29 #define GEOGRAPHICLIB_GEODESIC_ORDER 6
30 #define nA1 GEOGRAPHICLIB_GEODESIC_ORDER
31 #define nC1 GEOGRAPHICLIB_GEODESIC_ORDER
32 #define nC1p GEOGRAPHICLIB_GEODESIC_ORDER
33 #define nA2 GEOGRAPHICLIB_GEODESIC_ORDER
34 #define nC2 GEOGRAPHICLIB_GEODESIC_ORDER
35 #define nA3 GEOGRAPHICLIB_GEODESIC_ORDER
37 #define nC3 GEOGRAPHICLIB_GEODESIC_ORDER
38 #define nC3x ((nC3 * (nC3 - 1)) / 2)
39 #define nC4 GEOGRAPHICLIB_GEODESIC_ORDER
40 #define nC4x ((nC4 * (nC4 + 1)) / 2)
45 static unsigned init = 0;
46 static const int FALSE = 0;
47 static const int TRUE = 1;
48 static unsigned digits, maxit1, maxit2;
49 static real epsilon, realmin, pi, degree, NaN,
50 tiny, tol0, tol1, tol2, tolb, xthresh;
54 #if defined(__DBL_MANT_DIG__)
55 digits = __DBL_MANT_DIG__;
59 #if defined(__DBL_EPSILON__)
60 epsilon = __DBL_EPSILON__;
62 epsilon = pow(0.5, digits - 1);
64 #if defined(__DBL_MIN__)
65 realmin = __DBL_MIN__;
67 realmin = pow(0.5, 1022);
72 pi = atan2(0.0, -1.0);
75 maxit2 = maxit1 + digits + 10;
85 xthresh = 1000 * tol2;
103 static real sq(
real x) {
return x * x; }
112 return z == 0 ? x : x * log(y) / z;
117 y = log1px(2 * y/(1 - y))/2;
118 return x < 0 ? -y : y;
122 {
return sqrt(x * x + y * y); }
126 return x < 0 ? -y : y;
130 volatile real s = u + v;
131 volatile real up = s - v;
132 volatile real vpp = s - up;
143 real y = N < 0 ? 0 : *p++;
144 while (--N >= 0) y = y * x + *p++;
149 {
return x < y ? x : y; }
152 {
return x > y ? x : y; }
155 {
real t = *x; *x = *y; *y = t; }
157 static void norm2(
real* sinx,
real* cosx) {
158 real r = hypotx(*sinx, *cosx);
164 {
return x >= 180 ? x - 360 : (x < -180 ? x + 360 : x); }
166 {
return AngNormalize(fmod(x, (
real)(360))); }
169 real t, d = sumx(-x, y, &t);
170 if ((d - (
real)(180)) + t > (
real)(0))
172 else if ((d + (
real)(180)) + t <= (
real)(0))
179 volatile real y = fabs(x);
181 y = y < z ? z - (z - y) : y;
182 return x < 0 ? 0 - y : y;
188 static real SinCosSeries(boolx sinp,
190 const real c[],
int n);
197 boolx scalep,
real* pM12,
real* pM21,
221 boolx diffp,
real* pdlam12,
228 static void C1f(
real eps,
real c[]);
229 static void C1pf(
real eps,
real c[]);
231 static void C2f(
real eps,
real c[]);
232 static int transit(
real lon1,
real lon2);
233 static int transitdirect(
real lon1,
real lon2);
234 static void accini(
real s[]);
235 static void acccopy(
const real s[],
real t[]);
236 static void accadd(
real s[],
real y);
238 static void accneg(
real s[]);
243 g->
f = f <= 1 ? f : 1/
f;
245 g->e2 = g->
f * (2 - g->
f);
246 g->ep2 = g->e2 / sq(g->f1);
247 g->n = g->
f / ( 2 - g->
f);
249 g->c2 = (sq(g->
a) + sq(g->b) *
251 (g->e2 > 0 ? atanhx(sqrt(g->e2)) : atan(sqrt(-g->e2))) /
252 sqrt(fabs(g->e2))))/2;
262 g->etol2 = 0.1 * tol2 /
263 sqrt( maxx((
real)(0.001), fabs(g->
f)) * minx((
real)(1), 1 - g->
f/2) / 2 );
273 real alp1, cbet1, sbet1, phi, eps;
287 l->
azi1 = AngRound(AngNormalize(azi1));
289 alp1 = l->
azi1 * degree;
292 l->salp1 = l->
azi1 == -180 ? 0 : sin(alp1);
293 l->calp1 = fabs(l->
azi1) == 90 ? 0 : cos(alp1);
296 sbet1 = l->f1 * sin(phi);
297 cbet1 = fabs(lat1) == 90 ? tiny : cos(phi);
298 norm2(&sbet1, &cbet1);
299 l->dn1 = sqrt(1 + g->ep2 * sq(sbet1));
302 l->salp0 = l->salp1 * cbet1;
305 l->calp0 = hypotx(l->calp1, l->salp1 * sbet1);
