35# pragma warning (disable: 4701 4127)
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( max(real(0.001), abs(_f)) * min(real(1), 1 - _f/2) / 2 ))
86 if (!(isfinite(_a) && _a > 0))
88 if (!(isfinite(_b) && _b > 0))
100 const real c[],
int n) {
107 ar = 2 * (cosx - sinx) * (cosx + sinx),
108 y0 = n & 1 ? *--c : 0, y1 = 0;
113 y1 = ar * y0 - y1 + *--c;
114 y0 = ar * y1 - y0 + *--c;
116 return cosx * (y0 - y1);
120 unsigned caps)
const {
125 bool arcmode, real s12_a12,
127 real& lat2, real& lon2, real& azi2,
128 real& s12, real& m12,
129 real& M12, real& M21,
135 GenPosition(arcmode, s12_a12, outmask,
136 lat2, lon2, azi2, s12, m12, M12, M21, S12);
141 bool arcmode, real s12_a12,
142 unsigned caps)
const {
150 caps, arcmode, s12_a12);
155 unsigned caps)
const {
161 unsigned caps)
const {
167 unsigned outmask,
real& s12,
177 int lonsign = lon12 >= 0 ? 1 : -1;
195 int swapp = abs(lat1) < abs(lat2) ? -1 : 1;
201 int latsign = lat1 < 0 ? 1 : -1;
216 real sbet1, cbet1, sbet2, cbet2, s12x, m12x;
219 EllipticFunction E(-_ep2);
224 Math::norm(sbet1, cbet1); cbet1 = max(tiny_, cbet1);
228 Math::norm(sbet2, cbet2); cbet2 = max(tiny_, cbet2);
238 if (cbet1 < -sbet1) {
240 sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
242 if (abs(sbet2) == -sbet1)
247 dn1 = (_f >= 0 ? sqrt(1 + _ep2 *
Math::sq(sbet1)) :
248 sqrt(1 - _e2 * Math::sq(cbet1)) / _f1),
249 dn2 = (_f >= 0 ? sqrt(1 + _ep2 * Math::sq(sbet2)) :
250 sqrt(1 - _e2 * Math::sq(cbet2)) / _f1);
254 bool meridian = lat1 == -90 || slam12 == 0;
261 calp1 = clam12; salp1 = slam12;
262 calp2 = 1; salp2 = 0;
266 ssig1 = sbet1, csig1 = calp1 * cbet1,
267 ssig2 = sbet2, csig2 = calp2 * cbet2;
270 sig12 = atan2(max(
real(0), csig1 * ssig2 - ssig1 * csig2),
271 csig1 * csig2 + ssig1 * ssig2);
274 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
276 s12x, m12x, dummy, M12, M21);
285 if (sig12 < 1 || m12x >= 0) {
287 if (sig12 < 3 * tiny_ ||
289 (sig12 < tol0_ && (s12x < 0 || m12x < 0)))
290 sig12 = m12x = s12x = 0;
300 real omg12 = 0, somg12 = 2, comg12 = 0;
303 (_f <= 0 || lon12s >= _f * 180)) {
306 calp1 = calp2 = 0; salp1 = salp2 = 1;
308 sig12 = omg12 = lam12 / _f1;
309 m12x = _b * sin(sig12);
311 M12 = M21 = cos(sig12);
314 }
else if (!meridian) {
321 sig12 = InverseStart(E, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
322 lam12, slam12, clam12,
323 salp1, calp1, salp2, calp2, dnm);
327 s12x = sig12 * _b * dnm;
328 m12x =
Math::sq(dnm) * _b * sin(sig12 / dnm);
330 M12 = M21 = cos(sig12 / dnm);
332 omg12 = lam12 / (_f1 * dnm);
348 real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, domg12 = 0;
351 real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
352 for (
bool tripn =
false, tripb =
false;
377 real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
379 salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
380 E, domg12, numit < maxit1_, dv);
382 if (tripb || !(abs(v) >= (tripn ? 8 : 1) * tol0_))
break;
384 if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
385 { salp1b = salp1; calp1b = calp1; }
386 else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
387 { salp1a = salp1; calp1a = calp1; }
388 if (numit < maxit1_ && dv > 0) {
392 sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
393 nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
394 if (nsalp1 > 0 && abs(dalp1) <
Math::pi()) {
395 calp1 = calp1 * cdalp1 - salp1 * sdalp1;
401 tripn = abs(v) <= 16 * tol0_;
413 salp1 = (salp1a + salp1b)/2;
414 calp1 = (calp1a + calp1b)/2;
417 tripb = (abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
418 abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
