29 #define GEOGRAPHICLIB_GEODESIC_ORDER 6
30 #define nC1 GEOGRAPHICLIB_GEODESIC_ORDER
31 #define nC1p GEOGRAPHICLIB_GEODESIC_ORDER
32 #define nC2 GEOGRAPHICLIB_GEODESIC_ORDER
33 #define nA3 GEOGRAPHICLIB_GEODESIC_ORDER
35 #define nC3 GEOGRAPHICLIB_GEODESIC_ORDER
36 #define nC3x ((nC3 * (nC3 - 1)) / 2)
37 #define nC4 GEOGRAPHICLIB_GEODESIC_ORDER
38 #define nC4x ((nC4 * (nC4 + 1)) / 2)
43 static unsigned init = 0;
44 static const int FALSE = 0;
45 static const int TRUE = 1;
46 static unsigned digits, maxit1, maxit2;
47 static real epsilon, realmin, pi, degree, NaN,
48 tiny, tol0, tol1, tol2, tolb, xthresh;
52 #if defined(__DBL_MANT_DIG__)
53 digits = __DBL_MANT_DIG__;
57 #if defined(__DBL_EPSILON__)
58 epsilon = __DBL_EPSILON__;
60 epsilon = pow(0.5, digits - 1);
62 #if defined(__DBL_MIN__)
63 realmin = __DBL_MIN__;
65 realmin = pow(0.5, 1022);
70 pi = atan2(0.0, -1.0);
73 maxit2 = maxit1 + digits + 10;
83 xthresh = 1000 * tol2;
101 static real sq(
real x) {
return x * x; }
110 return z == 0 ? x : x * log(y) / z;
115 y = log1px(2 * y/(1 - y))/2;
116 return x < 0 ? -y : y;
120 {
return sqrt(x * x + y * y); }
124 return x < 0 ? -y : y;
128 volatile real s = u + v;
129 volatile real up = s - v;
130 volatile real vpp = s - up;
141 {
return x < y ? x : y; }
144 {
return x > y ? x : y; }
147 {
real t = *x; *x = *y; *y = t; }
149 static void SinCosNorm(
real* sinx,
real* cosx) {
150 real r = hypotx(*sinx, *cosx);
156 {
return x >= 180 ? x - 360 : (x < -180 ? x + 360 : x); }
158 {
return AngNormalize(fmod(x, (
real)(360))); }
161 real t, d = sumx(-x, y, &t);
162 if ((d - (
real)(180)) + t > (
real)(0))
164 else if ((d + (
real)(180)) + t <= (
real)(0))
171 volatile real y = fabs(x);
173 y = y < z ? z - (z - y) : y;
174 return x < 0 ? -y : y;
180 static real SinCosSeries(boolx sinp,
182 const real c[],
int n);
189 boolx scalep,
real* pM12,
real* pM21,
213 boolx diffp,
real* pdlam12,
220 static void C1f(
real eps,
real c[]);
221 static void C1pf(
real eps,
real c[]);
223 static void C2f(
real eps,
real c[]);
224 static int transit(
real lon1,
real lon2);
225 static void accini(
real s[]);
226 static void acccopy(
const real s[],
real t[]);
227 static void accadd(
real s[],
real y);
229 static void accneg(
real s[]);
234 g->
f = f <= 1 ? f : 1/
f;
236 g->e2 = g->
f * (2 - g->
f);
237 g->ep2 = g->e2 / sq(g->f1);
238 g->n = g->
f / ( 2 - g->
f);
240 g->c2 = (sq(g->
a) + sq(g->b) *
242 (g->e2 > 0 ? atanhx(sqrt(g->e2)) : atan(sqrt(-g->e2))) /
243 sqrt(fabs(g->e2))))/2;
253 g->etol2 = 0.1 * tol2 /
254 sqrt( maxx((
real)(0.001), fabs(g->
f)) * minx((
real)(1), 1 - g->
f/2) / 2 );
264 real alp1, cbet1, sbet1, phi, eps;
275 azi1 = AngRound(AngNormalize(azi1));
276 lon1 = AngNormalize(lon1);
281 alp1 = azi1 * degree;
284 l->salp1 = azi1 == -180 ? 0 : sin(alp1);
285 l->calp1 = fabs(azi1) == 90 ? 0 : cos(alp1);
288 sbet1 = l->f1 * sin(phi);
289 cbet1 = fabs(lat1) == 90 ? tiny : cos(phi);
290 SinCosNorm(&sbet1, &cbet1);
291 l->dn1 = sqrt(1 + g->ep2 * sq(sbet1));
294 l->salp0 = l->salp1 * cbet1;
297 l->calp0 = hypotx(l->calp1, l->salp1 * sbet1);
307 l->ssig1 = sbet1; l->somg1 = l->salp0 * sbet1;
308 l->csig1 = l->comg1 = sbet1 != 0 || l->calp1 != 0 ? cbet1 * l->calp1 : 1;
