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 ?
Math::asinh(sqrt(_ep2)) : atan(sqrt(-_e2))) /
83 , _etol2(0.1 * tol2_ /
84 sqrt( max(real(0.001), abs(_f)) * min(real(1), 1 - _f/2) / 2 ))
99 Math::real GeodesicExact::CosSeries(real sinx, real cosx,
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 {
165 Math::real GeodesicExact::GenInverse(real lat1, real lon1,
166 real lat2, real lon2,
167 unsigned outmask, real& s12,
168 real& salp1, real& calp1,
169 real& salp2, real& calp2,
170 real& m12, real& M12, real& M21,
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;
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_)
288 sig12 = m12x = s12x = 0;
298 real omg12 = 0, somg12 = 2, comg12 = 0;
301 (_f <= 0 || lon12s >= _f * 180)) {
304 calp1 = calp2 = 0; salp1 = salp2 = 1;
306 sig12 = omg12 = lam12 / _f1;
307 m12x = _b * sin(sig12);
309 M12 = M21 = cos(sig12);
312 }
else if (!meridian) {
319 sig12 = InverseStart(E, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
320 lam12, slam12, clam12,
321 salp1, calp1, salp2, calp2, dnm);
325 s12x = sig12 * _b * dnm;
326 m12x =
Math::sq(dnm) * _b * sin(sig12 / dnm);
328 M12 = M21 = cos(sig12 / dnm);
330 omg12 = lam12 / (_f1 * dnm);
346 real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0;
349 real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
350 for (
bool tripn =
false, tripb =
false;
375 real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
377 salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
378 E, somg12, comg12, numit < maxit1_, dv);
380 if (tripb || !(abs(v) >= (tripn ? 8 : 1) * tol0_))
break;
382 if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
383 { salp1b = salp1; calp1b = calp1; }
384 else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
385 { salp1a = salp1; calp1a = calp1; }
386 if (numit < maxit1_ && dv > 0) {
390 sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
391 nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
392 if (nsalp1 > 0 && abs(dalp1) <
Math::pi()) {
393 calp1 = calp1 * cdalp1 - salp1 * sdalp1;
399 tripn = abs(v) <= 16 * tol0_;
411 salp1 = (salp1a + salp1b)/2;
412 calp1 = (calp1a + calp1b)/2;
415 tripb = (abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
416 abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
420 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
421 cbet1, cbet2, outmask, s12x, m12x, dummy, M12, M21);
435 if (outmask &
AREA) {
438 salp0 = salp1 * cbet1,
441 if (calp0 != 0 && salp0 != 0) {
444 ssig1 = sbet1, csig1 = calp1 * cbet1,
445 ssig2 = sbet2, csig2 = calp2 * cbet2,
447 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
449 A4 =
Math::sq(_a) * calp0 * salp0 * _e2;
455 B41 = CosSeries(ssig1, csig1, C4a, nC4_),
456 B42 = CosSeries(ssig2, csig2, C4a, nC4_);
457 S12 = A4 * (B42 - B41);
464 somg12 = sin(omg12); comg12 = cos(omg12);
471 comg12 > -real(0.7071) &&
472 sbet2 - sbet1 < real(1.75)) {
476 real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
477 alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
478 domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
482 salp12 = salp2 * calp1 - calp2 * salp1,
483 calp12 = calp2 * calp1 + salp2 * salp1;
488 if (salp12 == 0 && calp12 < 0) {
489 salp12 = tiny_ * calp1;
492 alp12 = atan2(salp12, calp12);
495 S12 *= swapp * lonsign * latsign;
508 salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
509 salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
516 real lat2, real lon2,
518 real& s12, real& azi1, real& azi2,
519 real& m12, real& M12, real& M21,
523 real salp1, calp1, salp2, calp2,
524 a12 = GenInverse(lat1, lon1, lat2, lon2,
525 outmask, s12, salp1, calp1, salp2, calp2,
535 real lat2, real lon2,
536 unsigned caps)
const {
537 real t, salp1, calp1, salp2, calp2,
538 a12 = GenInverse(lat1, lon1, lat2, lon2,
540 0u, t, salp1, calp1, salp2, calp2,
551 real ssig1, real csig1, real dn1,
552 real ssig2, real csig2, real dn2,
553 real cbet1, real cbet2,
unsigned outmask,
554 real& s12b, real& m12b, real& m0,
555 real& M12, real& M21)
const {
570 (sig12 + (E.