422 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
423 cbet1, cbet2, outmask, s12x, m12x, dummy, M12, M21);
428 if (outmask &
AREA) {
430 real sdomg12 = sin(domg12), cdomg12 = cos(domg12);
431 somg12 = slam12 * cdomg12 - clam12 * sdomg12;
432 comg12 = clam12 * cdomg12 + slam12 * sdomg12;
443 if (outmask &
AREA) {
446 salp0 = salp1 * cbet1,
447 calp0 = hypot(calp1, salp1 * sbet1);
449 if (calp0 != 0 && salp0 != 0) {
452 ssig1 = sbet1, csig1 = calp1 * cbet1,
453 ssig2 = sbet2, csig2 = calp2 * cbet2,
455 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
457 A4 =
Math::sq(_a) * calp0 * salp0 * _e2;
463 B41 = CosSeries(ssig1, csig1, C4a, nC4_),
464 B42 = CosSeries(ssig2, csig2, C4a, nC4_);
465 S12 = A4 * (B42 - B41);
472 somg12 = sin(omg12); comg12 = cos(omg12);
478 comg12 > -
real(0.7071) &&
479 sbet2 - sbet1 <
real(1.75)) {
483 real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
484 alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
485 domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
489 salp12 = salp2 * calp1 - calp2 * salp1,
490 calp12 = calp2 * calp1 + salp2 * salp1;
495 if (salp12 == 0 && calp12 < 0) {
496 salp12 = tiny_ * calp1;
499 alp12 = atan2(salp12, calp12);
502 S12 *= swapp * lonsign * latsign;
515 salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
516 salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
523 real lat2, real lon2,
525 real& s12, real& azi1, real& azi2,
526 real& m12, real& M12, real& M21,
529 real salp1, calp1, salp2, calp2,
530 a12 = GenInverse(lat1, lon1, lat2, lon2,
531 outmask, s12, salp1, calp1, salp2, calp2,
541 real lat2, real lon2,
542 unsigned caps)
const {
543 real t, salp1, calp1, salp2, calp2,
544 a12 = GenInverse(lat1, lon1, lat2, lon2,
546 0u, t, salp1, calp1, salp2, calp2,
559 real cbet1,
real cbet2,
unsigned outmask,
576 (sig12 + (E.
deltaE(ssig2, csig2, dn2) - E.
deltaE(ssig1, csig1, dn1)));
581 (sig12 + (E.
deltaD(ssig2, csig2, dn2) - E.
deltaD(ssig1, csig1, dn1)));
587 m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
591 real csig12 = csig1 * csig2 + ssig1 * ssig2;
592 real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
593 M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
594 M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
607 if ( !(q == 0 && r <= 0) ) {
616 disc = S * (S + 2 * r3);
623 T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc);
627 u += T + (T != 0 ? r2 / T : 0);
630 real ang = atan2(sqrt(-disc), -(S + r3));
633 u += 2 * r * cos(ang / 3);
638 uv = u < 0 ? q / (v - u) : u + v,
639 w = (uv - q) / (2 * v);
642 k = uv / (sqrt(uv +
Math::sq(w)) + w);
651 Math::real GeodesicExact::InverseStart(EllipticFunction& E,
666 sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
667 cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
668 real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
669 bool shortline = cbet12 >= 0 && sbet12 <
real(0.5) &&
670 cbet2 * lam12 <
real(0.5);
676 sbetm2 /= sbetm2 +
Math::sq(cbet1 + cbet2);
677 dnm = sqrt(1 + _ep2 * sbetm2);
678 real omg12 = lam12 / (_f1 * dnm);
679 somg12 = sin(omg12); comg12 = cos(omg12);
681 somg12 = slam12; comg12 = clam12;
684 salp1 = cbet2 * somg12;
685 calp1 = comg12 >= 0 ?
686 sbet12 + cbet2 * sbet1 *
Math::sq(somg12) / (1 + comg12) :
687 sbet12a - cbet2 * sbet1 *
Math::sq(somg12) / (1 - comg12);
690 ssig12 = hypot(salp1, calp1),
691 csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
693 if (shortline && ssig12 < _etol2) {
695 salp2 = cbet1 * somg12;
696 calp2 = sbet12 - cbet1 * sbet2 *
697 (comg12 >= 0 ?
