= area0) + accadd(area, -area0); + else if (area[0] < 0) + accadd(area, +area0); + } + return 0 + area[0]; } double areareduceB(double area, double area0, - int crossings, boolx reverse, boolx sign) { - area = remainder(area, area0); + int crossings, boolx reverse, boolx sign) { + area = remainder(area, area0); if (crossings & 1) - area += (area < 0 ? 1 : -1) * area0/2; - /* area is with the clockwise sense. If !reverse convert to - * counter-clockwise convention. */ - if (!reverse) - area *= -1; - /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */ - if (sign) { - if (area > area0/2) - area -= area0; - else if (area <= -area0/2) - area += area0; - } else { - if (area >= area0) - area -= area0; - else if (area < 0) - area += area0; - } - return 0 + area; + area += (area < 0 ? 1 : -1) * area0/2; + /* area is with the clockwise sense. If !reverse convert to + * counter-clockwise convention. */ + if (!reverse) + area *= -1; + /* If sign put area in (-area0/2, area0/2], else put area in [0, area0) */ + if (sign) { + if (area > area0/2) + area -= area0; + else if (area <= -area0/2) + area += area0; + } else { + if (area >= area0) + area -= area0; + else if (area < 0) + area += area0; + } + return 0 + area; } /** @endcond */ diff --git a/Sources/geographiclib/include/geodesic.h b/Sources/geographiclib/include/geodesic.h index 8ecb771..26c9103 100644 --- a/Sources/geographiclib/include/geodesic.h +++ b/Sources/geographiclib/include/geodesic.h @@ -31,7 +31,7 @@ * The minor version of the geodesic library. (This tracks the version of * GeographicLib.) **********************************************************************/ -#define GEODESIC_VERSION_MINOR 0 +#define GEODESIC_VERSION_MINOR 1 /** * The patch level of the geodesic library. (This tracks the version of * GeographicLib.) @@ -79,25 +79,25 @@ extern "C" { #endif - /** - * The struct containing information about the ellipsoid. This must be - * initialized by geod_init() before use. - **********************************************************************/ - struct geod_geodesic { +/** + * The struct containing information about the ellipsoid. This must be + * initialized by geod_init() before use. + **********************************************************************/ +struct geod_geodesic { double a; /**< the equatorial radius */ double f; /**< the flattening */ /**< @cond SKIP */ double f1, e2, ep2, n, b, c2, etol2; double A3x[6], C3x[15], C4x[21]; /**< @endcond */ - }; - - /** - * The struct containing information about a single geodesic. This must be - * initialized by geod_lineinit(), geod_directline(), geod_gendirectline(), - * or geod_inverseline() before use. - **********************************************************************/ - struct geod_geodesicline { +}; + +/** + * The struct containing information about a single geodesic. This must be + * initialized by geod_lineinit(), geod_directline(), geod_gendirectline(), + * or geod_inverseline() before use. + **********************************************************************/ +struct geod_geodesicline { double lat1; /**< the starting latitude */ double lon1; /**< the starting longitude */ double azi1; /**< the starting azimuth */ @@ -109,19 +109,19 @@ extern "C" { double s13; /**< distance to reference point */ /**< @cond SKIP */ double b, c2, f1, salp0, calp0, k2, - ssig1, csig1, dn1, stau1, ctau1, somg1, comg1, - A1m1, A2m1, A3c, B11, B21, B31, A4, B41; + ssig1, csig1, dn1, stau1, ctau1, somg1, comg1, + A1m1, A2m1, A3c, B11, B21, B31, A4, B41; double C1a[6+1], C1pa[6+1], C2a[6+1], C3a[6], C4a[6]; /**< @endcond */ unsigned caps; /**< the capabilities */ - }; - - /** - * The struct for accumulating information about a geodesic polygon. This is - * used for computing the perimeter and area of a polygon. This must be - * initialized by geod_polygon_init() before use. - **********************************************************************/ - struct geod_polygon { +}; + +/** + * The struct for accumulating information about a geodesic polygon. This is + * used for computing the perimeter and area of a polygon. This must be + * initialized by geod_polygon_init() before use. + **********************************************************************/ +struct geod_polygon { double lat; /**< the current latitude */ double lon; /**< the current longitude */ /**< @cond SKIP */ @@ -133,681 +133,681 @@ extern "C" { int crossings; /**< @endcond */ unsigned num; /**< the number of points so far */ - }; - - /** - * Initialize a geod_geodesic object. - * - * @param[out] g a pointer to the object to be initialized. - * @param[in] a the equatorial radius (meters). - * @param[in] f the flattening. - **********************************************************************/ - void GEOD_DLL geod_init(struct geod_geodesic* g, double a, double f); - - /** - * Solve the direct geodesic problem. - * - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] lat1 latitude of point 1 (degrees). - * @param[in] lon1 longitude of point 1 (degrees). - * @param[in] azi1 azimuth at point 1 (degrees). - * @param[in] s12 distance from point 1 to point 2 (meters); it can be - * negative. - * @param[out] plat2 pointer to the latitude of point 2 (degrees). - * @param[out] plon2 pointer to the longitude of point 2 (degrees). - * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). - * - * \e g must have been initialized with a call to geod_init(). \e lat1 - * should be in the range [−90°, 90°]. The values of \e lon2 - * and \e azi2 returned are in the range [−180°, 180°]. Any of - * the "return" arguments \e plat2, etc., may be replaced by 0, if you do not - * need some quantities computed. - * - * If either point is at a pole, the azimuth is defined by keeping the - * longitude fixed, writing \e 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°.) - * - * Example, determine the point 10000 km NE of JFK: - @code{.c} - struct geod_geodesic g; - double lat, lon; - geod_init(&g, 6378137, 1/298.257223563); - geod_direct(&g, 40.64, -73.78, 45.0, 10e6, &lat, &lon, 0); - printf("%.5f %.5f\n", lat, lon); - @endcode - **********************************************************************/ - void GEOD_DLL geod_direct(const struct geod_geodesic* g, - double lat1, double lon1, double azi1, double s12, - double* plat2, double* plon2, double* pazi2); - - /** - * The general direct geodesic problem. - * - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] lat1 latitude of point 1 (degrees). - * @param[in] lon1 longitude of point 1 (degrees). - * @param[in] azi1 azimuth at point 1 (degrees). - * @param[in] flags bitor'ed combination of ::geod_flags; \e flags & - * ::GEOD_ARCMODE determines the meaning of \e s12_a12 and \e flags & - * ::GEOD_LONG_UNROLL "unrolls" \e lon2. - * @param[in] s12_a12 if \e flags & ::GEOD_ARCMODE is 0, this is the distance - * from point 1 to point 2 (meters); otherwise it is the arc length - * from point 1 to point 2 (degrees); it can be negative. - * @param[out] plat2 pointer to the latitude of point 2 (degrees). - * @param[out] plon2 pointer to the longitude of point 2 (degrees). - * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). - * @param[out] ps12 pointer to the distance from point 1 to point 2 - * (meters). - * @param[out] pm12 pointer to the reduced length of geodesic (meters). - * @param[out] pM12 pointer to the geodesic scale of point 2 relative to - * point 1 (dimensionless). - * @param[out] pM21 pointer to the geodesic scale of point 1 relative to - * point 2 (dimensionless). - * @param[out] pS12 pointer to the area under the geodesic - * (meters2). - * @return \e a12 arc length from point 1 to point 2 (degrees). - * - * \e g must have been initialized with a call to geod_init(). \e lat1 - * should be in the range [−90°, 90°]. The function value \e - * a12 equals \e s12_a12 if \e flags & ::GEOD_ARCMODE. Any of the "return" - * arguments, \e plat2, etc., may be replaced