309 SinCosNorm(&l->ssig1, &l->csig1);
312 l->k2 = sq(l->calp0) * g->ep2;
313 eps = l->k2 / (2 * (1 + sqrt(1 + l->k2)) + l->k2);
315 if (l->
caps & CAP_C1) {
317 l->A1m1 = A1m1f(eps);
319 l->B11 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C1a, nC1);
320 s = sin(l->B11); c = cos(l->B11);
322 l->stau1 = l->ssig1 * c + l->csig1 * s;
323 l->ctau1 = l->csig1 * c - l->ssig1 * s;
328 if (l->
caps & CAP_C1p)
331 if (l->
caps & CAP_C2) {
332 l->A2m1 = A2m1f(eps);
334 l->B21 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C2a, nC2);
337 if (l->
caps & CAP_C3) {
339 l->A3c = -l->
f * l->salp0 * A3f(g, eps);
340 l->B31 = SinCosSeries(TRUE, l->ssig1, l->csig1, l->C3a, nC3-1);
343 if (l->
caps & CAP_C4) {
346 l->A4 = sq(l->
a) * l->calp0 * l->salp0 * g->e2;
347 l->B41 = SinCosSeries(FALSE, l->ssig1, l->csig1, l->C4a, nC4);
352 boolx arcmode,
real s12_a12,
357 real lat2 = 0, lon2 = 0, azi2 = 0, s12 = 0,
358 m12 = 0, M12 = 0, M21 = 0, S12 = 0;
360 real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
361 real omg12, lam12, lon12;
362 real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2, dn2;
372 outmask &= l->
caps & OUT_ALL;
381 sig12 = s12_a12 * degree;
382 s12a = fabs(s12_a12);
383 s12a -= 180 * floor(s12a / 180);
384 ssig12 = s12a == 0 ? 0 : sin(sig12);
385 csig12 = s12a == 90 ? 0 : cos(sig12);
389 tau12 = s12_a12 / (l->b * (1 + l->A1m1)),
393 B12 = - SinCosSeries(TRUE,
394 l->stau1 * c + l->ctau1 * s,
395 l->ctau1 * c - l->stau1 * s,
397 sig12 = tau12 - (B12 - l->B11);
398 ssig12 = sin(sig12); csig12 = cos(sig12);
399 if (fabs(l->
f) > 0.01) {
422 ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12,
423 csig2 = l->csig1 * csig12 - l->ssig1 * ssig12,
425 B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1);
426 serr = (1 + l->A1m1) * (sig12 + (B12 - l->B11)) - s12_a12 / l->b;
427 sig12 = sig12 - serr / sqrt(1 + l->k2 * sq(ssig2));
428 ssig12 = sin(sig12); csig12 = cos(sig12);
434 ssig2 = l->ssig1 * csig12 + l->csig1 * ssig12;
435 csig2 = l->csig1 * csig12 - l->ssig1 * ssig12;
436 dn2 = sqrt(1 + l->k2 * sq(ssig2));
438 if (arcmode || fabs(l->
f) > 0.01)
439 B12 = SinCosSeries(TRUE, ssig2, csig2, l->C1a, nC1);
440 AB1 = (1 + l->A1m1) * (B12 - l->B11);
443 sbet2 = l->calp0 * ssig2;
445 cbet2 = hypotx(l->salp0, l->calp0 * csig2);
448 cbet2 = csig2 = tiny;
450 somg2 = l->salp0 * ssig2; comg2 = csig2;
452 salp2 = l->salp0; calp2 = l->calp0 * csig2;
454 omg12 = atan2(somg2 * l->comg1 - comg2 * l->somg1,
455 comg2 * l->comg1 + somg2 * l->somg1);
458 s12 = arcmode ? l->b * ((1 + l->A1m1) * sig12 + AB1) : s12_a12;
461 lam12 = omg12 + l->A3c *
462 ( sig12 + (SinCosSeries(TRUE, ssig2, csig2, l->C3a, nC3-1)
464 lon12 = lam12 / degree;
467 lon12 = AngNormalize2(lon12);
468 lon2 = AngNormalize(l->
lon1 + lon12);
472 lat2 = atan2(sbet2, l->f1 * cbet2) / degree;
476 azi2 = 0 - atan2(-salp2, calp2) / degree;
480 B22 = SinCosSeries(TRUE, ssig2, csig2, l->C2a, nC2),
481 AB2 = (1 + l->A2m1) * (B22 - l->B21),
482 J12 = (l->A1m1 - l->A2m1) * sig12 + (AB1 - AB2);
486 m12 = l->b * ((dn2 * (l->csig1 * ssig2) - l->dn1 * (l->ssig1 * csig2))
487 - l->csig1 * csig2 * J12);