Math::sq(somg12) / (1 + comg12) : 1 - comg12);
700 sig12 = atan2(ssig12, csig12);
701 }
else if (abs(_n) >
real(0.1) ||
708 real y, lamscale, betscale;
713 real lam12x = atan2(-slam12, -clam12);
718 E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
719 lamscale = _e2/_f1 * cbet1 * 2 * E.H();
721 betscale = lamscale * cbet1;
723 x = lam12x / lamscale;
724 y = sbet12a / betscale;
728 cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
729 bet12a = atan2(sbet12a, cbet12a);
730 real m12b, m0, dummy;
734 sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
736 x = -1 + m12b / (cbet1 * cbet2 * m0 *
Math::pi());
737 betscale = x < -
real(0.01) ? sbet12a / x :
739 lamscale = betscale / cbet1;
740 y = lam12x / lamscale;
743 if (y > -tol1_ && x > -1 - xthresh_) {
749 calp1 = max(
real(x > -tol1_ ? 0 : -1),
real(x));
787 real k = Astroid(x, y);
789 omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
790 somg12 = sin(omg12a); comg12 = -cos(omg12a);
792 salp1 = cbet2 * somg12;
793 calp1 = sbet12a - cbet2 * sbet1 *
Math::sq(somg12) / (1 - comg12);
800 salp1 = 1; calp1 = 0;
815 bool diffp,
real& dlam12)
const
818 if (sbet1 == 0 && calp1 == 0)
825 salp0 = salp1 * cbet1,
826 calp0 = hypot(calp1, salp1 * sbet1);
828 real somg1, comg1, somg2, comg2, somg12, comg12, cchi1, cchi2, lam12;
831 ssig1 = sbet1; somg1 = salp0 * sbet1;
832 csig1 = comg1 = calp1 * cbet1;
834 cchi1 = _f1 * dn1 * comg1;
843 salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
848 calp2 = cbet2 != cbet1 || abs(sbet2) != -sbet1 ?
851 (cbet2 - cbet1) * (cbet1 + cbet2) :
852 (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
856 ssig2 = sbet2; somg2 = salp0 * sbet2;
857 csig2 = comg2 = calp2 * cbet2;
859 cchi2 = _f1 * dn2 * comg2;
865 sig12 = atan2(max(
real(0), csig1 * ssig2 - ssig1 * csig2),
866 csig1 * csig2 + ssig1 * ssig2);
869 somg12 = max(
real(0), comg1 * somg2 - somg1 * comg2);
870 comg12 = comg1 * comg2 + somg1 * somg2;
872 E.Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
875 schi12 = max(
real(0), cchi1 * somg2 - somg1 * cchi2),
876 cchi12 = cchi1 * cchi2 + somg1 * somg2;
878 real eta = atan2(schi12 * clam120 - cchi12 * slam120,
879 cchi12 * clam120 + schi12 * slam120);
880 real deta12 = -_e2/_f1 * salp0 * E.H() / (
Math::pi() / 2) *
881 (sig12 + (E.deltaH(ssig2, csig2, dn2) - E.deltaH(ssig1, csig1, dn1)));
882 lam12 = eta + deta12;
884 domg12 = deta12 + atan2(schi12 * comg12 - cchi12 * somg12,
885 cchi12 * comg12 + schi12 * somg12);
888 dlam12 = - 2 * _f1 * dn1 / sbet1;
891 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
893 dummy, dlam12, dummy, dummy, dummy);
894 dlam12 *= _f1 / (calp2 * cbet2);
901 void GeodesicExact::C4f(
real eps,
real c[])
const {
906 for (
int l = 0; l < nC4_; ++l) {
907 int m = nC4_ - l - 1;
914 throw GeographicErr(
"C4 misalignment");
GeographicLib::Math::real real
Header for GeographicLib::GeodesicExact class.
Header for GeographicLib::GeodesicLineExact class.
#define GEOGRAPHICLIB_VOLATILE
#define GEOGRAPHICLIB_PANIC
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 T AngNormalize(T x)
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 polyval(int N, const T p[], T x)
static T AngDiff(T x, T y, T &e)
Namespace for GeographicLib.
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)