by 0, if you do not need some - * quantities computed. - * - * With \e flags & ::GEOD_LONG_UNROLL bit set, the longitude is "unrolled" so - * that the quantity \e lon2 − \e lon1 indicates how many times and in - * what sense the geodesic encircles the ellipsoid. - **********************************************************************/ - double GEOD_DLL 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); - - /** - * Solve the inverse geodesic problem. - * - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] lat1 latitude of point 1 (degrees). - * @param[in] lon1 longitude of point 1 (degrees). - * @param[in] lat2 latitude of point 2 (degrees). - * @param[in] lon2 longitude of point 2 (degrees). - * @param[out] ps12 pointer to the distance from point 1 to point 2 - * (meters). - * @param[out] pazi1 pointer to the azimuth at point 1 (degrees). - * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). - * - * \e g must have been initialized with a call to geod_init(). \e lat1 and - * \e lat2 should be in the range [−90°, 90°]. The values of - * \e azi1 and \e azi2 returned are in the range [−180°, 180°]. - * Any of the "return" arguments, \e ps12, etc., may be replaced by 0, if you - * do not need some quantities computed. - * - * If either point is at a pole, the azimuth is defined by keeping the - * longitude fixed, writing \e lat = ±(90° − ε), and - * 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. - * - * Example, determine the distance between JFK and Singapore Changi Airport: - @code{.c} - struct geod_geodesic g; - double s12; - geod_init(&g, 6378137, 1/298.257223563); - geod_inverse(&g, 40.64, -73.78, 1.36, 103.99, &s12, 0, 0); - printf("%.3f\n", s12); - @endcode - **********************************************************************/ - void GEOD_DLL geod_inverse(const struct geod_geodesic* g, - double lat1, double lon1, - double lat2, double lon2, - double* ps12, double* pazi1, double* pazi2); - - /** - * The general inverse geodesic calculation. - * - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] lat1 latitude of point 1 (degrees). - * @param[in] lon1 longitude of point 1 (degrees). - * @param[in] lat2 latitude of point 2 (degrees). - * @param[in] lon2 longitude of point 2 (degrees). - * @param[out] ps12 pointer to the distance from point 1 to point 2 - * (meters). - * @param[out] pazi1 pointer to the azimuth at point 1 (degrees). - * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). - * @param[out] pm12 pointer to the reduced length of geodesic (meters). - * @param[out] pM12 pointer to the geodesic scale of point 2 relative to - * point 1 (dimensionless). - * @param[out] pM21 pointer to the geodesic scale of point 1 relative to - * point 2 (dimensionless). - * @param[out] pS12 pointer to the area under the geodesic - * (meters2). - * @return \e a12 arc length from point 1 to point 2 (degrees). - * - * \e g must have been initialized with a call to geod_init(). \e lat1 and - * \e lat2 should be in the range [−90°, 90°]. Any of the - * "return" arguments \e ps12, etc., may be replaced by 0, if you do not need - * some quantities computed. - **********************************************************************/ - double GEOD_DLL geod_geninverse(const struct geod_geodesic* g, - double lat1, double lon1, - double lat2, double lon2, - double* ps12, double* pazi1, double* pazi2, - double* pm12, double* pM12, double* pM21, - double* pS12); - - /** - * Initialize a geod_geodesicline object. - * - * @param[out] l a pointer to the object to be initialized. - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] lat1 latitude of point 1 (degrees). - * @param[in] lon1 longitude of point 1 (degrees). - * @param[in] azi1 azimuth at point 1 (degrees). - * @param[in] caps bitor'ed combination of ::geod_mask values specifying the - * capabilities the geod_geodesicline object should possess, i.e., which - * quantities can be returned in calls to geod_position() and - * geod_genposition(). - * - * \e g must have been initialized with a call to geod_init(). \e lat1 - * should be in the range [−90°, 90°]. - * - * The ::geod_mask values are: - * - \e caps |= ::GEOD_LATITUDE for the latitude \e lat2; this is - * added automatically, - * - \e caps |= ::GEOD_LONGITUDE for the latitude \e lon2, - * - \e caps |= ::GEOD_AZIMUTH for the latitude \e azi2; this is - * added automatically, - * - \e caps |= ::GEOD_DISTANCE for the distance \e s12, - * - \e caps |= ::GEOD_REDUCEDLENGTH for the reduced length \e m12, - * - \e caps |= ::GEOD_GEODESICSCALE for the geodesic scales \e M12 - * and \e M21, - * - \e caps |= ::GEOD_AREA for the area \e S12, - * - \e caps |= ::GEOD_DISTANCE_IN permits the length of the - * geodesic to be given in terms of \e s12; without this capability the - * length can only be specified in terms of arc length. - * . - * A value of \e caps = 0 is treated as ::GEOD_LATITUDE | ::GEOD_LONGITUDE | - * ::GEOD_AZIMUTH | ::GEOD_DISTANCE_IN (to support the solution of the - * "standard" direct problem). - * - * When initialized by this function, point 3 is undefined (l->s13 = l->a13 = - * NaN). - **********************************************************************/ - void GEOD_DLL geod_lineinit(struct geod_geodesicline* l, - const struct geod_geodesic* g, - double lat1, double lon1, double azi1, - unsigned caps); - - /** - * Initialize a geod_geodesicline object in terms of the direct geodesic - * problem. - * - * @param[out] l a pointer to the object to be initialized. - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] lat1 latitude of point 1 (degrees). - * @param[in] lon1 longitude of point 1 (degrees). - * @param[in] azi1 azimuth at point 1 (degrees). - * @param[in] s12 distance from point 1 to point 2 (meters); it can be - * negative. - * @param[in] caps bitor'ed combination of ::geod_mask values specifying the - * capabilities the geod_geodesicline object should possess, i.e., which - * quantities can be returned in calls to geod_position() and - * geod_genposition(). - * - * This function sets point 3 of the geod_geodesicline to correspond to point - * 2 of the direct geodesic problem. See geod_lineinit() for more - * information. - **********************************************************************/ - void GEOD_DLL geod_directline(struct geod_geodesicline* l, - const struct geod_geodesic* g, - double lat1, double lon1, - double azi1, double s12, - unsigned caps); - - /** - * Initialize a geod_geodesicline object in terms of the direct geodesic - * problem specified in terms of either distance or arc length. - * - * @param[out] l a pointer to the object to be initialized. - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] lat1 latitude of point 1 (degrees). - * @param[in] lon1 longitude of point 1 (degrees). - * @param[in] azi1 azimuth at point 1 (degrees). - * @param[in] flags either ::GEOD_NOFLAGS or ::GEOD_ARCMODE to determining - * the meaning of the \e s12_a12. - * @param[in] s12_a12 if \e flags = ::GEOD_NOFLAGS, this is the distance - * from point 1 to point 2 (meters); if \e flags = ::GEOD_ARCMODE, it is - * the arc length from point 1 to point 2 (degrees); it can be - * negative. - * @param[in] caps bitor'ed combination of ::geod_mask values specifying the - * capabilities the geod_geodesicline object should possess, i.e., which - * quantities can be returned in calls to geod_position() and - * geod_genposition(). - * - * This function sets point 3 of the geod_geodesicline to correspond to point - * 2 of the direct geodesic problem. See geod_lineinit() for more - * information. - **********************************************************************/ - void GEOD_DLL geod_gendirectline(struct geod_geodesicline* l, - const struct geod_geodesic* g, - double lat1, double lon1, double azi1, - unsigned flags, double s12_a12, - unsigned caps); - - /** - * Initialize a geod_geodesicline object in terms of the inverse geodesic - * problem. - * - * @param[out] l a pointer to the object to be initialized. - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] lat1 latitude of point 1 (degrees). - * @param[in] lon1 longitude of point 1 (degrees). - * @param[in] lat2 latitude of point 2 (degrees). - * @param[in] lon2 longitude of point 2 (degrees). - * @param[in] caps bitor'ed combination of ::geod_mask values specifying the - * capabilities the geod_geodesicline object should possess, i.e., which - * quantities can be returned in calls to geod_position() and - * geod_genposition(). - * - * This function sets point 3 of the geod_geodesicline to correspond to point - * 2 of the inverse geodesic problem. See geod_lineinit() for more - * information. - **********************************************************************/ - void GEOD_DLL geod_inverseline(struct geod_geodesicline* l, - const struct geod_geodesic* g, - double lat1, double lon1, - double lat2, double lon2, - unsigned caps); - - /** - * Compute the position along a geod_geodesicline. - * - * @param[in] l a pointer to the geod_geodesicline object specifying the - * geodesic line. - * @param[in] s12 distance from point 1 to point 2 (meters); it can be - * negative. - * @param[out] plat2 pointer to the latitude of point 2 (degrees). - * @param[out] plon2 pointer to the longitude of point 2 (degrees); requires - * that \e l was initialized with \e caps |= ::GEOD_LONGITUDE. - * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). - * - * \e l must have been initialized with a call, e.g., to geod_lineinit(), - * with \e caps |= ::GEOD_DISTANCE_IN (or \e caps = 0). The values of \e - * lon2 and \e azi2 returned are in the range [−180°, 180°]. - * Any of the "return" arguments \e plat2, etc., may be replaced by 0, if you - * do not need some quantities computed. - * - * Example, compute way points between JFK and Singapore Changi Airport - * the "obvious" way using geod_direct(): - @code{.c} - struct geod_geodesic g; - double s12, azi1, lat[101], lon[101]; - int i; - geod_init(&g, 6378137, 1/298.257223563); - geod_inverse(&g, 40.64, -73.78, 1.36, 103.99, &s12, &azi1, 0); - for (i = 0; i < 101; ++i) { - geod_direct(&g, 40.64, -73.78, azi1, i * s12 * 0.01, lat + i, lon + i, 0); - printf("%.5f %.5f\n", lat[i], lon[i]); - } - @endcode - * A faster way using geod_position(): - @code{.c} - struct geod_geodesic g; - struct geod_geodesicline l; - double lat[101], lon[101]; - int i; - geod_init(&g, 6378137, 1/298.257223563); - geod_inverseline(&l, &g, 40.64, -73.78, 1.36, 103.99, 0); - for (i = 0; i <= 100; ++i) { - geod_position(&l, i * l.s13 * 0.01, lat + i, lon + i, 0); - printf("%.5f %.5f\n", lat[i], lon[i]); - } - @endcode - **********************************************************************/ - void GEOD_DLL geod_position(const struct geod_geodesicline* l, double s12, - double* plat2, double* plon2, double* pazi2); - - /** - * The general position function. - * - * @param[in] l a pointer to the geod_geodesicline object specifying the - * geodesic line. - * @param[in] flags bitor'ed combination of ::geod_flags; \e flags & - * ::GEOD_ARCMODE determines the meaning of \e s12_a12 and \e flags & - * ::GEOD_LONG_UNROLL "unrolls" \e lon2; if \e flags & ::GEOD_ARCMODE is 0, - * then \e l must have been initialized with \e caps |= ::GEOD_DISTANCE_IN. - * @param[in] s12_a12 if \e flags & ::GEOD_ARCMODE is 0, this is the - * distance from point 1 to point 2 (meters); otherwise it is the - * arc length from point 1 to point 2 (degrees); it can be - * negative. - * @param[out] plat2 pointer to the latitude of point 2 (degrees). - * @param[out] plon2 pointer to the longitude of point 2 (degrees); requires - * that \e l was initialized with \e caps |= ::GEOD_LONGITUDE. - * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). - * @param[out] ps12 pointer to the distance from point 1 to point 2 - * (meters); requires that \e l was initialized with \e caps |= - * ::GEOD_DISTANCE. - * @param[out] pm12 pointer to the reduced length of geodesic (meters); - * requires that \e l was initialized with \e caps |= ::GEOD_REDUCEDLENGTH. - * @param[out] pM12 pointer to the geodesic scale of point 2 relative to - * point 1 (dimensionless); requires that \e l was initialized with \e caps - * |= ::GEOD_GEODESICSCALE. - * @param[out] pM21 pointer to the geodesic scale of point 1 relative to - * point 2 (dimensionless); requires that \e l was initialized with \e caps - * |= ::GEOD_GEODESICSCALE. - * @param[out] pS12 pointer to the area under the geodesic - * (meters2); requires that \e l was initialized with \e caps |= - * ::GEOD_AREA. - * @return \e a12 arc length from point 1 to point 2 (degrees). - * - * \e l must have been initialized with a call to geod_lineinit() with \e - * caps |= ::GEOD_DISTANCE_IN. The value \e azi2 returned is in the range - * [−180°, 180°]. Any of the "return" arguments \e plat2, - * etc., may be replaced by 0, if you do not need some quantities - * computed. Requesting a value which \e l is not capable of computing - * is not an error; the corresponding argument will not be altered. - * - * With \e flags & ::GEOD_LONG_UNROLL bit set, the longitude is "unrolled" so - * that the quantity \e lon2 − \e lon1 indicates how many times and in - * what sense the geodesic encircles the ellipsoid. - * - * Example, compute way points between JFK and Singapore Changi Airport using - * geod_genposition(). In this example, the points are evenly spaced in arc - * length (and so only approximately equally spaced in distance). This is - * faster than using geod_position() and would be appropriate if drawing the - * path on a map. - @code{.c} - struct geod_geodesic g; - struct geod_geodesicline l; - double lat[101], lon[101]; - int i; - geod_init(&g, 6378137, 1/298.257223563); - geod_inverseline(&l, &g, 40.64, -73.78, 1.36, 103.99, - GEOD_LATITUDE | GEOD_LONGITUDE); - for (i = 0; i <= 100; ++i) { - geod_genposition(&l, GEOD_ARCMODE, i * l.a13 * 0.01, - lat + i, lon + i, 0, 0, 0, 0, 0, 0); - printf("%.5f %.5f\n", lat[i], lon[i]); - } - @endcode - **********************************************************************/ - double GEOD_DLL 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); - - /** - * Specify position of point 3 in terms of distance. - * - * @param[in,out] l a pointer to the geod_geodesicline object. - * @param[in] s13 the distance from point 1 to point 3 (meters); it - * can be negative. - * - * This is only useful if the geod_geodesicline object has been constructed - * with \e caps |= ::GEOD_DISTANCE_IN. - **********************************************************************/ - void GEOD_DLL geod_setdistance(struct geod_geodesicline* l, double s13); - - /** - * Specify position of point 3 in terms of either distance or arc length. - * - * @param[in,out] l a pointer to the geod_geodesicline object. - * @param[in] flags either ::GEOD_NOFLAGS or ::GEOD_ARCMODE to determining - * the meaning of the \e s13_a13. - * @param[in] s13_a13 if \e flags = ::GEOD_NOFLAGS, this is the distance - * from point 1 to point 3 (meters); if \e flags = ::GEOD_ARCMODE, it is - * the arc length from point 1 to point 3 (degrees); it can be - * negative. - * - * If flags = ::GEOD_NOFLAGS, this calls geod_setdistance(). If flags = - * ::GEOD_ARCMODE, the \e s13 is only set if the geod_geodesicline object has - * been constructed with \e caps |= ::GEOD_DISTANCE. - **********************************************************************/ - void GEOD_DLL geod_gensetdistance(struct geod_geodesicline* l, - unsigned flags, double s13_a13); - - /** - * Initialize a geod_polygon object. - * - * @param[out] p a pointer to the object to be initialized. - * @param[in] polylinep non-zero if a polyline instead of a polygon. - * - * If \e polylinep is zero, then the sequence of vertices and edges added by - * geod_polygon_addpoint() and geod_polygon_addedge() define a polygon and - * the perimeter and