315 l->ssig1 = sbet1; l->somg1 = l->salp0 * sbet1;
316 l->csig1 = l->comg1 = sbet1 != 0 || l->calp1 != 0 ? cbet1 * l->calp1 : 1;
317 norm2(&l->ssig1, &l->csig1);
320 l->k2 = sq(l->calp0) * g->ep2;
321 eps = l->k2 / (2 * (1 + sqrt(1 + l->k2)) + l->k2);
323 if (l->
caps & CAP_C1) {
325 l->A1m1 = A1m1f(eps);
327 l->B11 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C1a, nC1);
328 s = sin(l->B11); c = cos(l->B11);
330 l->stau1 = l->ssig1 * c + l->csig1 * s;
331 l->ctau1 = l->csig1 * c - l->ssig1 * s;
336 if (l->
caps & CAP_C1p)
339 if (l->
caps & CAP_C2) {
340 l->A2m1 = A2m1f(eps);
342 l->B21 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C2a, nC2);
345 if (l->
caps & CAP_C3) {
347 l->A3c = -l->
f * l->salp0 * A3f(g, eps);
348 l->B31 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C3a, nC3-1);
351 if (l->
caps & CAP_C4) {
354 l->A4 = sq(l->
a) * l->calp0 * l->salp0 * g->e2;
355 l->B41 = SinCosSeries(FALSE, l->ssig1, l->csig1, l->C4a, nC4);
360 unsigned flags,
real s12_a12,
365 real lat2 = 0, lon2 = 0, azi2 = 0, s12 = 0,
366 m12 = 0, M12 = 0, M21 = 0, S12 = 0;
368 real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
369 real omg12, lam12, lon12;
370 real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2, dn2;
380 outmask &= l->
caps & OUT_ALL;
389 sig12 = s12_a12 * degree;
390 s12a = fabs(s12_a12);
391 s12a -= 180 * floor(s12a / 180);
392 ssig12 = s12a == 0 ? 0 : sin(sig12);
393 csig12 = s12a == 90 ? 0 : cos(sig12);
397 tau12 = s12_a12 / (l->b * (1 + l->A1m1)),
401 B12 = - SinCosSeries(TRUE,
402 l->stau1 * c + l->ctau1 * s,
403 l->ctau1 * c - l->stau1 * s,
405 sig12 = tau12 - (B12 - l->B11);
406 ssig12 = sin(sig12); csig12 = cos(sig12);
407 if (fabs(l->
f) > 0.01) {
430 ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12,
431 csig2 = l->csig1 * csig12 - l->ssig1 * ssig12,
433 B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1);
434 serr = (1 + l->A1m1) * (sig12 + (B12 - l->B11)) - s12_a12 / l->b;
435 sig12 = sig12 - serr / sqrt(1 + l->k2 * sq(ssig2));
436 ssig12 = sin(sig12); csig12 = cos(sig12);
442 ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12;
443 csig2 = l->csig1 * csig12 - l->ssig1 * ssig12;
444 dn2 = sqrt(1 + l->k2 * sq(ssig2));
446 if (flags & GEOD_ARCMODE || fabs(l->
f) > 0.01)
447 B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1);
448 AB1 = (1 + l->A1m1) * (B12 - l->B11);
451 sbet2 = l->calp0 * ssig2;
453 cbet2 = hypotx(l->salp0, l->calp0 * csig2);
456 cbet2 = csig2 = tiny;
458 salp2 = l->salp0; calp2 = l->calp0 * csig2;
461 s12 = flags & GEOD_ARCMODE ? l->b * ((1 + l->A1m1) * sig12 + AB1) : s12_a12;
464 int E = l->salp0 < 0 ? -1 : 1;
466 somg2 = l->salp0 * ssig2; comg2 = csig2;
470 - (atan2( ssig2, csig2) - atan2( l->ssig1, l->csig1))
471 + (atan2(E * somg2, comg2) - atan2(E * l->somg1, l->comg1)))
472 : atan2(somg2 * l->comg1 - comg2 * l->somg1,
473 comg2 * l->comg1 + somg2 * l->somg1);
474 lam12 = omg12 + l->A3c *
475 ( sig12 + (SinCosSeries(TRUE, ssig2, csig2, l->C3a, nC3-1)
477 lon12 = lam12 / degree;
481 AngNormalize(AngNormalize(l->
lon1) + AngNormalize2(lon12));
485 lat2 = atan2(sbet2, l->f1 * cbet2) / degree;
489 azi2 = 0 - atan2(-salp2, calp2) / degree;
493 B22 = SinCosSeries(TRUE, ssig2, csig2, l->C2a, nC2),