489 real t = l->k2 * (ssig2 - l->ssig1) * (ssig2 + l->ssig1) / (l->dn1 + dn2);
490 M12 = csig12 + (t * ssig2 - csig2 * J12) * l->ssig1 / l->dn1;
491 M21 = csig12 - (t * l->ssig1 - l->csig1 * J12) * ssig2 / dn2;
497 B42 = SinCosSeries(FALSE, ssig2, csig2, l->C4a, nC4);
499 if (l->calp0 == 0 || l->salp0 == 0) {
501 salp12 = salp2 * l->calp1 - calp2 * l->salp1;
502 calp12 = calp2 * l->calp1 + salp2 * l->salp1;
507 if (salp12 == 0 && calp12 < 0) {
508 salp12 = tiny * l->calp1;
520 salp12 = l->calp0 * l->salp0 *
521 (csig12 <= 0 ? l->csig1 * (1 - csig12) + ssig12 * l->ssig1 :
522 ssig12 * (l->csig1 * ssig12 / (1 + csig12) + l->ssig1));
523 calp12 = sq(l->salp0) + sq(l->calp0) * l->csig1 * csig2;
525 S12 = l->c2 * atan2(salp12, calp12) + l->A4 * (B42 - l->B41);
528 if (outmask & GEOD_LATITUDE)
530 if (outmask & GEOD_LONGITUDE)
532 if (outmask & GEOD_AZIMUTH)
534 if (outmask & GEOD_DISTANCE)
536 if (outmask & GEOD_REDUCEDLENGTH)
538 if (outmask & GEOD_GEODESICSCALE) {
539 if (pM12) *pM12 = M12;
540 if (pM21) *pM21 = M21;
542 if (outmask & GEOD_AREA)
545 return arcmode ? s12_a12 : sig12 / degree;
550 geod_genposition(l, FALSE, s12, plat2, plon2, pazi2, 0, 0, 0, 0, 0);
555 boolx arcmode,
real s12_a12,
561 (plat2 ? GEOD_LATITUDE : 0U) |
562 (plon2 ? GEOD_LONGITUDE : 0U) |
563 (pazi2 ? GEOD_AZIMUTH : 0U) |
564 (ps12 ? GEOD_DISTANCE : 0U) |
565 (pm12 ? GEOD_REDUCEDLENGTH : 0U) |
566 (pM12 || pM21 ? GEOD_GEODESICSCALE : 0U) |
567 (pS12 ? GEOD_AREA : 0U);
573 plat2, plon2, pazi2, ps12, pm12, pM12, pM21, pS12);
580 geod_gendirect(g, lat1, lon1, azi1, FALSE, s12, plat2, plon2, pazi2,
588 real s12 = 0, azi1 = 0, azi2 = 0, m12 = 0, M12 = 0, M21 = 0, S12 = 0;
590 int latsign, lonsign, swapp;
591 real phi, sbet1, cbet1, sbet2, cbet2, s12x = 0, m12x = 0;
592 real dn1, dn2, lam12, slam12, clam12;
593 real a12 = 0, sig12, calp1 = 0, salp1 = 0, calp2 = 0, salp2 = 0;
600 (ps12 ? GEOD_DISTANCE : 0U) |
601 (pazi1 || pazi2 ? GEOD_AZIMUTH : 0U) |
602 (pm12 ? GEOD_REDUCEDLENGTH : 0U) |
603 (pM12 || pM21 ? GEOD_GEODESICSCALE : 0U) |
604 (pS12 ? GEOD_AREA : 0U);
610 lon12 = AngDiff(AngNormalize(lon1), AngNormalize(lon2));
612 lon12 = AngRound(lon12);
614 lonsign = lon12 >= 0 ? 1 : -1;
617 lat1 = AngRound(lat1);
618 lat2 = AngRound(lat2);
620 swapp = fabs(lat1) >= fabs(lat2) ? 1 : -1;
626 latsign = lat1 < 0 ? 1 : -1;
643 sbet1 = g->f1 * sin(phi);
644 cbet1 = lat1 == -90 ? tiny : cos(phi);
645 SinCosNorm(&sbet1, &cbet1);
649 sbet2 = g->f1 * sin(phi);
650 cbet2 = fabs(lat2) == 90 ? tiny : cos(phi);
651 SinCosNorm(&sbet2, &cbet2);
661 if (cbet1 < -sbet1) {
663 sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
665 if (fabs(sbet2) == -sbet1)
669 dn1 = sqrt(1 + g->ep2 * sq(sbet1));
670 dn2 = sqrt(1 + g->ep2 * sq(sbet2));
672 lam12 = lon12 * degree;
673 slam12 = lon12 == 180 ? 0 : sin(lam12);
676 meridian = lat1 == -90 || slam12 == 0;
683 real ssig1, csig1, ssig2, csig2;
684 calp1 = clam12; salp1 = slam12;
685 calp2 = 1; salp2 = 0;
688 ssig1 = sbet1; csig1 = calp1 * cbet1;
689 ssig2 = sbet2; csig2 = calp2 * cbet2;
692 sig12 = atan2(maxx(csig1 * ssig2 - ssig1 * csig2, (
real)(0)),
693 csig1 * csig2 + ssig1 * ssig2);
696 Lengths(g, g->n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