area are returned by geod_polygon_compute(). If \e - * polylinep is non-zero, then the vertices and edges define a polyline and - * only the perimeter is returned by geod_polygon_compute(). - * - * The area and perimeter are accumulated at two times the standard floating - * point precision to guard against the loss of accuracy with many-sided - * polygons. At any point you can ask for the perimeter and area so far. - * - * An example of the use of this function is given in the documentation for - * geod_polygon_compute(). - **********************************************************************/ - void GEOD_DLL geod_polygon_init(struct geod_polygon* p, int polylinep); - - /** - * Clear the polygon, allowing a new polygon to be started. - * - * @param[in,out] p a pointer to the object to be cleared. - **********************************************************************/ - void GEOD_DLL geod_polygon_clear(struct geod_polygon* p); - - /** - * Add a point to the polygon or polyline. - * - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in,out] p a pointer to the geod_polygon object specifying the - * polygon. - * @param[in] lat the latitude of the point (degrees). - * @param[in] lon the longitude of the point (degrees). - * - * \e g and \e p must have been initialized with calls to geod_init() and - * geod_polygon_init(), respectively. The same \e g must be used for all the - * points and edges in a polygon. \e lat should be in the range - * [−90°, 90°]. - * - * An example of the use of this function is given in the documentation for - * geod_polygon_compute(). - **********************************************************************/ - void GEOD_DLL geod_polygon_addpoint(const struct geod_geodesic* g, - struct geod_polygon* p, - double lat, double lon); - - /** - * Add an edge to the polygon or polyline. - * - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in,out] p a pointer to the geod_polygon object specifying the - * polygon. - * @param[in] azi azimuth at current point (degrees). - * @param[in] s distance from current point to next point (meters). - * - * \e g and \e p must have been initialized with calls to geod_init() and - * geod_polygon_init(), respectively. The same \e g must be used for all the - * points and edges in a polygon. This does nothing if no points have been - * added yet. The \e lat and \e lon fields of \e p give the location of the - * new vertex. - **********************************************************************/ - void GEOD_DLL geod_polygon_addedge(const struct geod_geodesic* g, - struct geod_polygon* p, - double azi, double s); - - /** - * Return the results for a polygon. - * - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] p a pointer to the geod_polygon object specifying the polygon. - * @param[in] reverse if non-zero then clockwise (instead of - * counter-clockwise) traversal counts as a positive area. - * @param[in] sign if non-zero then return a signed result for the area if - * the polygon is traversed in the "wrong" direction instead of returning - * the area for the rest of the earth. - * @param[out] pA pointer to the area of the polygon (meters2); - * only set if \e polyline is non-zero in the call to geod_polygon_init(). - * @param[out] pP pointer to the perimeter of the polygon or length of the - * polyline (meters). - * @return the number of points. - * - * The area and perimeter are accumulated at two times the standard floating - * point precision to guard against the loss of accuracy with many-sided - * polygons. Arbitrarily complex polygons are allowed. In the case of - * self-intersecting polygons the area is accumulated "algebraically", e.g., - * the areas of the 2 loops in a figure-8 polygon will partially cancel. - * There's no need to "close" the polygon by repeating the first vertex. Set - * \e pA or \e pP to zero, if you do not want the corresponding quantity - * returned. - * - * More points can be added to the polygon after this call. - * - * Example, compute the perimeter and area of the geodesic triangle with - * vertices (0°N,0°E), (0°N,90°E), (90°N,0°E). - @code{.c} - double A, P; - int n; - struct geod_geodesic g; - struct geod_polygon p; - geod_init(&g, 6378137, 1/298.257223563); - geod_polygon_init(&p, 0); - - geod_polygon_addpoint(&g, &p, 0, 0); - geod_polygon_addpoint(&g, &p, 0, 90); - geod_polygon_addpoint(&g, &p, 90, 0); - n = geod_polygon_compute(&g, &p, 0, 1, &A, &P); - printf("%d %.8f %.3f\n", n, P, A); - @endcode - **********************************************************************/ - unsigned GEOD_DLL geod_polygon_compute(const struct geod_geodesic* g, - const struct geod_polygon* p, - int reverse, int sign, - double* pA, double* pP); - - /** - * Return the results assuming a tentative final test point is added; - * however, the data for the test point is not saved. This lets you report a - * running result for the perimeter and area as the user moves the mouse - * cursor. Ordinary floating point arithmetic is used to accumulate the data - * for the test point; thus the area and perimeter returned are less accurate - * than if geod_polygon_addpoint() and geod_polygon_compute() are used. - * - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] p a pointer to the geod_polygon object specifying the polygon. - * @param[in] lat the latitude of the test point (degrees). - * @param[in] lon the longitude of the test point (degrees). - * @param[in] reverse if non-zero then clockwise (instead of - * counter-clockwise) traversal counts as a positive area. - * @param[in] sign if non-zero then return a signed result for the area if - * the polygon is traversed in the "wrong" direction instead of returning - * the area for the rest of the earth. - * @param[out] pA pointer to the area of the polygon (meters2); - * only set if \e polyline is non-zero in the call to geod_polygon_init(). - * @param[out] pP pointer to the perimeter of the polygon or length of the - * polyline (meters). - * @return the number of points. - * - * \e lat should be in the range [−90°, 90°]. - **********************************************************************/ - unsigned GEOD_DLL geod_polygon_testpoint(const struct geod_geodesic* g, - const struct geod_polygon* p, - double lat, double lon, - int reverse, int sign, - double* pA, double* pP); - - /** - * Return the results assuming a tentative final test point is added via an - * azimuth and distance; however, the data for the test point is not saved. - * This lets you report a running result for the perimeter and area as the - * user moves the mouse cursor. Ordinary floating point arithmetic is used - * to accumulate the data for the test point; thus the area and perimeter - * returned are less accurate than if geod_polygon_addedge() and - * geod_polygon_compute() are used. - * - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] p a pointer to the geod_polygon object specifying the polygon. - * @param[in] azi azimuth at current point (degrees). - * @param[in] s distance from current point to final test point (meters). - * @param[in] reverse if non-zero then clockwise (instead of - * counter-clockwise) traversal counts as a positive area. - * @param[in] sign if non-zero then return a signed result for the area if - * the polygon is traversed in the "wrong" direction instead of returning - * the area for the rest of the earth. - * @param[out] pA pointer to the area of the polygon (meters2); - * only set if \e polyline is non-zero in the call to geod_polygon_init(). - * @param[out] pP pointer to the perimeter of the polygon or length of the - * polyline (meters). - * @return the number of points. - **********************************************************************/ - unsigned GEOD_DLL geod_polygon_testedge(const struct geod_geodesic* g, - const struct geod_polygon* p, - double azi, double s, - int reverse, int sign, - double* pA, double* pP); - - /** - * A simple