494 AB2 = (1 + l->A2m1) * (B22 - l->B21),
495 J12 = (l->A1m1 - l->A2m1) * sig12 + (AB1 - AB2);
499 m12 = l->b * ((dn2 * (l->csig1 * ssig2) - l->dn1 * (l->ssig1 * csig2))
500 - l->csig1 * csig2 * J12);
502 real t = l->k2 * (ssig2 - l->ssig1) * (ssig2 + l->ssig1) / (l->dn1 + dn2);
503 M12 = csig12 + (t * ssig2 - csig2 * J12) * l->ssig1 / l->dn1;
504 M21 = csig12 - (t * l->ssig1 - l->csig1 * J12) * ssig2 / dn2;
510 B42 = SinCosSeries(FALSE, ssig2, csig2, l->C4a, nC4);
512 if (l->calp0 == 0 || l->salp0 == 0) {
514 salp12 = salp2 * l->calp1 - calp2 * l->salp1;
515 calp12 = calp2 * l->calp1 + salp2 * l->salp1;
520 if (salp12 == 0 && calp12 < 0) {
521 salp12 = tiny * l->calp1;
533 salp12 = l->calp0 * l->salp0 *
534 (csig12 <= 0 ? l->csig1 * (1 - csig12) + ssig12 * l->ssig1 :
535 ssig12 * (l->csig1 * ssig12 / (1 + csig12) + l->ssig1));
536 calp12 = sq(l->salp0) + sq(l->calp0) * l->csig1 * csig2;
538 S12 = l->c2 * atan2(salp12, calp12) + l->A4 * (B42 - l->B41);
541 if (outmask & GEOD_LATITUDE)
543 if (outmask & GEOD_LONGITUDE)
545 if (outmask & GEOD_AZIMUTH)
547 if (outmask & GEOD_DISTANCE)
549 if (outmask & GEOD_REDUCEDLENGTH)
551 if (outmask & GEOD_GEODESICSCALE) {
552 if (pM12) *pM12 = M12;
553 if (pM21) *pM21 = M21;
555 if (outmask & GEOD_AREA)
558 return flags & GEOD_ARCMODE ? s12_a12 : sig12 / degree;
563 geod_genposition(l, FALSE, s12, plat2, plon2, pazi2, 0, 0, 0, 0, 0);
568 unsigned flags,
real s12_a12,
574 (plat2 ? GEOD_LATITUDE : 0U) |
575 (plon2 ? GEOD_LONGITUDE : 0U) |
576 (pazi2 ? GEOD_AZIMUTH : 0U) |
577 (ps12 ? GEOD_DISTANCE : 0U) |
578 (pm12 ? GEOD_REDUCEDLENGTH : 0U) |
579 (pM12 || pM21 ? GEOD_GEODESICSCALE : 0U) |
580 (pS12 ? GEOD_AREA : 0U);
587 plat2, plon2, pazi2, ps12, pm12, pM12, pM21, pS12);
602 real s12 = 0, azi1 = 0, azi2 = 0, m12 = 0, M12 = 0, M21 = 0, S12 = 0;
604 int latsign, lonsign, swapp;
605 real phi, sbet1, cbet1, sbet2, cbet2, s12x = 0, m12x = 0;
606 real dn1, dn2, lam12, slam12, clam12;
607 real a12 = 0, sig12, calp1 = 0, salp1 = 0, calp2 = 0, salp2 = 0;
609 real C1a[nC1 + 1], C2a[nC2 + 1], C3a[nC3];
614 (ps12 ? GEOD_DISTANCE : 0U) |
615 (pazi1 || pazi2 ? GEOD_AZIMUTH : 0U) |
616 (pm12 ? GEOD_REDUCEDLENGTH : 0U) |
617 (pM12 || pM21 ? GEOD_GEODESICSCALE : 0U) |
618 (pS12 ? GEOD_AREA : 0U);
624 lon12 = AngDiff(AngNormalize(lon1), AngNormalize(lon2));
626 lon12 = AngRound(lon12);
628 lonsign = lon12 >= 0 ? 1 : -1;
631 lat1 = AngRound(lat1);
632 lat2 = AngRound(lat2);
634 swapp = fabs(lat1) >= fabs(lat2) ? 1 : -1;
640 latsign = lat1 < 0 ? 1 : -1;
657 sbet1 = g->f1 * sin(phi);
658 cbet1 = lat1 == -90 ? tiny : cos(phi);
659 norm2(&sbet1, &cbet1);
663 sbet2 = g->f1 * sin(phi);
664 cbet2 = fabs(lat2) == 90 ? tiny : cos(phi);
665 norm2(&sbet2, &cbet2);
675 if (cbet1 < -sbet1) {
677 sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
679 if (fabs(sbet2) == -sbet1)
683 dn1 = sqrt(1 + g->ep2 * sq(sbet1));
684 dn2 = sqrt(1 + g->ep2 * sq(sbet2));
686 lam12 = lon12 * degree;
687 slam12 = lon12 == 180 ? 0 : sin(lam12);
690 meridian = lat1 == -90 || slam12 == 0;
697 real ssig1, csig1, ssig2, csig2;
698 calp1 = clam12; salp1 = slam12;
699 calp2 = 1; salp2 = 0;
702 ssig1 = sbet1; csig1 = calp1 * cbet1;
703 ssig2 = sbet2; csig2 = calp2 * cbet2;
