35 # pragma warning (disable: 4701 4127 5055 5054)
43 : maxit2_(maxit1_ +
Math::digits() + 10)
47 , tiny_(sqrt(numeric_limits<real>::min()))
48 , tol0_(numeric_limits<real>::epsilon())
54 , tolb_(tol0_ * tol2_)
55 , xthresh_(1000 * tol2_)
60 , _ep2(_e2 /
Math::sq(_f1))
71 (_f > 0 ? asinh(sqrt(_ep2)) : atan(sqrt(-_e2))) /
83 , _etol2(real(0.1) * tol2_ /
84 sqrt( fmax(real(0.001), fabs(_f)) * fmin(real(1), 1 - _f/2) / 2 ))
86 if (!(isfinite(_a) && _a > 0))
88 if (!(isfinite(_b) && _b > 0))
204 #if GEOGRAPHICLIB_PRECISION == 1
205 if (n >= -10 && n <= 11) N = 4;
206 else if (n >= -21 && n <= 26) N = 6;
207 else if (n >= -30 && n <= 41) N = 8;
208 else if (n >= -45 && n <= 62) N = 12;
209 else if (n >= -55 && n <= 76) N = 16;
211 else if (n >= -67 && n <= 99) N = 24;
212 else if (n >= -74 && n <= 99) N = 32;
213 else if (n >= -82 && n <= 99) N = 48;
214 else if (n >= -86 && n <= 99) N = 64;
215 else if (n >= -90 && n <= 99) N = 96;
216 else if (n >= -92 && n <= 99) N = 128;
217 else if (n >= -95 && n <= 99) N = 192;
218 else if (n >= -96 && n <= 99) N = 256;
219 else if (n >= -97 && n <= 99) N = 384;
220 else if (n >= -98 && n <= 99) N = 512;
222 #elif GEOGRAPHICLIB_PRECISION == 2
223 if (n >= - 1 && n <= 1) N = 6;
224 else if (n >= - 3 && n <= 3) N = 8;
225 else if (n >= -11 && n <= 11) N = 12;
226 else if (n >= -19 && n <= 20) N = 16;
227 else if (n >= -33 && n <= 36) N = 24;
228 else if (n >= -43 && n <= 48) N = 32;
229 else if (n >= -57 && n <= 63) N = 48;
230 else if (n >= -65 && n <= 72) N = 64;
231 else if (n >= -75 && n <= 82) N = 96;
232 else if (n >= -81 && n <= 87) N = 128;
233 else if (n >= -87 && n <= 92) N = 192;
234 else if (n >= -90 && n <= 94) N = 256;
235 else if (n >= -93 && n <= 96) N = 384;
236 else if (n >= -94 && n <= 97) N = 512;
237 else if (n >= -96 && n <= 98) N = 768;
238 else if (n >= -97 && n <= 99) N = 1024;
239 else if (n >= -98 && n <= 99) N = 1536;
241 #elif GEOGRAPHICLIB_PRECISION == 3
242 if (n >= - 1 && n <= 1) N = 8;
243 else if (n >= - 6 && n <= 6) N = 12;
244 else if (n >= -12 && n <= 13) N = 16;
245 else if (n >= -23 && n <= 26) N = 24;
246 else if (n >= -35 && n <= 38) N = 32;
247 else if (n >= -49 && n <= 54) N = 48;
248 else if (n >= -59 && n <= 64) N = 64;
249 else if (n >= -70 && n <= 75) N = 96;
250 else if (n >= -76 && n <= 81) N = 128;
251 else if (n >= -83 && n <= 88) N = 192;
252 else if (n >= -87 && n <= 91) N = 256;
253 else if (n >= -91 && n <= 94) N = 384;
254 else if (n >= -93 && n <= 96) N = 512;
255 else if (n >= -95 && n <= 97) N = 768;
256 else if (n >= -96 && n <= 98) N = 1024;
257 else if (n >= -97 && n <= 99) N = 1536;
258 else if (n >= -98 && n <= 99) N = 2048;
260 #elif GEOGRAPHICLIB_PRECISION == 4
261 if (n >= - 1 && n <= 1) N = 16;
262 else if (n >= - 6 && n <= 6) N = 24;
263 else if (n >= -12 && n <= 13) N = 32;
264 else if (n >= -25 && n <= 26) N = 48;
265 else if (n >= -35 && n <= 37) N = 64;
266 else if (n >= -50 && n <= 52) N = 96;
267 else if (n >= -59 && n <= 62) N = 128;
268 else if (n >= -71 && n <= 73) N = 192;
269 else if (n >= -77 && n <= 79) N = 256;
270 else if (n >= -84 && n <= 86) N = 384;