interface for computing the area of a geodesic polygon. - * - * @param[in] g a pointer to the geod_geodesic object specifying the - * ellipsoid. - * @param[in] lats an array of latitudes of the polygon vertices (degrees). - * @param[in] lons an array of longitudes of the polygon vertices (degrees). - * @param[in] n the number of vertices. - * @param[out] pA pointer to the area of the polygon (meters2). - * @param[out] pP pointer to the perimeter of the polygon (meters). - * - * \e lats should be in the range [−90°, 90°]. - * - * Arbitrarily complex polygons are allowed. In the case self-intersecting - * of polygons the area is accumulated "algebraically", e.g., the areas of - * the 2 loops in a figure-8 polygon will partially cancel. There's no need - * to "close" the polygon by repeating the first vertex. The area returned - * is signed with counter-clockwise traversal being treated as positive. - * - * Example, compute the area of Antarctica: - @code{.c} - double - lats[] = {-72.9, -71.9, -74.9, -74.3, -77.5, -77.4, -71.7, -65.9, -65.7, - -66.6, -66.9, -69.8, -70.0, -71.0, -77.3, -77.9, -74.7}, - lons[] = {-74, -102, -102, -131, -163, 163, 172, 140, 113, - 88, 59, 25, -4, -14, -33, -46, -61}; - struct geod_geodesic g; - double A, P; - geod_init(&g, 6378137, 1/298.257223563); - geod_polygonarea(&g, lats, lons, (sizeof lats) / (sizeof lats[0]), &A, &P); - printf("%.0f %.2f\n", A, P); - @endcode - **********************************************************************/ - void GEOD_DLL geod_polygonarea(const struct geod_geodesic* g, - double lats[], double lons[], int n, - double* pA, double* pP); - - /** - * mask values for the \e caps argument to geod_lineinit(). - **********************************************************************/ - enum geod_mask { +}; + +/** + * Initialize a geod_geodesic object. + * + * @param[out] g a pointer to the object to be initialized. + * @param[in] a the equatorial radius (meters). + * @param[in] f the flattening. + **********************************************************************/ +void GEOD_DLL geod_init(struct geod_geodesic* g, double a, double f); + +/** + * Solve the direct geodesic problem. + * + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] lat1 latitude of point 1 (degrees). + * @param[in] lon1 longitude of point 1 (degrees). + * @param[in] azi1 azimuth at point 1 (degrees). + * @param[in] s12 distance from point 1 to point 2 (meters); it can be + * negative. + * @param[out] plat2 pointer to the latitude of point 2 (degrees). + * @param[out] plon2 pointer to the longitude of point 2 (degrees). + * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). + * + * \e g must have been initialized with a call to geod_init(). \e lat1 + * should be in the range [−90°, 90°]. The values of \e lon2 + * and \e azi2 returned are in the range [−180°, 180°]. Any of + * the "return" arguments \e plat2, etc., may be replaced by 0, if you do not + * need some quantities computed. + * + * If either point is at a pole, the azimuth is defined by keeping the + * longitude fixed, writing \e 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°.) + * + * Example, determine the point 10000 km NE of JFK: + @code{.c} + struct geod_geodesic g; + double lat, lon; + geod_init(&g, 6378137, 1/298.257223563); + geod_direct(&g, 40.64, -73.78, 45.0, 10e6, &lat, &lon, 0); + printf("%.5f %.5f\n", lat, lon); + @endcode + **********************************************************************/ +void GEOD_DLL geod_direct(const struct geod_geodesic* g, + double lat1, double lon1, double azi1, double s12, + double* plat2, double* plon2, double* pazi2); + +/** + * The general direct geodesic problem. + * + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] lat1 latitude of point 1 (degrees). + * @param[in] lon1 longitude of point 1 (degrees). + * @param[in] azi1 azimuth at point 1 (degrees). + * @param[in] flags bitor'ed combination of ::geod_flags; \e flags & + * ::GEOD_ARCMODE determines the meaning of \e s12_a12 and \e flags & + * ::GEOD_LONG_UNROLL "unrolls" \e lon2. + * @param[in] s12_a12 if \e flags & ::GEOD_ARCMODE is 0, this is the distance + * from point 1 to point 2 (meters); otherwise it is the arc length + * from point 1 to point 2 (degrees); it can be negative. + * @param[out] plat2 pointer to the latitude of point 2 (degrees). + * @param[out] plon2 pointer to the longitude of point 2 (degrees). + * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). + * @param[out] ps12 pointer to the distance from point 1 to point 2 + * (meters). + * @param[out] pm12 pointer to the reduced length of geodesic (meters). + * @param[out] pM12 pointer to the geodesic scale of point 2 relative to + * point 1 (dimensionless). + * @param[out] pM21 pointer to the geodesic scale of point 1 relative to + * point 2 (dimensionless). + * @param[out] pS12 pointer to the area under the geodesic + * (meters2). + * @return \e a12 arc length from point 1 to point 2 (degrees). + * + * \e g must have been initialized with a call to geod_init(). \e lat1 + * should be in the range [−90°, 90°]. The function value \e + * a12 equals \e s12_a12 if \e flags & ::GEOD_ARCMODE. Any of the "return" + * arguments, \e plat2, etc., may be replaced by 0, if you do not need some + * quantities computed. + * + * With \e flags & ::GEOD_LONG_UNROLL bit set, the longitude is "unrolled" so + * that the quantity \e lon2 − \e lon1 indicates how many times and in + * what sense the geodesic encircles the ellipsoid. + **********************************************************************/ +double GEOD_DLL 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); + +/** + * Solve the inverse geodesic problem. + * + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] lat1 latitude of point 1 (degrees). + * @param[in] lon1 longitude of point 1 (degrees). + * @param[in] lat2 latitude of point 2 (degrees). + * @param[in] lon2 longitude of point 2 (degrees). + * @param[out] ps12 pointer to the distance from point 1 to point 2 + * (meters). + * @param[out] pazi1 pointer to the azimuth at point 1 (degrees). + * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). + * + * \e g must have been initialized with a call to geod_init(). \e lat1 and + * \e lat2 should be in the range [−90°, 90°]. The values of + * \e azi1 and \e azi2 returned are in the range [−180°, 180°]. + * Any of the "return" arguments, \e ps12, etc., may be replaced by 0, if you + * do not need some quantities computed. + * + * If either point is at a pole, the azimuth is defined by keeping the + * longitude fixed, writing \e lat = ±(90° − ε), and + * 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. + * + * Example, determine the distance between JFK and Singapore Changi Airport: + @code{.c} + struct geod_geodesic g; + double s12; + geod_init(&g, 6378137, 1/298.257223563); + geod_inverse(&g, 40.64, -73.78, 1.36, 103.99, &s12, 0, 0); + printf("%.3f\n", s12); + @endcode + **********************************************************************/ +void GEOD_DLL geod_inverse(const struct geod_geodesic* g, + double lat1, double lon1, + double lat2, double lon2, + double* ps12, double* pazi1, double* pazi2); + +/** + * The general inverse geodesic calculation. + * + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] lat1 latitude of point 1 (degrees). + * @param[in] lon1 longitude of point 1 (degrees). + * @param[in] lat2 latitude of point 2 (degrees). + * @param[in] lon2 longitude of point 2 (degrees). + * @param[out] ps12 pointer to the distance from point 1 to point 2 + * (meters). + * @param[out] pazi1 pointer to the azimuth at point 1 (degrees). + * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). + * @param[out] pm12 pointer to the reduced length of geodesic (meters). + * @param[out] pM12 pointer to the geodesic scale of point 2 relative to + * point 1 (dimensionless). + * @param[out] pM21 pointer to the geodesic scale of point 1 relative to + * point 2 (dimensionless). + * @param[out] pS12 pointer to the area under the geodesic + * (meters2). + * @return \e a12 arc length from point 1 to point 2 (degrees). + * + * \e g must have been initialized with a call to geod_init(). \e lat1 and + * \e lat2 should be in the range [−90°, 90°]. Any of the + * "return" arguments \e ps12, etc., may be replaced by 0, if you do not need + * some quantities computed. + **********************************************************************/ +double GEOD_DLL geod_geninverse(const struct geod_geodesic* g, + double lat1, double lon1, + double lat2, double lon2, + double* ps12, double* pazi1, double* pazi2, + double* pm12, double* pM12, double* pM21, + double* pS12); + +/** + * Initialize a geod_geodesicline object. + * + * @param[out] l a pointer to the object to be initialized. + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] lat1 latitude of point 1 (degrees). + * @param[in] lon1 longitude of point 1 (degrees). + * @param[in] azi1 azimuth at point 1 (degrees). + * @param[in] caps bitor'ed combination of ::geod_mask values specifying the + * capabilities the geod_geodesicline object should possess, i.e., which + * quantities can be returned in calls to geod_position() and + * geod_genposition(). + * + * \e g must have been initialized with a call to geod_init(). \e lat1 + * should be in the range [−90°, 90°]. + * + * The ::geod_mask values are: + * - \e caps |= ::GEOD_LATITUDE for the latitude \e lat2; this is + * added automatically, + * - \e caps |= ::GEOD_LONGITUDE for the latitude \e lon2, + * - \e caps |= ::GEOD_AZIMUTH for the latitude \e azi2; this is + * added automatically, + * - \e caps |= ::GEOD_DISTANCE for the distance \e s12, + * - \e caps |= ::GEOD_REDUCEDLENGTH for the reduced length \e m12, + * - \e caps |= ::GEOD_GEODESICSCALE for the geodesic scales \e M12 + * and \e M21, + * - \e caps |= ::GEOD_AREA for the area \e S12, + * - \e caps |= ::GEOD_DISTANCE_IN permits the length of the + * geodesic to be given in terms of \e s12; without this capability the + * length can only be specified in terms of arc length. + * . + * A value of \e caps = 0 is treated as ::GEOD_LATITUDE | ::GEOD_LONGITUDE | + * ::GEOD_AZIMUTH | ::GEOD_DISTANCE_IN (to support the solution of the + * "standard" direct problem). + * + * When initialized by this function, point 3 is undefined (l->s13 = l->a13 = + * NaN). + **********************************************************************/ +void GEOD_DLL geod_lineinit(struct geod_geodesicline* l, + const struct geod_geodesic* g, + double lat1, double lon1, double azi1, + unsigned caps); + +/** + * Initialize a geod_geodesicline object in terms of the direct geodesic + * problem. + * + * @param[out] l a pointer to the object to be initialized. + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] lat1 latitude of point 1 (degrees). + * @param[in] lon1 longitude of point 1 (degrees). + * @param[in] azi1 azimuth at point 1 (degrees). + * @param[in] s12 distance from point 1 to point 2 (meters); it can be + * negative. + * @param[in] caps bitor'ed combination of ::geod_mask values specifying the + * capabilities the geod_geodesicline object should possess, i.e., which + * quantities can be returned in calls to geod_position() and + * geod_genposition(). + * + * This function sets point 3 of the geod_geodesicline to correspond to point + * 2 of the direct geodesic problem. See geod_lineinit() for more + * information. + **********************************************************************/ +void GEOD_DLL geod_directline(struct geod_geodesicline* l, + const struct geod_geodesic* g, + double lat1, double lon1, + double azi1, double s12, + unsigned caps); + +/** + * Initialize a geod_geodesicline object in terms of the direct geodesic + * problem specified in terms of either distance or arc length. + * + * @param[out] l a pointer to the object to be initialized. + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] lat1 latitude of point 1 (degrees). + * @param[in] lon1 longitude of point 1 (degrees). + * @param[in] azi1 azimuth at point 1 (degrees). + * @param[in] flags either ::GEOD_NOFLAGS or ::GEOD_ARCMODE to determining + * the meaning of the \e s12_a12. + * @param[in] s12_a12 if \e flags = ::GEOD_NOFLAGS, this is the distance + * from point 1 to point 2 (meters); if \e flags = ::GEOD_ARCMODE, it is + * the arc length from point 1 to point 2 (degrees); it can be + * negative. + * @param[in] caps bitor'ed combination of ::geod_mask values specifying the + * capabilities the geod_geodesicline object should possess, i.e., which + * quantities can be returned in calls to geod_position() and + * geod_genposition(). + * + * This function sets point 3 of the geod_geodesicline to correspond to point + * 2 of the direct geodesic problem. See geod_lineinit() for more + * information. + **********************************************************************/ +void GEOD_DLL geod_gendirectline(struct geod_geodesicline* l, + const struct geod_geodesic* g, + double lat1, double lon1, double azi1, + unsigned flags, double s12_a12, + unsigned caps); + +/** + * Initialize a geod_geodesicline object in terms of the inverse geodesic + * problem. + * + * @param[out] l a pointer to the object to be initialized. + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] lat1 latitude of point 1 (degrees). + * @param[in] lon1 longitude of point 1 (degrees). + * @param[in] lat2 latitude of point 2 (degrees). + * @param[in] lon2 longitude of point 2 (degrees). + * @param[in] caps bitor'ed combination of ::geod_mask values specifying the + * capabilities the geod_geodesicline object should possess, i.e., which + * quantities can be returned in calls to geod_position() and + * geod_genposition(). + * + * This function sets point 3 of the geod_geodesicline to correspond to point + * 2 of the inverse geodesic problem. See geod_lineinit() for more + * information. + **********************************************************************/ +void GEOD_DLL geod_inverseline(struct geod_geodesicline* l, + const struct geod_geodesic* g, + double lat1, double lon1, + double lat2, double lon2, + unsigned caps); + +/** + * Compute the position along a geod_geodesicline. + * + * @param[in] l a pointer to the geod_geodesicline object specifying the + * geodesic line. + * @param[in] s12 distance from point 1 to point 2 (meters); it can be + * negative. + * @param[out] plat2 pointer to the latitude of point 2 (degrees). + * @param[out] plon2 pointer to the longitude of point 2 (degrees); requires + * that \e l was initialized with \e caps |= ::GEOD_LONGITUDE. + * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). + * + * \e l must have been initialized with a call, e.g., to geod_lineinit(), + * with \e caps |= ::GEOD_DISTANCE_IN (or \e caps = 0). The values of \e + * lon2 and \e azi2 returned are in the range [−180°, 180°]. + * Any of the "return" arguments \e plat2, etc., may be replaced by 0, if you + * do not need some quantities computed. + * + * Example, compute way points between JFK and Singapore Changi Airport + * the "obvious" way using geod_direct(): + @code{.c} + struct geod_geodesic g; + double s12, azi1, lat[101], lon[101]; + int i; + geod_init(&g, 6378137, 1/298.257223563); + geod_inverse(&g, 40.64, -73.78, 1.36, 103.99, &s12, &azi1, 0); + for (i = 0; i < 101; ++i) { + geod_direct(&g, 40.64, -73.78, azi1, i * s12 * 0.01, lat + i, lon + i, 0); + printf("%.5f %.5f\n", lat[i], lon[i]); + } + @endcode + * A faster way using geod_position(): + @code{.c} + struct geod_geodesic g; + struct geod_geodesicline l; + double lat[101], lon[101]; + int i; + geod_init(&g, 6378137, 1/298.257223563); + geod_inverseline(&l, &g, 40.64, -73.78, 1.36, 103.99, 0); + for (i = 0; i <= 100; ++i) { + geod_position(&l, i * l.s13 * 0.01, lat + i, lon + i, 0); + printf("%.5f %.5f\n", lat[i], lon[i]); + } + @endcode + **********************************************************************/ +void GEOD_DLL geod_position(const struct geod_geodesicline* l, double s12, + double* plat2, double* plon2, double* pazi2); + +/** + * The general position function. + * + * @param[in] l a pointer to the geod_geodesicline object specifying the + * geodesic line. + * @param[in] flags bitor'ed combination of ::geod_flags; \e flags & + * ::GEOD_ARCMODE determines the meaning of \e s12_a12 and \e flags & + * ::GEOD_LONG_UNROLL "unrolls" \e lon2; if \e flags & ::GEOD_ARCMODE is 0, + * then \e l must have been initialized with \e caps |= ::GEOD_DISTANCE_IN. + * @param[in] s12_a12 if \e flags & ::GEOD_ARCMODE is 0, this is the + * distance from point 1 to point 2 (meters); otherwise it is the + * arc length from point 1 to point 2 (degrees); it can be + * negative. + * @param[out] plat2 pointer to the latitude of point 2 (degrees). + * @param[out] plon2 pointer to the longitude of point 2 (degrees); requires + * that \e l was initialized with \e caps |= ::GEOD_LONGITUDE. + * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees). + * @param[out] ps12 pointer to the distance from point 1 to point 2 + * (meters); requires that \e l was initialized with \e caps |= + * ::GEOD_DISTANCE. + * @param[out] pm12 pointer to the reduced length of geodesic (meters); + * requires that \e l was initialized with \e caps |= ::GEOD_REDUCEDLENGTH. + * @param[out] pM12 pointer to the geodesic scale of point 2 relative to + * point 1 (dimensionless); requires that \e l was initialized with \e caps + * |= ::GEOD_GEODESICSCALE. + * @param[out] pM21 pointer to the geodesic scale of point 1 relative to + * point 2 (dimensionless); requires that \e l was initialized with \e caps + * |= ::GEOD_GEODESICSCALE. + * @param[out] pS12 pointer to the area under the geodesic + * (meters2); requires that \e l was initialized with \e caps |= + * ::GEOD_AREA. + * @return \e a12 arc length from point 1 to point 2 (degrees). + * + * \e l must have been initialized with a call to geod_lineinit() with \e + * caps |= ::GEOD_DISTANCE_IN. The value \e azi2 returned is in the range + * [−180°, 180°]. Any of the "return" arguments \e plat2, + * etc., may be replaced by 0, if you do not need some quantities + * computed. Requesting a value which \e l is not capable of computing + * is not an error; the corresponding argument will not be altered. + * + * With \e flags & ::GEOD_LONG_UNROLL bit set, the longitude is "unrolled" so + * that the quantity \e lon2 − \e lon1 indicates how many times and in + * what sense the geodesic encircles the ellipsoid. + * + * Example, compute way points between JFK and Singapore Changi Airport using + * geod_genposition(). In this example, the points are evenly spaced in arc + * length (and so only approximately equally spaced in distance). This is + * faster than using geod_position() and would be appropriate if drawing the + * path on a map. + @code{.c} + struct geod_geodesic g; + struct geod_geodesicline l; + double lat[101], lon[101]; + int i; + geod_init(&g, 6378137, 1/298.257223563); + geod_inverseline(&l, &g, 40.64, -73.78, 1.36, 103.99, + GEOD_LATITUDE | GEOD_LONGITUDE); + for (i = 0; i <= 100; ++i) { + geod_genposition(&l, GEOD_ARCMODE, i * l.a13 * 0.01, + lat + i, lon + i, 0, 0, 0, 0, 0, 0); + printf("%.5f %.5f\n", lat[i], lon[i]); + } + @endcode + **********************************************************************/ +double GEOD_DLL 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); + +/** + * Specify position of point 3 in terms of distance. + * + * @param[in,out] l a pointer to the geod_geodesicline object. + * @param[in] s13 the distance from point 1 to point 3 (meters); it + * can be negative. + * + * This is only useful if the geod_geodesicline object has been constructed + * with \e caps |= ::GEOD_DISTANCE_IN. + **********************************************************************/ +void GEOD_DLL geod_setdistance(struct geod_geodesicline* l, double s13); + +/** + * Specify position of point 3 in terms of either distance or arc length. + * + * @param[in,out] l a pointer to the geod_geodesicline object. + * @param[in] flags either ::GEOD_NOFLAGS or ::GEOD_ARCMODE to determining + * the meaning of the \e s13_a13. + * @param[in] s13_a13 if \e flags = ::GEOD_NOFLAGS, this is the distance + * from point 1 to point 3 (meters); if \e flags = ::GEOD_ARCMODE, it is + * the arc length from point 1 to point 3 (degrees); it can be + * negative. + * + * If flags = ::GEOD_NOFLAGS, this calls geod_setdistance(). If flags = + * ::GEOD_ARCMODE, the \e s13 is only set if the geod_geodesicline object has + * been constructed with \e caps |= ::GEOD_DISTANCE. + **********************************************************************/ +void GEOD_DLL geod_gensetdistance(struct geod_geodesicline* l, + unsigned flags, double s13_a13); + +/** + * Initialize a geod_polygon object. + * + * @param[out] p a pointer to the object to be initialized. + * @param[in] polylinep non-zero if a polyline instead of a polygon. + * + * If \e polylinep is zero, then the sequence of vertices and edges added by + * geod_polygon_addpoint() and geod_polygon_addedge() define a polygon and + * the perimeter and area are returned by geod_polygon_compute(). If \e + * polylinep is non-zero, then the vertices and edges define a polyline and + * only the perimeter is returned by geod_polygon_compute(). + * + * The area and perimeter are accumulated at two times the standard floating + * point precision to guard against the loss of accuracy with many-sided + * polygons. At any point you can ask for the perimeter and area so far. + * + * An example of the use of this function is given in the documentation for + * geod_polygon_compute(). + **********************************************************************/ +void GEOD_DLL geod_polygon_init(struct geod_polygon* p, int polylinep); + +/** + * Clear the polygon, allowing a new polygon to be started. + * + * @param[in,out] p a pointer to the object to be cleared. + **********************************************************************/ +void GEOD_DLL geod_polygon_clear(struct geod_polygon* p); + +/** + * Add a point to the polygon or polyline. + * + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in,out] p a pointer to the geod_polygon object specifying the + * polygon. + * @param[in] lat the latitude of the point (degrees). + * @param[in] lon the longitude of the point (degrees). + * + * \e g and \e p must have been initialized with calls to geod_init() and + * geod_polygon_init(), respectively. The same \e g must be used for all the + * points and edges in a polygon. \e lat should be in the range + * [−90°, 90°]. + * + * An example of the use of this function is given in the documentation for + * geod_polygon_compute(). + **********************************************************************/ +void GEOD_DLL geod_polygon_addpoint(const struct geod_geodesic* g, + struct geod_polygon* p, + double lat, double lon); + +/** + * Add an edge to the polygon or polyline. + * + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in,out] p a pointer to the geod_polygon object specifying the + * polygon. + * @param[in] azi azimuth at current point (degrees). + * @param[in] s distance from current point to next point (meters). + * + * \e g and \e p must have been initialized with calls to geod_init() and + * geod_polygon_init(), respectively. The same \e g must be used for all the + * points and edges in a polygon. This does nothing if no points have been + * added yet. The \e lat and \e lon fields of \e p give the location of the + * new vertex. + **********************************************************************/ +void GEOD_DLL geod_polygon_addedge(const struct geod_geodesic* g, + struct geod_polygon* p, + double azi, double s); + +/** + * Return the results for a polygon. + * + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] p a pointer to