706 sig12 = atan2(maxx(csig1 * ssig2 - ssig1 * csig2, (
real)(0)),
707 csig1 * csig2 + ssig1 * ssig2);
710 Lengths(g, g->n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
711 cbet1, cbet2, &s12x, &m12x, &dummy,
712 (outmask & GEOD_GEODESICSCALE) != 0U, &M12, &M21, C1a, C2a);
721 if (sig12 < 1 || m12x >= 0) {
724 a12 = sig12 / degree;
733 (g->
f <= 0 || lam12 <= pi - g->f * pi)) {
736 calp1 = calp2 = 0; salp1 = salp2 = 1;
738 sig12 = omg12 = lam12 / g->f1;
739 m12x = g->b * sin(sig12);
740 if (outmask & GEOD_GEODESICSCALE)
741 M12 = M21 = cos(sig12);
744 }
else if (!meridian) {
751 sig12 = InverseStart(g, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
753 &salp1, &calp1, &salp2, &calp2, &dnm,
758 s12x = sig12 * g->b * dnm;
759 m12x = sq(dnm) * g->b * sin(sig12 / dnm);
760 if (outmask & GEOD_GEODESICSCALE)
761 M12 = M21 = cos(sig12 / dnm);
762 a12 = sig12 / degree;
763 omg12 = lam12 / (g->f1 * dnm);
777 real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0;
780 real salp1a = tiny, calp1a = 1, salp1b = tiny, calp1b = -1;
782 for (tripn = FALSE, tripb = FALSE; numit < maxit2; ++numit) {
786 v = (Lambda12(g, sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
787 &salp2, &calp2, &sig12, &ssig1, &csig1, &ssig2, &csig2,
788 &eps, &omg12, numit < maxit1, &dv, C1a, C2a, C3a)
792 if (tripb || !(fabs(v) >= (tripn ? 8 : 2) * tol0))
break;
794 if (v > 0 && (numit > maxit1 || calp1/salp1 > calp1b/salp1b))
795 { salp1b = salp1; calp1b = calp1; }
796 else if (v < 0 && (numit > maxit1 || calp1/salp1 < calp1a/salp1a))
797 { salp1a = salp1; calp1a = calp1; }
798 if (numit < maxit1 && dv > 0) {
802 sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
803 nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
804 if (nsalp1 > 0 && fabs(dalp1) < pi) {
805 calp1 = calp1 * cdalp1 - salp1 * sdalp1;
807 norm2(&salp1, &calp1);
811 tripn = fabs(v) <= 16 * tol0;
823 salp1 = (salp1a + salp1b)/2;
824 calp1 = (calp1a + calp1b)/2;
825 norm2(&salp1, &calp1);
827 tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb ||
828 fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb);
832 Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
833 cbet1, cbet2, &s12x, &m12x, &dummy,
834 (outmask & GEOD_GEODESICSCALE) != 0U, &M12, &M21, C1a, C2a);
838 a12 = sig12 / degree;
839 omg12 = lam12 - omg12;
843 if (outmask & GEOD_DISTANCE)
846 if (outmask & GEOD_REDUCEDLENGTH)
849 if (outmask & GEOD_AREA) {
852 salp0 = salp1 * cbet1,
853 calp0 = hypotx(calp1, salp1 * sbet1);
855 if (calp0 != 0 && salp0 != 0) {
858 ssig1 = sbet1, csig1 = calp1 * cbet1,
859 ssig2 = sbet2, csig2 = calp2 * cbet2,
860 k2 = sq(calp0) * g->ep2,
861 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
863 A4 = sq(g->
a) * calp0 * salp0 * g->e2;
866 norm2(&ssig1, &csig1);
867 norm2(&ssig2, &csig2);
869 B41 = SinCosSeries(FALSE, ssig1, csig1, C4a, nC4);
870 B42 = SinCosSeries(FALSE, ssig2, csig2, C4a, nC4);
871 S12 = A4 * (B42 - B41);
877 omg12 < (
real)(0.75) * pi &&
878 sbet2 - sbet1 < (
real)(1.75)) {
883 somg12 = sin(omg12), domg12 = 1 + cos(omg12),
884 dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
885 alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
886 domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