697 cbet1, cbet2, &s12x, &m12x, &dummy,
698 (outmask & GEOD_GEODESICSCALE) != 0U, &M12, &M21, C1a, C2a);
707 if (sig12 < 1 || m12x >= 0) {
710 a12 = sig12 / degree;
719 (g->
f <= 0 || lam12 <= pi - g->f * pi)) {
722 calp1 = calp2 = 0; salp1 = salp2 = 1;
724 sig12 = omg12 = lam12 / g->f1;
725 m12x = g->b * sin(sig12);
726 if (outmask & GEOD_GEODESICSCALE)
727 M12 = M21 = cos(sig12);
730 }
else if (!meridian) {
737 sig12 = InverseStart(g, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
739 &salp1, &calp1, &salp2, &calp2, &dnm,
744 s12x = sig12 * g->b * dnm;
745 m12x = sq(dnm) * g->b * sin(sig12 / dnm);
746 if (outmask & GEOD_GEODESICSCALE)
747 M12 = M21 = cos(sig12 / dnm);
748 a12 = sig12 / degree;
749 omg12 = lam12 / (g->f1 * dnm);
763 real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0;
766 real salp1a = tiny, calp1a = 1, salp1b = tiny, calp1b = -1;
768 for (tripn = FALSE, tripb = FALSE; numit < maxit2; ++numit) {
772 v = (Lambda12(g, sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
773 &salp2, &calp2, &sig12, &ssig1, &csig1, &ssig2, &csig2,
774 &eps, &omg12, numit < maxit1, &dv, C1a, C2a, C3a)
778 if (tripb || !(fabs(v) >= (tripn ? 8 : 2) * tol0))
break;
780 if (v > 0 && (numit > maxit1 || calp1/salp1 > calp1b/salp1b))
781 { salp1b = salp1; calp1b = calp1; }
782 else if (v < 0 && (numit > maxit1 || calp1/salp1 < calp1a/salp1a))
783 { salp1a = salp1; calp1a = calp1; }
784 if (numit < maxit1 && dv > 0) {
788 sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
789 nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
790 if (nsalp1 > 0 && fabs(dalp1) < pi) {
791 calp1 = calp1 * cdalp1 - salp1 * sdalp1;
793 SinCosNorm(&salp1, &calp1);
797 tripn = fabs(v) <= 16 * tol0;
809 salp1 = (salp1a + salp1b)/2;
810 calp1 = (calp1a + calp1b)/2;
811 SinCosNorm(&salp1, &calp1);
813 tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb ||
814 fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb);
818 Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
819 cbet1, cbet2, &s12x, &m12x, &dummy,
820 (outmask & GEOD_GEODESICSCALE) != 0U, &M12, &M21, C1a, C2a);
824 a12 = sig12 / degree;
825 omg12 = lam12 - omg12;
829 if (outmask & GEOD_DISTANCE)
832 if (outmask & GEOD_REDUCEDLENGTH)
835 if (outmask & GEOD_AREA) {
838 salp0 = salp1 * cbet1,
839 calp0 = hypotx(calp1, salp1 * sbet1);
841 if (calp0 != 0 && salp0 != 0) {
844 ssig1 = sbet1, csig1 = calp1 * cbet1,
845 ssig2 = sbet2, csig2 = calp2 * cbet2,
846 k2 = sq(calp0) * g->ep2,
847 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
849 A4 = sq(g->
a) * calp0 * salp0 * g->e2;
852 SinCosNorm(&ssig1, &csig1);
853 SinCosNorm(&ssig2, &csig2);
855 B41 = SinCosSeries(FALSE, ssig1, csig1, C4a, nC4);
856 B42 = SinCosSeries(FALSE, ssig2, csig2, C4a, nC4);
857 S12 = A4 * (B42 - B41);
863 omg12 < (
real)(0.75) * pi &&
864 sbet2 - sbet1 < (
real)(1.75)) {
869 somg12 = sin(omg12), domg12 = 1 + cos(omg12),
870 dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
871 alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
872 domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
876 salp12 = salp2 * calp1 - calp2 * salp1,
877 calp12 = calp2 * calp1 + salp2 * salp1;