deltaE(ssig2, csig2, dn2) - E.
deltaE(ssig1, csig1, dn1)));
575 (sig12 + (E.
deltaD(ssig2, csig2, dn2) - E.
deltaD(ssig1, csig1, dn1)));
581 m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
585 real csig12 = csig1 * csig2 + ssig1 * ssig2;
586 real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
587 M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
588 M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
593 Math::real GeodesicExact::Astroid(real x, real y) {
601 if ( !(q == 0 && r <= 0) ) {
610 disc = S * (S + 2 * r3);
617 T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc);
621 u += T + (T ? r2 / T : 0);
624 real ang = atan2(sqrt(-disc), -(S + r3));
627 u += 2 * r * cos(ang / 3);
632 uv = u < 0 ? q / (v - u) : u + v,
633 w = (uv - q) / (2 * v);
636 k = uv / (sqrt(uv +
Math::sq(w)) + w);
646 real sbet1, real cbet1, real dn1,
647 real sbet2, real cbet2, real dn2,
648 real lam12, real slam12, real clam12,
649 real& salp1, real& calp1,
651 real& salp2, real& calp2,
661 sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
662 cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
663 #if defined(__GNUC__) && __GNUC__ == 4 && \ 664 (__GNUC_MINOR__ < 6 || defined(__MINGW32__)) 678 real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
680 bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
681 cbet2 * lam12 < real(0.5);
684 real sbetm2 =
Math::sq(sbet1 + sbet2);
687 sbetm2 /= sbetm2 +
Math::sq(cbet1 + cbet2);
688 dnm = sqrt(1 + _ep2 * sbetm2);
689 real omg12 = lam12 / (_f1 * dnm);
690 somg12 = sin(omg12); comg12 = cos(omg12);
692 somg12 = slam12; comg12 = clam12;
695 salp1 = cbet2 * somg12;
696 calp1 = comg12 >= 0 ?
697 sbet12 + cbet2 * sbet1 *
Math::sq(somg12) / (1 + comg12) :
698 sbet12a - cbet2 * sbet1 *
Math::sq(somg12) / (1 - comg12);
702 csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
704 if (shortline && ssig12 < _etol2) {
706 salp2 = cbet1 * somg12;
707 calp2 = sbet12 - cbet1 * sbet2 *
708 (comg12 >= 0 ?
Math::sq(somg12) / (1 + comg12) : 1 - comg12);
711 sig12 = atan2(ssig12, csig12);
712 }
else if (abs(_n) > real(0.1) ||
719 real y, lamscale, betscale;
724 real lam12x = atan2(-slam12, -clam12);
729 E.
Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
730 lamscale = _e2/_f1 * cbet1 * 2 * E.