890 salp12 = salp2 * calp1 - calp2 * salp1,
891 calp12 = calp2 * calp1 + salp2 * salp1;
896 if (salp12 == 0 && calp12 < 0) {
897 salp12 = tiny * calp1;
900 alp12 = atan2(salp12, calp12);
902 S12 += g->c2 * alp12;
903 S12 *= swapp * lonsign * latsign;
910 swapx(&salp1, &salp2);
911 swapx(&calp1, &calp2);
912 if (outmask & GEOD_GEODESICSCALE)
916 salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
917 salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
919 if (outmask & GEOD_AZIMUTH) {
921 azi1 = 0 - atan2(-salp1, calp1) / degree;
922 azi2 = 0 - atan2(-salp2, calp2) / degree;
925 if (outmask & GEOD_DISTANCE)
927 if (outmask & GEOD_AZIMUTH) {
928 if (pazi1) *pazi1 =
azi1;
929 if (pazi2) *pazi2 = azi2;
931 if (outmask & GEOD_REDUCEDLENGTH)
933 if (outmask & GEOD_GEODESICSCALE) {
934 if (pM12) *pM12 = M12;
935 if (pM21) *pM21 = M21;
937 if (outmask & GEOD_AREA)
947 geod_geninverse(g, lat1, lon1, lat2, lon2, ps12, pazi1, pazi2, 0, 0, 0, 0);
950 real SinCosSeries(boolx sinp,
real sinx,
real cosx,
const real c[],
int n) {
958 ar = 2 * (cosx - sinx) * (cosx + sinx);
959 y0 = n & 1 ? *--c : 0; y1 = 0;
964 y1 = ar * y0 - y1 + *--c;
965 y0 = ar * y1 - y0 + *--c;
968 ? 2 * sinx * cosx * y0
978 boolx scalep,
real* pM12,
real* pM21,
981 real s12b = 0, m12b = 0, m0 = 0, M12 = 0, M21 = 0;
982 real A1m1, AB1, A2m1, AB2, J12;
989 AB1 = (1 + A1m1) * (SinCosSeries(TRUE, ssig2, csig2, C1a, nC1) -
990 SinCosSeries(TRUE, ssig1, csig1, C1a, nC1));
992 AB2 = (1 + A2m1) * (SinCosSeries(TRUE, ssig2, csig2, C2a, nC2) -
993 SinCosSeries(TRUE, ssig1, csig1, C2a, nC2));
995 J12 = m0 * sig12 + (AB1 - AB2);
999 m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - csig1 * csig2 * J12;
1001 s12b = (1 + A1m1) * sig12 + AB1;
1003 real csig12 = csig1 * csig2 + ssig1 * ssig2;
1004 real t = g->ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
1005 M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
1006 M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
1024 r = (p + q - 1) / 6;
1025 if ( !(q == 0 && r <= 0) ) {
1034 disc = S * (S + 2 * r3);
1038 real T3 = S + r3, T;
1042 T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc);
1046 u += T + (T != 0 ? r2 / T : 0);
1049 real ang = atan2(sqrt(-disc), -(S + r3));
1052 u += 2 * r * cos(ang / 3);
1054 v = sqrt(sq(u) + q);
1056 uv = u < 0 ? q / (v - u) : u + v;
1057 w = (uv - q) / (2 * v);
1060 k = uv / (sqrt(uv + sq(w)) + w);
1080 real salp1 = 0, calp1 = 0, salp2 = 0, calp2 = 0, dnm = 0;
1088 sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
1089 cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
1090 #if defined(__GNUC__) && __GNUC__ == 4 && \
1091 (__GNUC_MINOR__ < 6 || defined(__MINGW32__))
1100 volatile real xx1 = sbet2 * cbet1;
1101 volatile real xx2 = cbet2 * sbet1;
1102 sbet12a = xx1 + xx2;
1105 real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
1107 boolx shortline = cbet12 >= 0 && sbet12 < (
real)(0.5) &&
1108 cbet2 * lam12 < (
real)(0.5);
1109 real omg12 = lam12, somg12, comg12, ssig12, csig12;
1111 real sbetm2 = sq(sbet1 + sbet2);
1114 sbetm2 /= sbetm2 + sq(cbet1 + cbet2);
1115 dnm = sqrt(1 + g->ep2 * sbetm2);
1116 omg12 /= g->f1 * dnm;
1118 somg12 = sin(omg12); comg12 = cos(omg12);
1120 salp1 = cbet2 * somg12;
1121 calp1 = comg12 >= 0 ?