882 if (salp12 == 0 && calp12 < 0) {
883 salp12 = tiny * calp1;
886 alp12 = atan2(salp12, calp12);
888 S12 += g->c2 * alp12;
889 S12 *= swapp * lonsign * latsign;
896 swapx(&salp1, &salp2);
897 swapx(&calp1, &calp2);
898 if (outmask & GEOD_GEODESICSCALE)
902 salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
903 salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
905 if (outmask & GEOD_AZIMUTH) {
907 azi1 = 0 - atan2(-salp1, calp1) / degree;
908 azi2 = 0 - atan2(-salp2, calp2) / degree;
911 if (outmask & GEOD_DISTANCE)
913 if (outmask & GEOD_AZIMUTH) {
914 if (pazi1) *pazi1 =
azi1;
915 if (pazi2) *pazi2 = azi2;
917 if (outmask & GEOD_REDUCEDLENGTH)
919 if (outmask & GEOD_GEODESICSCALE) {
920 if (pM12) *pM12 = M12;
921 if (pM21) *pM21 = M21;
923 if (outmask & GEOD_AREA)
933 geod_geninverse(g, lat1, lon1, lat2, lon2, ps12, pazi1, pazi2, 0, 0, 0, 0);
936 real SinCosSeries(boolx sinp,
real sinx,
real cosx,
const real c[],
int n) {
944 ar = 2 * (cosx - sinx) * (cosx + sinx);
945 y0 = n & 1 ? *--c : 0; y1 = 0;
950 y1 = ar * y0 - y1 + *--c;
951 y0 = ar * y1 - y0 + *--c;
954 ? 2 * sinx * cosx * y0
964 boolx scalep,
real* pM12,
real* pM21,
967 real s12b = 0, m12b = 0, m0 = 0, M12 = 0, M21 = 0;
968 real A1m1, AB1, A2m1, AB2, J12;
975 AB1 = (1 + A1m1) * (SinCosSeries(TRUE, ssig2, csig2, C1a, nC1) -
976 SinCosSeries(TRUE, ssig1, csig1, C1a, nC1));
978 AB2 = (1 + A2m1) * (SinCosSeries(TRUE, ssig2, csig2, C2a, nC2) -
979 SinCosSeries(TRUE, ssig1, csig1, C2a, nC2));
981 J12 = m0 * sig12 + (AB1 - AB2);
985 m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - csig1 * csig2 * J12;
987 s12b = (1 + A1m1) * sig12 + AB1;
989 real csig12 = csig1 * csig2 + ssig1 * ssig2;
990 real t = g->ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
991 M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
992 M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
1010 r = (p + q - 1) / 6;
1011 if ( !(q == 0 && r <= 0) ) {
1020 disc = S * (S + 2 * r3);
1024 real T3 = S + r3, T;
1028 T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc);
1032 u += T + (T != 0 ? r2 / T : 0);
1035 real ang = atan2(sqrt(-disc), -(S + r3));
1038 u += 2 * r * cos(ang / 3);
1040 v = sqrt(sq(u) + q);
1042 uv = u < 0 ? q / (v - u) : u + v;
1043 w = (uv - q) / (2 * v);
1046 k = uv / (sqrt(uv + sq(w)) + w);
1066 real salp1 = 0, calp1 = 0, salp2 = 0, calp2 = 0, dnm = 0;
1074 sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
1075 cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
1076 #if defined(__GNUC__) && __GNUC__ == 4 && \
1077 (__GNUC_MINOR__ < 6 || defined(__MINGW32__))
1086 volatile real xx1 = sbet2 * cbet1;
1087 volatile real xx2 = cbet2 * sbet1;
1088 sbet12a = xx1 + xx2;
1091 real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
1093 boolx shortline = cbet12 >= 0 && sbet12 < (
real)(0.5) &&
1094 cbet2 * lam12 < (
real)(0.5);
1095 real omg12 = lam12, somg12, comg12, ssig12, csig12;
1097 real sbetm2 = sq(sbet1 + sbet2);
1100 sbetm2 /= sbetm2 + sq(cbet1 + cbet2);
1101 dnm = sqrt(1 + g->ep2 * sbetm2);
1102 omg12 /= g->f1 * dnm;
1104 somg12 = sin(omg12); comg12 = cos(omg12);
1106 salp1 = cbet2 * somg12;
1107 calp1 = comg12 >= 0 ?