the geod_polygon object specifying the polygon. + * @param[in] reverse if non-zero then clockwise (instead of + * counter-clockwise) traversal counts as a positive area. + * @param[in] sign if non-zero then return a signed result for the area if + * the polygon is traversed in the "wrong" direction instead of returning + * the area for the rest of the earth. + * @param[out] pA pointer to the area of the polygon (meters2); + * only set if \e polyline is non-zero in the call to geod_polygon_init(). + * @param[out] pP pointer to the perimeter of the polygon or length of the + * polyline (meters). + * @return the number of points. + * + * The area and perimeter are accumulated at two times the standard floating + * point precision to guard against the loss of accuracy with many-sided + * polygons. Arbitrarily complex polygons are allowed. In the case of + * self-intersecting polygons the area is accumulated "algebraically", e.g., + * the areas of the 2 loops in a figure-8 polygon will partially cancel. + * There's no need to "close" the polygon by repeating the first vertex. Set + * \e pA or \e pP to zero, if you do not want the corresponding quantity + * returned. + * + * More points can be added to the polygon after this call. + * + * Example, compute the perimeter and area of the geodesic triangle with + * vertices (0°N,0°E), (0°N,90°E), (90°N,0°E). + @code{.c} + double A, P; + int n; + struct geod_geodesic g; + struct geod_polygon p; + geod_init(&g, 6378137, 1/298.257223563); + geod_polygon_init(&p, 0); + + geod_polygon_addpoint(&g, &p, 0, 0); + geod_polygon_addpoint(&g, &p, 0, 90); + geod_polygon_addpoint(&g, &p, 90, 0); + n = geod_polygon_compute(&g, &p, 0, 1, &A, &P); + printf("%d %.8f %.3f\n", n, P, A); + @endcode + **********************************************************************/ +unsigned GEOD_DLL geod_polygon_compute(const struct geod_geodesic* g, + const struct geod_polygon* p, + int reverse, int sign, + double* pA, double* pP); + +/** + * Return the results assuming a tentative final test point is added; + * however, the data for the test point is not saved. This lets you report a + * running result for the perimeter and area as the user moves the mouse + * cursor. Ordinary floating point arithmetic is used to accumulate the data + * for the test point; thus the area and perimeter returned are less accurate + * than if geod_polygon_addpoint() and geod_polygon_compute() are used. + * + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] p a pointer to the geod_polygon object specifying the polygon. + * @param[in] lat the latitude of the test point (degrees). + * @param[in] lon the longitude of the test point (degrees). + * @param[in] reverse if non-zero then clockwise (instead of + * counter-clockwise) traversal counts as a positive area. + * @param[in] sign if non-zero then return a signed result for the area if + * the polygon is traversed in the "wrong" direction instead of returning + * the area for the rest of the earth. + * @param[out] pA pointer to the area of the polygon (meters2); + * only set if \e polyline is non-zero in the call to geod_polygon_init(). + * @param[out] pP pointer to the perimeter of the polygon or length of the + * polyline (meters). + * @return the number of points. + * + * \e lat should be in the range [−90°, 90°]. + **********************************************************************/ +unsigned GEOD_DLL geod_polygon_testpoint(const struct geod_geodesic* g, + const struct geod_polygon* p, + double lat, double lon, + int reverse, int sign, + double* pA, double* pP); + +/** + * Return the results assuming a tentative final test point is added via an + * azimuth and distance; however, the data for the test point is not saved. + * This lets you report a running result for the perimeter and area as the + * user moves the mouse cursor. Ordinary floating point arithmetic is used + * to accumulate the data for the test point; thus the area and perimeter + * returned are less accurate than if geod_polygon_addedge() and + * geod_polygon_compute() are used. + * + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] p a pointer to the geod_polygon object specifying the polygon. + * @param[in] azi azimuth at current point (degrees). + * @param[in] s distance from current point to final test point (meters). + * @param[in] reverse if non-zero then clockwise (instead of + * counter-clockwise) traversal counts as a positive area. + * @param[in] sign if non-zero then return a signed result for the area if + * the polygon is traversed in the "wrong" direction instead of returning + * the area for the rest of the earth. + * @param[out] pA pointer to the area of the polygon (meters2); + * only set if \e polyline is non-zero in the call to geod_polygon_init(). + * @param[out] pP pointer to the perimeter of the polygon or length of the + * polyline (meters). + * @return the number of points. + **********************************************************************/ +unsigned GEOD_DLL geod_polygon_testedge(const struct geod_geodesic* g, + const struct geod_polygon* p, + double azi, double s, + int reverse, int sign, + double* pA, double* pP); + +/** + * A simple interface for computing the area of a geodesic polygon. + * + * @param[in] g a pointer to the geod_geodesic object specifying the + * ellipsoid. + * @param[in] lats an array of latitudes of the polygon vertices (degrees). + * @param[in] lons an array of longitudes of the polygon vertices (degrees). + * @param[in] n the number of vertices. + * @param[out] pA pointer to the area of the polygon (meters2). + * @param[out] pP pointer to the perimeter of the polygon (meters). + * + * \e lats should be in the range [−90°, 90°]. + * + * Arbitrarily complex polygons are allowed. In the case self-intersecting + * of polygons the area is accumulated "algebraically", e.g., the areas of + * the 2 loops in a figure-8 polygon will partially cancel. There's no need + * to "close" the polygon by repeating the first vertex. The area returned + * is signed with counter-clockwise traversal being treated as positive. + * + * Example, compute the area of Antarctica: + @code{.c} + double + lats[] = {-72.9, -71.9, -74.9, -74.3, -77.5, -77.4, -71.7, -65.9, -65.7, + -66.6, -66.9, -69.8, -70.0, -71.0, -77.3, -77.9, -74.7}, + lons[] = {-74, -102, -102, -131, -163, 163, 172, 140, 113, + 88, 59, 25, -4, -14, -33, -46, -61}; + struct geod_geodesic g; + double A, P; + geod_init(&g, 6378137, 1/298.257223563); + geod_polygonarea(&g, lats, lons, (sizeof lats) / (sizeof lats[0]), &A, &P); + printf("%.0f %.2f\n", A, P); + @endcode + **********************************************************************/ +void GEOD_DLL geod_polygonarea(const struct geod_geodesic* g, + double lats[], double lons[], int n, + double* pA, double* pP); + +/** + * mask values for the \e caps argument to geod_lineinit(). + **********************************************************************/ +enum geod_mask { GEOD_NONE = 0U, /**< Calculate nothing */ GEOD_LATITUDE = 1U<<7 | 0U, /**< Calculate latitude */ GEOD_LONGITUDE = 1U<<8 | 1U<<3, /**< Calculate longitude */ @@ -818,17 +818,17 @@ extern "C" { GEOD_GEODESICSCALE= 1U<<13 | 1U<<0 | 1U<<2,/**< Calculate geodesic scale */ GEOD_AREA = 1U<<14 | 1U<<4, /**< Calculate reduced length */ GEOD_ALL = 0x7F80U| 0x1FU /**< Calculate everything */ - }; +}; - /** - * flag values for the \e flags argument to geod_gendirect() and - * geod_genposition() - **********************************************************************/ - enum geod_flags { +/** + * flag values for the \e flags argument to geod_gendirect() and + * geod_genposition() + **********************************************************************/ +enum geod_flags { GEOD_NOFLAGS = 0U, /**< No flags */ GEOD_ARCMODE = 1U<<0, /**< Position given in terms of arc distance */ GEOD_LONG_UNROLL = 1U<<15 /**< Unroll the longitude */ - }; +}; #if defined(__cplusplus) }