1122 sbet12 + cbet2 * sbet1 * sq(somg12) / (1 + comg12) :
1123 sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12);
1125 ssig12 = hypotx(salp1, calp1);
1126 csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
1128 if (shortline && ssig12 < g->etol2) {
1130 salp2 = cbet1 * somg12;
1131 calp2 = sbet12 - cbet1 * sbet2 *
1132 (comg12 >= 0 ? sq(somg12) / (1 + comg12) : 1 - comg12);
1133 norm2(&salp2, &calp2);
1135 sig12 = atan2(ssig12, csig12);
1136 }
else if (fabs(g->n) > (
real)(0.1) ||
1138 ssig12 >= 6 * fabs(g->n) * pi * sq(cbet1)) {
1143 real y, lamscale, betscale;
1152 k2 = sq(sbet1) * g->ep2,
1153 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
1154 lamscale = g->
f * cbet1 * A3f(g, eps) * pi;
1156 betscale = lamscale * cbet1;
1158 x = (lam12 - pi) / lamscale;
1159 y = sbet12a / betscale;
1163 cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
1164 bet12a = atan2(sbet12a, cbet12a);
1165 real m12b, m0, dummy;
1168 Lengths(g, g->n, pi + bet12a,
1169 sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
1170 cbet1, cbet2, &dummy, &m12b, &m0, FALSE,
1171 &dummy, &dummy, C1a, C2a);
1172 x = -1 + m12b / (cbet1 * cbet2 * m0 * pi);
1173 betscale = x < -(
real)(0.01) ? sbet12a / x :
1174 -g->
f * sq(cbet1) * pi;
1175 lamscale = betscale / cbet1;
1176 y = (lam12 - pi) / lamscale;
1179 if (y > -tol1 && x > -1 - xthresh) {
1182 salp1 = minx((
real)(1), -(
real)(x)); calp1 = - sqrt(1 - sq(salp1));
1184 calp1 = maxx((
real)(x > -tol1 ? 0 : -1), (
real)(x));
1185 salp1 = sqrt(1 - sq(calp1));
1222 real k = Astroid(x, y);
1224 omg12a = lamscale * ( g->
f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
1225 somg12 = sin(omg12a); comg12 = -cos(omg12a);
1227 salp1 = cbet2 * somg12;
1228 calp1 = sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12);
1233 norm2(&salp1, &calp1);
1235 salp1 = 1; calp1 = 0;
1258 boolx diffp,
real* pdlam12,
1261 real salp2 = 0, calp2 = 0, sig12 = 0,
1262 ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, domg12 = 0, dlam12 = 0;
1264 real somg1, comg1, somg2, comg2, omg12, lam12;
1267 if (sbet1 == 0 && calp1 == 0)
1273 salp0 = salp1 * cbet1;
1274 calp0 = hypotx(calp1, salp1 * sbet1);
1278 ssig1 = sbet1; somg1 = salp0 * sbet1;
1279 csig1 = comg1 = calp1 * cbet1;
1280 norm2(&ssig1, &csig1);
1287 salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
1292 calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ?