1108 sbet12 + cbet2 * sbet1 * sq(somg12) / (1 + comg12) :
1109 sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12);
1111 ssig12 = hypotx(salp1, calp1);
1112 csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
1114 if (shortline && ssig12 < g->etol2) {
1116 salp2 = cbet1 * somg12;
1117 calp2 = sbet12 - cbet1 * sbet2 *
1118 (comg12 >= 0 ? sq(somg12) / (1 + comg12) : 1 - comg12);
1119 SinCosNorm(&salp2, &calp2);
1121 sig12 = atan2(ssig12, csig12);
1122 }
else if (fabs(g->n) > (
real)(0.1) ||
1124 ssig12 >= 6 * fabs(g->n) * pi * sq(cbet1)) {
1129 real y, lamscale, betscale;
1138 k2 = sq(sbet1) * g->ep2,
1139 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
1140 lamscale = g->
f * cbet1 * A3f(g, eps) * pi;
1142 betscale = lamscale * cbet1;
1144 x = (lam12 - pi) / lamscale;
1145 y = sbet12a / betscale;
1149 cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
1150 bet12a = atan2(sbet12a, cbet12a);
1151 real m12b, m0, dummy;
1154 Lengths(g, g->n, pi + bet12a,
1155 sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
1156 cbet1, cbet2, &dummy, &m12b, &m0, FALSE,
1157 &dummy, &dummy, C1a, C2a);
1158 x = -1 + m12b / (cbet1 * cbet2 * m0 * pi);
1159 betscale = x < -(
real)(0.01) ? sbet12a / x :
1160 -g->
f * sq(cbet1) * pi;
1161 lamscale = betscale / cbet1;
1162 y = (lam12 - pi) / lamscale;
1165 if (y > -tol1 && x > -1 - xthresh) {
1168 salp1 = minx((
real)(1), -(
real)(x)); calp1 = - sqrt(1 - sq(salp1));
1170 calp1 = maxx((
real)(x > -tol1 ? 0 : -1), (
real)(x));
1171 salp1 = sqrt(1 - sq(calp1));
1208 real k = Astroid(x, y);
1210 omg12a = lamscale * ( g->
f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
1211 somg12 = sin(omg12a); comg12 = -cos(omg12a);
1213 salp1 = cbet2 * somg12;
1214 calp1 = sbet12a - cbet2 * sbet1 * sq(somg12) / (1 - comg12);
1218 SinCosNorm(&salp1, &calp1);
1220 salp1 = 1; calp1 = 0;
1243 boolx diffp,
real* pdlam12,
1246 real salp2 = 0, calp2 = 0, sig12 = 0,
1247 ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, domg12 = 0, dlam12 = 0;
1249 real somg1, comg1, somg2, comg2, omg12, lam12;
1252 if (sbet1 == 0 && calp1 == 0)
1258 salp0 = salp1 * cbet1;
1259 calp0 = hypotx(calp1, salp1 * sbet1);
1263 ssig1 = sbet1; somg1 = salp0 * sbet1;
1264 csig1 = comg1 = calp1 * cbet1;
1265 SinCosNorm(&ssig1, &csig1);
1272 salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
1277 calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ?