271 else if (n >= -87 && n <= 89) N = 512;
272 else if (n >= -91 && n <= 93) N = 768;
273 else if (n >= -93 && n <= 95) N = 1024;
274 else if (n >= -95 && n <= 96) N = 1536;
275 else if (n >= -96 && n <= 97) N = 2048;
276 else if (n >= -97 && n <= 98) N = 3072;
277 else if (n >= -98 && n <= 98) N = 4096;
278 else if (n >= -98 && n <= 99) N = 6144;
280 #elif GEOGRAPHICLIB_PRECISION == 5
281 if (n >= - 3 && n <= 3) N = 48;
282 else if (n >= - 7 && n <= 8) N = 64;
283 else if (n >= -18 && n <= 18) N = 96;
284 else if (n >= -28 && n <= 28) N = 128;
285 else if (n >= -42 && n <= 43) N = 192;
286 else if (n >= -53 && n <= 54) N = 256;
287 else if (n >= -65 && n <= 66) N = 384;
288 else if (n >= -72 && n <= 73) N = 512;
289 else if (n >= -80 && n <= 81) N = 768;
290 else if (n >= -85 && n <= 86) N = 1024;
291 else if (n >= -89 && n <= 90) N = 1536;
292 else if (n >= -92 && n <= 92) N = 2048;
293 else if (n >= -94 && n <= 95) N = 3072;
294 else if (n >= -96 && n <= 96) N = 4096;
295 else if (n >= -97 && n <= 97) N = 6144;
296 else if (n >= -98 && n <= 98) N = 8192;
302 while (N < M) N = N % 3 == 0 ? 4*N/3 : 3*N/2;
305 #error "Bad value for GEOGRAPHICLIB_PRECISION"
318 unsigned caps)
const {
323 bool arcmode, real s12_a12,
325 real& lat2, real& lon2, real& azi2,
326 real& s12, real& m12,
327 real& M12, real& M21,
333 GenPosition(arcmode, s12_a12, outmask,
334 lat2, lon2, azi2, s12, m12, M12, M21, S12);
339 bool arcmode, real s12_a12,
340 unsigned caps)
const {
348 caps, arcmode, s12_a12);
353 unsigned caps)
const {
359 unsigned caps)
const {
365 unsigned outmask,
real& s12,
376 int lonsign = signbit(lon12) ? -1 : 1;
377 lon12 *= lonsign; lon12s *= lonsign;
384 lon12s = (
Math::hd - lon12) - lon12s;
391 int swapp = fabs(lat1) < fabs(lat2) || isnan(lat2) ? -1 : 1;
397 int latsign = signbit(lat1) ? 1 : -1;
412 real sbet1, cbet1, sbet2, cbet2, s12x, m12x;
415 EllipticFunction E(-_ep2);
420 Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1);
424 Math::norm(sbet2, cbet2); cbet2 = fmax(tiny_, cbet2);
434 if (cbet1 < -sbet1) {
436 sbet2 = copysign(sbet1, sbet2);
438 if (fabs(sbet2) == -sbet1)
443 dn1 = (_f >= 0 ? sqrt(1 + _ep2 *
Math::sq(sbet1)) :
444 sqrt(1 - _e2 * Math::sq(cbet1)) / _f1),
445 dn2 = (_f >= 0 ? sqrt(1 + _ep2 * Math::sq(sbet2)) :
446 sqrt(1 - _e2 * Math::sq(cbet2)) / _f1);
450 bool meridian = lat1 == -
Math::qd || slam12 == 0;
457 calp1 = clam12; salp1 = slam12;
458 calp2 = 1; salp2 = 0;
462 ssig1 = sbet1, csig1 = calp1 * cbet1,
463 ssig2 = sbet2, csig2 = calp2 * cbet2;
466 sig12 = atan2(fmax(
real(0), csig1 * ssig2 - ssig1 * csig2),
467 csig1 * csig2 + ssig1 * ssig2);
470 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
472 s12x, m12x, dummy, M12, M21);
481 if (sig12 < 1 || m12x >= 0) {
483 if (sig12 < 3 * tiny_ ||
485 (sig12 < tol0_ && (s12x < 0 || m12x < 0)))
486 sig12 = m12x = s12x = 0;
496 real omg12 = 0, somg12 = 2, comg12 = 0;
499 (_f <= 0 || lon12s >= _f *
Math::hd)) {
502 calp1 = calp2 = 0; salp1 = salp2 = 1;
504 sig12 = omg12 = lam12 / _f1;
505 m12x = _b * sin(sig12);
507 M12 = M21 = cos(sig12);
510 }
else if (!meridian) {