H();
732 betscale = lamscale * cbet1;
734 x = lam12x / lamscale;
735 y = sbet12a / betscale;
739 cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
740 bet12a = atan2(sbet12a, cbet12a);
741 real m12b, m0, dummy;
745 sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
747 x = -1 + m12b / (cbet1 * cbet2 * m0 *
Math::pi());
748 betscale = x < -real(0.01) ? sbet12a / x :
750 lamscale = betscale / cbet1;
751 y = lam12x / lamscale;
754 if (y > -tol1_ && x > -1 - xthresh_) {
758 salp1 = min(real(1), -real(x)); calp1 = - sqrt(1 -
Math::sq(salp1));
760 calp1 = max(real(x > -tol1_ ? 0 : -1), real(x));
798 real k = Astroid(x, y);
800 omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
801 somg12 = sin(omg12a); comg12 = -cos(omg12a);
803 salp1 = cbet2 * somg12;
804 calp1 = sbet12a - cbet2 * sbet1 *
Math::sq(somg12) / (1 - comg12);
811 salp1 = 1; calp1 = 0;
816 Math::real GeodesicExact::Lambda12(real sbet1, real cbet1, real dn1,
817 real sbet2, real cbet2, real dn2,
818 real salp1, real calp1,
819 real slam120, real clam120,
820 real& salp2, real& calp2,
822 real& ssig1, real& csig1,
823 real& ssig2, real& csig2,
825 real& somg12, real& comg12,
826 bool diffp, real& dlam12)
const 829 if (sbet1 == 0 && calp1 == 0)
836 salp0 = salp1 * cbet1,
839 real somg1, comg1, somg2, comg2, cchi1, cchi2, lam12;
842 ssig1 = sbet1; somg1 = salp0 * sbet1;
843 csig1 = comg1 = calp1 * cbet1;
845 cchi1 = _f1 * dn1 * comg1;
854 salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
859 calp2 = cbet2 != cbet1 || abs(sbet2) != -sbet1 ?
862 (cbet2 - cbet1) * (cbet1 + cbet2) :
863 (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
867 ssig2 = sbet2; somg2 = salp0 * sbet2;
868 csig2 = comg2 = calp2 * cbet2;
870 cchi2 = _f1 * dn2 * comg2;
876 sig12 = atan2(max(real(0), csig1 * ssig2 - ssig1 * csig2),
877 csig1 * csig2 + ssig1 * ssig2);
880 somg12 = max(real(0), comg1 * somg2 - somg1 * comg2);
881 comg12 = comg1 * comg2 + somg1 * somg2;
883 E.
Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
886 schi12 = max(real(0), cchi1 * somg2 - somg1 * cchi2),
887 cchi12 = cchi1 * cchi2 + somg1 * somg2;
889 real eta = atan2(schi12 * clam120 - cchi12 * slam120,
890 cchi12 * clam120 + schi12 * slam120);
893 _e2/_f1 * salp0 * E.
H() / (
Math::pi() / 2) *
894 (sig12 + (E.
deltaH(ssig2, csig2, dn2) - E.
deltaH(ssig1, csig1, dn1)));
898 dlam12 = - 2 * _f1 * dn1 / sbet1;
901 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
903 dummy, dlam12, dummy, dummy, dummy);
904 dlam12 *= _f1 / (calp2 * cbet2);
911 void GeodesicExact::C4f(real eps, real c[])
const {
916 for (
int l = 0; l < nC4_; ++l) {
917 int m = nC4_ - l - 1;
static T AngNormalize(T x)
GeodesicLineExact InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const
void Reset(real k2=0, real alpha2=0)
static bool isfinite(T x)
GeodesicLineExact ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const
Mathematical functions needed by GeographicLib.
static void sincosd(T x, T &sinx, T &cosx)
static T AngDiff(T x, T y, T &e)
Elliptic integrals and functions.
static void norm(T &x, T &y)
#define GEOGRAPHICLIB_VOLATILE
GeodesicExact(real a, real f)
GeodesicLineExact Line(real lat1, real lon1, real azi1, unsigned caps=ALL) const
Header for GeographicLib::GeodesicLineExact class.
static T atan2d(T y, T x)
static T polyval(int N, const T p[], T x)
Namespace for GeographicLib.
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
Exact geodesic calculations.
Math::real deltaE(real sn, real cn, real dn) const
Math::real deltaH(real sn, real cn, real dn) const
Header for GeographicLib::GeodesicExact class.
GeodesicLineExact GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const
Exception handling for GeographicLib.
friend class GeodesicLineExact
Math::real deltaD(real sn, real cn, real dn) const
#define GEOGRAPHICLIB_PANIC
static const GeodesicExact & WGS84()