1278 sqrt(sq(calp1 * cbet1) +
1280 (cbet2 - cbet1) * (cbet1 + cbet2) :
1281 (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
1285 ssig2 = sbet2; somg2 = salp0 * sbet2;
1286 csig2 = comg2 = calp2 * cbet2;
1287 SinCosNorm(&ssig2, &csig2);
1291 sig12 = atan2(maxx(csig1 * ssig2 - ssig1 * csig2, (
real)(0)),
1292 csig1 * csig2 + ssig1 * ssig2);
1295 omg12 = atan2(maxx(comg1 * somg2 - somg1 * comg2, (
real)(0)),
1296 comg1 * comg2 + somg1 * somg2);
1297 k2 = sq(calp0) * g->ep2;
1298 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
1300 B312 = (SinCosSeries(TRUE, ssig2, csig2, C3a, nC3-1) -
1301 SinCosSeries(TRUE, ssig1, csig1, C3a, nC3-1));
1302 h0 = -g->
f * A3f(g, eps);
1303 domg12 = salp0 * h0 * (sig12 + B312);
1304 lam12 = omg12 + domg12;
1308 dlam12 = - 2 * g->f1 * dn1 / sbet1;
1311 Lengths(g, eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
1312 cbet1, cbet2, &dummy, &dlam12, &dummy,
1313 FALSE, &dummy, &dummy, C1a, C2a);
1314 dlam12 *= g->f1 / (calp2 * cbet2);
1338 v = eps * v + g->A3x[--i];
1347 for (j = nC3x, k = nC3 - 1; k; ) {
1349 for (i = nC3 - k; i; --i)
1350 t = eps * t + g->C3x[--j];
1354 for (k = 1; k <
nC3; ) {
1365 for (j = nC4x, k = nC4; k; ) {
1367 for (i = nC4 - k + 1; i; --i)
1368 t = eps * t + g->C4x[--j];
1372 for (k = 1; k <
nC4; ) {
1384 t = eps2*(eps2*(eps2+4)+64)/256;
1385 return (t + eps) / (1 - eps);
1393 c[1] = d*((6-eps2)*eps2-16)/32;
1395 c[2] = d*((64-9*eps2)*eps2-128)/2048;
1397 c[3] = d*(9*eps2-16)/768;
1399 c[4] = d*(3*eps2-5)/512;
1411 c[1] = d*(eps2*(205*eps2-432)+768)/1536;
1413 c[2] = d*(eps2*(4005*eps2-4736)+3840)/12288;
1415 c[3] = d*(116-225*eps2)/384;
1417 c[4] = d*(2695-7173*eps2)/7680;
1421 c[6] = 38081*d/61440;
1428 t = eps2*(eps2*(25*eps2+36)+64)/256;
1429 return t * (1 - eps) - eps;
1437 c[1] = d*(eps2*(eps2+2)+16)/32;
1439 c[2] = d*(eps2*(35*eps2+64)+384)/2048;
1441 c[3] = d*(15*eps2+80)/768;
1443 c[4] = d*(7*eps2+35)/512;
1453 g->A3x[1] = (g->n-1)/2;
1454 g->A3x[2] = (g->n*(3*g->n-1)-2)/8;
1455 g->A3x[3] = ((-g->n-3)*g->n-1)/16;
1456 g->A3x[4] = (-2*g->n-3)/64;
1457 g->A3x[5] = -3/(
real)(128);
1462 g->C3x[0] = (1-g->n)/4;
1463 g->C3x[1] = (1-g->n*g->n)/8;
1464 g->C3x[2] = ((3-g->n)*g->n+3)/64;
1465 g->C3x[3] = (2*g->n+5)/128;
1466 g->C3x[4] = 3/(
real)(128);
1467 g->C3x[5] = ((g->n-3)*g->n+2)/32;
1468 g->C3x[6] = ((-3*g->n-2)*g->n+3)/64;
1469 g->C3x[7] = (g->n+3)/128;
1470 g->C3x[8] = 5/(
real)(256);
1471 g->C3x[9] = (g->n*(5*g->n-9)+5)/192;
1472 g->C3x[10] = (9-10*g->n)/384;
1473 g->C3x[11] = 7/(
real)(512);
1474 g->C3x[12] = (7-14*g->n)/512;
1475 g->C3x[13] = 7/(
real)(512);
1476 g->C3x[14] = 21/(
real)(2560);
1483 g->C4x[0] = (g->n*(g->n*(g->n*(g->n*(100*g->n+208)+572)+3432)-12012)+30030)/
1485 g->C4x[1] = (g->n*(g->n*(g->n*(64*g->n+624)-4576)+6864)-3003)/15015;
1486 g->C4x[2] = (g->n*((14144-10656*g->n)*g->n-4576)-858)/45045;
1487 g->C4x[3] = ((-224*g->n-4784)*g->n+1573)/45045;
1488 g->C4x[4] = (1088*g->n+156)/45045;
1489 g->C4x[5] = 97/(
real)(15015);
1490 g->C4x[6] = (g->n*(g->n*((-64*g->n-624)*g->n+4576)-6864)+3003)/135135;
1491 g->C4x[7] = (g->n*(g->n*(5952*g->n-11648)+9152)-2574)/135135;
1492 g->C4x[8] = (g->n*(5792*g->n+1040)-1287)/135135;
1493 g->C4x[9] = (468-2944*g->n)/135135;
1494 g->C4x[10] = 1/(
real)(9009);