517 sig12 = InverseStart(E, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
518 lam12, slam12, clam12,
519 salp1, calp1, salp2, calp2, dnm);
523 s12x = sig12 * _b * dnm;
524 m12x =
Math::sq(dnm) * _b * sin(sig12 / dnm);
526 M12 = M21 = cos(sig12 / dnm);
528 omg12 = lam12 / (_f1 * dnm);
544 real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, domg12 = 0;
547 real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
548 for (
bool tripn =
false, tripb =
false;
573 real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
575 salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
576 E, domg12, numit < maxit1_, dv);
578 if (tripb || !(fabs(v) >= (tripn ? 8 : 1) * tol0_))
break;
580 if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
581 { salp1b = salp1; calp1b = calp1; }
582 else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
583 { salp1a = salp1; calp1a = calp1; }
584 if (numit < maxit1_ && dv > 0) {
592 sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
593 nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
595 calp1 = calp1 * cdalp1 - salp1 * sdalp1;
601 tripn = fabs(v) <= 16 * tol0_;
614 salp1 = (salp1a + salp1b)/2;
615 calp1 = (calp1a + calp1b)/2;
618 tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
619 fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
623 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
624 cbet1, cbet2, outmask, s12x, m12x, dummy, M12, M21);
629 if (outmask &
AREA) {
631 real sdomg12 = sin(domg12), cdomg12 = cos(domg12);
632 somg12 = slam12 * cdomg12 - clam12 * sdomg12;
633 comg12 = clam12 * cdomg12 + slam12 * sdomg12;
639 s12 =
real(0) + s12x;
642 m12 =
real(0) + m12x;
644 if (outmask &
AREA) {
647 salp0 = salp1 * cbet1,
648 calp0 = hypot(calp1, salp1 * sbet1);
651 A4 =
Math::sq(_a) * calp0 * salp0 * _e2;
656 ssig1 = sbet1, csig1 = calp1 * cbet1,
657 ssig2 = sbet2, csig2 = calp2 * cbet2;
660 I4Integrand i4(_ep2, k2);
661 vector<real> C4a(_nC4);
666 S12 = A4 * (B42 - B41);
671 if (!meridian && somg12 == 2) {
672 somg12 = sin(omg12); comg12 = cos(omg12);
677 comg12 > -
real(0.7071) &&
678 sbet2 - sbet1 <
real(1.75)) {
682 real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
683 alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
684 domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
688 salp12 = salp2 * calp1 - calp2 * salp1,
689 calp12 = calp2 * calp1 + salp2 * salp1;
694 if (salp12 == 0 && calp12 < 0) {
695 salp12 = tiny_ * calp1;
698 alp12 = atan2(salp12, calp12);
701 S12 *= swapp * lonsign * latsign;
714 salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
715 salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
722 real lat2, real lon2,
724 real& s12, real& azi1, real& azi2,
725 real& m12, real& M12, real& M21,
728 real salp1, calp1, salp2, calp2,
729 a12 = GenInverse(lat1, lon1, lat2, lon2,
730 outmask, s12, salp1, calp1, salp2, calp2,
740 real lat2, real lon2,
741 unsigned caps)
const {
742 real t, salp1, calp1, salp2, calp2,
743 a12 = GenInverse(lat1, lon1, lat2, lon2,
745 0u, t, salp1, calp1, salp2, calp2,
758 real cbet1,
real cbet2,
unsigned outmask,
775 (sig12 + (E.
deltaE(ssig2, csig2, dn2) - E.