1293 sqrt(sq(calp1 * cbet1) +
1295 (cbet2 - cbet1) * (cbet1 + cbet2) :
1296 (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
1300 ssig2 = sbet2; somg2 = salp0 * sbet2;
1301 csig2 = comg2 = calp2 * cbet2;
1302 norm2(&ssig2, &csig2);
1306 sig12 = atan2(maxx(csig1 * ssig2 - ssig1 * csig2, (
real)(0)),
1307 csig1 * csig2 + ssig1 * ssig2);
1310 omg12 = atan2(maxx(comg1 * somg2 - somg1 * comg2, (
real)(0)),
1311 comg1 * comg2 + somg1 * somg2);
1312 k2 = sq(calp0) * g->ep2;
1313 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
1315 B312 = (SinCosSeries(TRUE, ssig2, csig2, C3a, nC3-1) -
1316 SinCosSeries(TRUE, ssig1, csig1, C3a, nC3-1));
1317 h0 = -g->
f * A3f(g, eps);
1318 domg12 = salp0 * h0 * (sig12 + B312);
1319 lam12 = omg12 + domg12;
1323 dlam12 = - 2 * g->f1 * dn1 / sbet1;
1326 Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
1327 cbet1, cbet2, &dummy, &dlam12, &dummy,
1328 FALSE, &dummy, &dummy, C1a, C2a);
1329 dlam12 *= g->f1 / (calp2 * cbet2);
1350 return polyval(nA3 - 1, g->A3x, eps);
1358 for (l = 1; l < nC3; ++l) {
1359 int m = nC3 - l - 1;
1361 c[l] = mult * polyval(m, g->C3x + o, eps);
1371 for (l = 0; l < nC4; ++l) {
1372 int m = nC4 - l - 1;
1373 c[l] = mult * polyval(m, g->C4x + o, eps);
1381 static const real coeff[] = {
1386 real t = polyval(m, coeff, sq(eps)) / coeff[m + 1];
1387 return (t + eps) / (1 - eps);
1392 static const real coeff[] = {
1410 for (l = 1; l <= nC1; ++l) {
1411 int m = (nC1 - l) / 2;
1412 c[l] = d * polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1420 static const real coeff[] = {
1422 205, -432, 768, 1536,
1424 4005, -4736, 3840, 12288,
1438 for (l = 1; l <= nC1p; ++l) {
1439 int m = (nC1p - l) / 2;
1440 c[l] = d * polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1448 static const real coeff[] = {
1453 real t = polyval(m, coeff, sq(eps)) / coeff[m + 1];
1454 return t * (1 - eps) - eps;
1459 static const real coeff[] = {
1477 for (l = 1; l <= nC2; ++l) {
1478 int m = (nC2 - l) / 2;
1479 c[l] = d * polyval(m, coeff + o, eps2) / coeff[o + m + 1];
1487 static const real coeff[] = {
1501 int o = 0, k = 0, j;
1502 for (j = nA3 - 1; j >= 0; --j) {
1503 int m = nA3 - j - 1 < j ? nA3 - j - 1 : j;
1504 g->A3x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1];
1511 static const real coeff[] = {
1543 int o = 0, k = 0, l, j;
1544 for (l = 1; l < nC3; ++l) {
1545 for (j = nC3 - 1; j >= l; --j) {
1546 int m = nC3 - j - 1 < j ? nC3 - j - 1 : j;
1547 g->C3x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1];
1555 static const real coeff[] = {
1561 -224, -4784, 1573, 45045,
1563 -10656, 14144, -4576, -858, 45045,
1565 64, 624, -4576, 6864, -3003, 15015,
1567 100, 208, 572, 3432, -12012, 30030, 45045,
1573 5792, 1040, -1287, 135135,
1575 5952, -11648, 9152, -2574, 135135,
1577 -64, -624, 4576, -6864, 3003, 135135,
1583 -8448, 4992, -1144, 225225,
1585 -1440, 4160, -4576, 1716, 225225,
1591 3584, -3328, 1144, 315315,
1599 int o = 0, k = 0, l, j;
1600 for (l = 0; l < nC4; ++l) {
1601 for (j = nC4 - 1; j >= l; --j) {
1602 int m = nC4 - j - 1;
1603 g->C4x[k++] = polyval(m, coeff + o, g->n) / coeff[o + m + 1];
1609 int transit(
real lon1,
real lon2) {
1614 lon1 = AngNormalize(lon1);
1615 lon2 = AngNormalize(lon2);
1616 lon12 = AngDiff(lon1, lon2);
1617 return lon1 < 0 && lon2 >= 0 && lon12 > 0 ? 1 :
1618 (lon2 < 0 && lon1 >= 0 && lon12 < 0 ? -1 : 0);
1621 int transitdirect(
real lon1,
real lon2) {
1622 lon1 = fmod(lon1, (
real)(720));
1623 lon2 = fmod(lon2, (
real)(720));
1624 return ( ((lon2 >= 0 && lon2 < 360) || lon2 < -360 ? 0 : 1) -
1625 ((lon1 >= 0 && lon1 < 360) || lon1 < -360 ? 0 : 1) );
1628 void accini(
real s[]) {
1633 void acccopy(
const real s[],
real t[]) {
1635 t[0] = s[0]; t[1] = s[1];
1640 real u, z = sumx(y, s[1], &u);
1641 s[0] = sumx(z, s[0], &s[1]);
1656 void accneg(
real s[]) {