1495 g->C4x[11] = (g->n*((4160-1440*g->n)*g->n-4576)+1716)/225225;
1496 g->C4x[12] = ((4992-8448*g->n)*g->n-1144)/225225;
1497 g->C4x[13] = (1856*g->n-936)/225225;
1498 g->C4x[14] = 8/(
real)(10725);
1499 g->C4x[15] = (g->n*(3584*g->n-3328)+1144)/315315;
1500 g->C4x[16] = (1024*g->n-208)/105105;
1501 g->C4x[17] = -136/(
real)(63063);
1502 g->C4x[18] = (832-2560*g->n)/405405;
1503 g->C4x[19] = -128/(
real)(135135);
1504 g->C4x[20] = 128/(
real)(99099);
1507 int transit(
real lon1,
real lon2) {
1512 lon1 = AngNormalize(lon1);
1513 lon2 = AngNormalize(lon2);
1514 lon12 = AngDiff(lon1, lon2);
1515 return lon1 < 0 && lon2 >= 0 && lon12 > 0 ? 1 :
1516 (lon2 < 0 && lon1 >= 0 && lon12 < 0 ? -1 : 0);
1519 void accini(
real s[]) {
1524 void acccopy(
const real s[],
real t[]) {
1526 t[0] = s[0]; t[1] = s[1];
1531 real u, z = sumx(y, s[1], &u);
1532 s[0] = sumx(z, s[0], &s[1]);
1547 void accneg(
real s[]) {
1549 s[0] = -s[0]; s[1] = -s[1];
1553 p->lat0 = p->lon0 = p->
lat = p->
lon = NaN;
1554 p->polyline = (polylinep != 0);
1557 p->
num = p->crossings = 0;
1563 lon = AngNormalize(lon);
1565 p->lat0 = p->
lat = lat;
1566 p->lon0 = p->
lon = lon;
1570 &s12, 0, 0, 0, 0, 0, p->polyline ? 0 : &S12);
1574 p->crossings += transit(p->
lon, lon);
1576 p->
lat = lat; p->
lon = lon;
1588 0, 0, 0, 0, p->polyline ? 0 : &S12);
1592 p->crossings += transit(p->
lon, lon);
1594 p->
lat = lat; p->
lon = lon;
1601 boolx reverse, boolx sign,
1603 real s12, S12, t[2], area0;
1607 if (!p->polyline && pA) *pA = 0;
1611 if (pP) *pP = p->P[0];
1615 &s12, 0, 0, 0, 0, 0, &S12);
1616 if (pP) *pP = accsum(p->P, s12);
1619 crossings = p->crossings + transit(p->
lon, p->lon0);
1620 area0 = 4 * pi * g->c2;
1622 accadd(t, (t[0] < 0 ? 1 : -1) * area0/2);
1631 else if (t[0] <= -area0/2)
1639 if (pA) *pA = 0 + t[0];
1646 boolx reverse, boolx sign,
1648 real perimeter, tempsum, area0;
1650 unsigned num = p->
num + 1;
1653 if (!p->polyline && pA) *pA = 0;
1656 perimeter = p->P[0];
1657 tempsum = p->polyline ? 0 : p->A[0];
1658 crossings = p->crossings;
1659 for (i = 0; i < (p->polyline ? 1 : 2); ++i) {
1662 i == 0 ? p->
lat : lat, i == 0 ? p->
lon : lon,
1663 i != 0 ? p->lat0 : lat, i != 0 ? p->lon0 : lon,
1664 &s12, 0, 0, 0, 0, 0, p->polyline ? 0 : &S12);
1668 crossings += transit(i == 0 ? p->
lon : lon,
1669 i != 0 ? p->lon0 : lon);
1673 if (pP) *pP = perimeter;
1677 area0 = 4 * pi * g->c2;
1679 tempsum += (tempsum < 0 ? 1 : -1) * area0/2;
1686 if (tempsum > area0/2)
1688 else if (tempsum <= -area0/2)
1691 if (tempsum >= area0)
1693 else if (tempsum < 0)
1696 if (pA) *pA = 0 + tempsum;
1703 boolx reverse, boolx sign,
1705 real perimeter, tempsum, area0;
1707 unsigned num = p->
num + 1;
1710 if (!p->polyline && pA) *pA = NaN;
1713 perimeter = p->P[0] + s;
1715 if (pP) *pP = perimeter;
1720 crossings = p->crossings;
1722 real lat, lon, s12, S12;
1727 crossings += transit(p->
lon, lon);
1729 &s12, 0, 0, 0, 0, 0, &S12);
1732 crossings += transit(lon, p->lon0);
1735 area0 = 4 * pi * g->c2;
1737 tempsum += (tempsum < 0 ? 1 : -1) * area0/2;
1744 if (tempsum > area0/2)
1746 else if (tempsum <= -area0/2)
1749 if (tempsum >= area0)
1751 else if (tempsum < 0)
1754 if (pP) *pP = perimeter;
1755 if (pA) *pA = 0 + tempsum;
1765 for (i = 0; i < n; ++i)