deltaE(ssig1, csig1, dn1)));
780 (sig12 + (E.
deltaD(ssig2, csig2, dn2) - E.
deltaD(ssig1, csig1, dn1)));
786 m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
790 real csig12 = csig1 * csig2 + ssig1 * ssig2;
791 real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
792 M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
793 M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
806 if ( !(q == 0 && r <= 0) ) {
815 disc = S * (S + 2 * r3);
822 T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc);
826 u += T + (T != 0 ? r2 / T : 0);
829 real ang = atan2(sqrt(-disc), -(S + r3));
832 u += 2 * r * cos(ang / 3);
837 uv = u < 0 ? q / (v - u) : u + v,
838 w = (uv - q) / (2 * v);
841 k = uv / (sqrt(uv +
Math::sq(w)) + w);
850 Math::real GeodesicExact::InverseStart(EllipticFunction& E,
865 sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
866 cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
867 real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
868 bool shortline = cbet12 >= 0 && sbet12 <
real(0.5) &&
869 cbet2 * lam12 <
real(0.5);
875 sbetm2 /= sbetm2 +
Math::sq(cbet1 + cbet2);
876 dnm = sqrt(1 + _ep2 * sbetm2);
877 real omg12 = lam12 / (_f1 * dnm);
878 somg12 = sin(omg12); comg12 = cos(omg12);
880 somg12 = slam12; comg12 = clam12;
883 salp1 = cbet2 * somg12;
884 calp1 = comg12 >= 0 ?
885 sbet12 + cbet2 * sbet1 *
Math::sq(somg12) / (1 + comg12) :
886 sbet12a - cbet2 * sbet1 *
Math::sq(somg12) / (1 - comg12);
889 ssig12 = hypot(salp1, calp1),
890 csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
892 if (shortline && ssig12 < _etol2) {
894 salp2 = cbet1 * somg12;
895 calp2 = sbet12 - cbet1 * sbet2 *
896 (comg12 >= 0 ?
Math::sq(somg12) / (1 + comg12) : 1 - comg12);
899 sig12 = atan2(ssig12, csig12);
900 }
else if (fabs(_n) >
real(0.1) ||
907 real x, y, lamscale, betscale;
908 real lam12x = atan2(-slam12, -clam12);
913 E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
914 lamscale = _e2/_f1 * cbet1 * 2 * E.H();
916 betscale = lamscale * cbet1;
918 x = lam12x / lamscale;
919 y = sbet12a / betscale;
923 cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
924 bet12a = atan2(sbet12a, cbet12a);
925 real m12b, m0, dummy;
929 sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
931 x = -1 + m12b / (cbet1 * cbet2 * m0 *
Math::pi());
932 betscale = x < -
real(0.01) ? sbet12a / x :
934 lamscale = betscale / cbet1;
935 y = lam12x / lamscale;
938 if (y > -tol1_ && x > -1 - xthresh_) {
942 salp1 = fmin(
real(1), -x); calp1 = - sqrt(1 -
Math::sq(salp1));
944 calp1 = fmax(
real(x > -tol1_ ? 0 : -1), x);
982 real k = Astroid(x, y);
984 omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
985 somg12 = sin(omg12a); comg12 = -cos(omg12a);
987 salp1 = cbet2 * somg12;
988 calp1 = sbet12a - cbet2 * sbet1 *
Math::sq(somg12) / (1 - comg12);
995 salp1 = 1; calp1 = 0;
1008 EllipticFunction& E,
1010 bool diffp,
real& dlam12)
const
1013 if (sbet1 == 0 && calp1 == 0)
1020 salp0 = salp1 * cbet1,
1021 calp0 = hypot(calp1, salp1 * sbet1);
1023 real somg1, comg1, somg2, comg2, somg12, comg12, cchi1, cchi2, lam12;
1026 ssig1 = sbet1; somg1 = salp0 * sbet1;
1027 csig1 = comg1 = calp1 * cbet1;
1029 cchi1 = _f1 * dn1 * comg1;
1038 salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
1043 calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ?