1658 s[0] = -s[0]; s[1] = -s[1];
1662 p->lat0 = p->lon0 = p->
lat = p->
lon = NaN;
1663 p->polyline = (polylinep != 0);
1666 p->
num = p->crossings = 0;
1672 lon = AngNormalize(lon);
1674 p->lat0 = p->
lat = lat;
1675 p->lon0 = p->
lon = lon;
1679 &s12, 0, 0, 0, 0, 0, p->polyline ? 0 : &S12);
1683 p->crossings += transit(p->
lon, lon);
1685 p->
lat = lat; p->
lon = lon;
1697 0, 0, 0, 0, p->polyline ? 0 : &S12);
1701 p->crossings += transitdirect(p->
lon, lon);
1703 p->
lat = lat; p->
lon = lon;
1710 boolx reverse, boolx sign,
1712 real s12, S12, t[2], area0;
1716 if (!p->polyline && pA) *pA = 0;
1720 if (pP) *pP = p->P[0];
1724 &s12, 0, 0, 0, 0, 0, &S12);
1725 if (pP) *pP = accsum(p->P, s12);
1728 crossings = p->crossings + transit(p->
lon, p->lon0);
1729 area0 = 4 * pi * g->c2;
1731 accadd(t, (t[0] < 0 ? 1 : -1) * area0/2);
1740 else if (t[0] <= -area0/2)
1748 if (pA) *pA = 0 + t[0];
1755 boolx reverse, boolx sign,
1757 real perimeter, tempsum, area0;
1759 unsigned num = p->
num + 1;
1762 if (!p->polyline && pA) *pA = 0;
1765 perimeter = p->P[0];
1766 tempsum = p->polyline ? 0 : p->A[0];
1767 crossings = p->crossings;
1768 for (i = 0; i < (p->polyline ? 1 : 2); ++i) {
1771 i == 0 ? p->
lat : lat, i == 0 ? p->
lon : lon,
1772 i != 0 ? p->lat0 : lat, i != 0 ? p->lon0 : lon,
1773 &s12, 0, 0, 0, 0, 0, p->polyline ? 0 : &S12);
1777 crossings += transit(i == 0 ? p->
lon : lon,
1778 i != 0 ? p->lon0 : lon);
1782 if (pP) *pP = perimeter;
1786 area0 = 4 * pi * g->c2;
1788 tempsum += (tempsum < 0 ? 1 : -1) * area0/2;
1795 if (tempsum > area0/2)
1797 else if (tempsum <= -area0/2)
1800 if (tempsum >= area0)
1802 else if (tempsum < 0)
1805 if (pA) *pA = 0 + tempsum;
1812 boolx reverse, boolx sign,
1814 real perimeter, tempsum, area0;
1816 unsigned num = p->
num + 1;
1819 if (!p->polyline && pA) *pA = NaN;
1822 perimeter = p->P[0] + s;
1824 if (pP) *pP = perimeter;
1829 crossings = p->crossings;
1831 real lat, lon, s12, S12;
1836 crossings += transitdirect(p->
lon, lon);
1838 &s12, 0, 0, 0, 0, 0, &S12);
1841 crossings += transit(lon, p->lon0);
1844 area0 = 4 * pi * g->c2;
1846 tempsum += (tempsum < 0 ? 1 : -1) * area0/2;
1853 if (tempsum > area0/2)
1855 else if (tempsum <= -area0/2)
1858 if (tempsum >= area0)
1860 else if (tempsum < 0)
1863 if (pP) *pP = perimeter;
1864 if (pA) *pA = 0 + tempsum;
1874 for (i = 0; i < n; ++i)
unsigned 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)
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)
GeographicLib::Math::real real
void geod_polygon_addedge(const struct geod_geodesic *g, struct geod_polygon *p, double azi, double s)
void geod_position(const struct geod_geodesicline *l, double s12, double *plat2, double *plon2, double *pazi2)
void geod_lineinit(struct geod_geodesicline *l, const struct geod_geodesic *g, double lat1, double lon1, double azi1, unsigned caps)
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)
void geod_polygon_addpoint(const struct geod_geodesic *g, struct geod_polygon *p, double lat, double lon)
void geod_polygon_init(struct geod_polygon *p, int polylinep)
void geod_direct(const struct geod_geodesic *g, double lat1, double lon1, double azi1, double s12, double *plat2, double *plon2, double *pazi2)
unsigned geod_polygon_compute(const struct geod_geodesic *g, const struct geod_polygon *p, int reverse, int sign, double *pA, double *pP)
void geod_polygonarea(const struct geod_geodesic *g, double lats[], double lons[], int n, double *pA, double *pP)
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)
unsigned 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)
void geod_inverse(const struct geod_geodesic *g, double lat1, double lon1, double lat2, double lon2, double *ps12, double *pazi1, double *pazi2)
void geod_init(struct geod_geodesic *g, double a, double f)
Header for the geodesic routines in C.