1046 (cbet2 - cbet1) * (cbet1 + cbet2) :
1047 (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
1051 ssig2 = sbet2; somg2 = salp0 * sbet2;
1052 csig2 = comg2 = calp2 * cbet2;
1054 cchi2 = _f1 * dn2 * comg2;
1060 sig12 = atan2(fmax(
real(0), csig1 * ssig2 - ssig1 * csig2),
1061 csig1 * csig2 + ssig1 * ssig2);
1064 somg12 = fmax(
real(0), comg1 * somg2 - somg1 * comg2);
1065 comg12 = comg1 * comg2 + somg1 * somg2;
1067 E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
1070 schi12 = fmax(
real(0), cchi1 * somg2 - somg1 * cchi2),
1071 cchi12 = cchi1 * cchi2 + somg1 * somg2;
1073 real eta = atan2(schi12 * clam120 - cchi12 * slam120,
1074 cchi12 * clam120 + schi12 * slam120);
1075 real deta12 = -_e2/_f1 * salp0 * E.H() / (
Math::pi() / 2) *
1076 (sig12 + (E.deltaH(ssig2, csig2, dn2) - E.deltaH(ssig1, csig1, dn1)));
1077 lam12 = eta + deta12;
1079 domg12 = deta12 + atan2(schi12 * comg12 - cchi12 * somg12,
1080 cchi12 * comg12 + schi12 * somg12);
1083 dlam12 = - 2 * _f1 * dn1 / sbet1;
1086 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
1088 dummy, dlam12, dummy, dummy, dummy);
1089 dlam12 *= _f1 / (calp2 * cbet2);
1098 using std::sqrt;
using std::asinh;
using std::asin;
1100 (x > 0 ? asinh(sqrt(x))/sqrt(x) :
1101 asin(sqrt(-x))/sqrt(-x));
1111 return x + (sqrt(1 + x) * asinhsqrt(x) - 1);
1116 return x == 0 ? 4/
real(3) :
1118 1 + (1 - asinhsqrt(x) / sqrt(1+x)) / (2*x);
1139 using std::sqrt;
using std::fabs;
using std::asinh;
using std::asin;
1140 if (X == y)
return tdX;
1141 if (X * y <= 0)
return ( tX - t(y) ) / (X - y);
1143 sy = sqrt(fabs(y)), sy1 = sqrt(1 + y),
1144 z = (X - y) / (sX * sy1 + sy * sX1),
1146 d2 = 2 * (X * sy * sy1 + y * sXX1);
1148 ( 1 + (asinh(z)/z) / d1 - (asinhsX + asinh(sy)) / d2 ) :
1150 ( 1 - (asin (z)/z) / d1 - (asinhsX + asin (sy)) / d2 );
1152 GeodesicExact::I4Integrand::I4Integrand(
real ep2,
real k2)
1158 using std::fabs;
using std::sqrt;
using std::asinh;
using std::asin;
1162 asinhsX = X > 0 ? asinh(sX) : asin(sX);
1164 Math::real GeodesicExact::I4Integrand::operator()(
real sig)
const {
1166 real ssig = sin(sig);
1167 return - DtX(_k2 *
Math::sq(ssig)) * ssig/2;
GeographicLib::Math::real real
Header for GeographicLib::GeodesicExact class.
Header for GeographicLib::GeodesicLineExact class.
#define GEOGRAPHICLIB_PANIC
void transform(std::function< real(real)> f, real F[]) const
static real integral(real sinx, real cosx, const real F[], int N)
Elliptic integrals and functions.
Math::real deltaE(real sn, real cn, real dn) const
Math::real deltaD(real sn, real cn, real dn) const
Exact geodesic calculations.
GeodesicLineExact InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const
GeodesicLineExact GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const
GeodesicLineExact DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const
Math::real GenDirect(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned outmask, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21, real &S12) const
friend class GeodesicLineExact
GeodesicLineExact Line(real lat1, real lon1, real azi1, unsigned caps=ALL) const
GeodesicExact(real a, real f)
static const GeodesicExact & WGS84()
GeodesicLineExact ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const
Exception handling for GeographicLib.
Mathematical functions needed by GeographicLib.
static void sincosd(T x, T &sinx, T &cosx)
static T atan2d(T y, T x)
static void norm(T &x, T &y)
static T AngNormalize(T x)
static void sincosde(T x, T t, T &sinx, T &cosx)
static T AngDiff(T x, T y, T &e)
@ hd
degrees per half turn
@ qd
degrees per quarter turn
Namespace for GeographicLib.
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)