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())
55 , tolb_(tol0_ * tol2_)
56 , xthresh_(1000 * tol2_)
58 , _f(f <= 1 ? f : 1/f)
61 , _ep2(_e2 /
Math::sq(_f1))
72 (_f > 0 ?
Math::asinh(sqrt(_ep2)) : atan(sqrt(-_e2))) /
84 , _etol2(0.1 * tol2_ /
85 sqrt( max(real(0.001), abs(_f)) * min(real(1), 1 - _f/2) / 2 ))
100 Math::real GeodesicExact::CosSeries(real sinx, real cosx,
101 const real c[],
int n) {
108 ar = 2 * (cosx - sinx) * (cosx + sinx),
109 y0 = n & 1 ? *--c : 0, y1 = 0;
114 y1 = ar * y0 - y1 + *--c;
115 y0 = ar * y1 - y0 + *--c;
117 return cosx * (y0 - y1);
121 unsigned caps)
const {
126 bool arcmode, real s12_a12,
128 real& lat2, real& lon2, real& azi2,
129 real& s12, real& m12,
130 real& M12, real& M21,
136 GenPosition(arcmode, s12_a12, outmask,
137 lat2, lon2, azi2, s12, m12, M12, M21, S12);
141 real lat2, real lon2,
143 real& s12, real& azi1, real& azi2,
144 real& m12, real& M12, real& M21,
155 int lonsign = lon12 >= 0 ? 1 : -1;
161 int swapp = abs(lat1) >= abs(lat2) ? 1 : -1;
167 int latsign = lat1 < 0 ? 1 : -1;
182 real phi, sbet1, cbet1, sbet2, cbet2, s12x, m12x;
189 sbet1 = _f1 * sin(phi);
190 cbet1 = lat1 == -90 ? tiny_ : cos(phi);
195 sbet2 = _f1 * sin(phi);
196 cbet2 = abs(lat2) == 90 ? tiny_ : cos(phi);
207 if (cbet1 < -sbet1) {
209 sbet2 = sbet2 < 0 ? sbet1 : -sbet1;
211 if (abs(sbet2) == -sbet1)
216 dn1 = (_f >= 0 ? sqrt(1 + _ep2 *
Math::sq(sbet1)) :
217 sqrt(1 - _e2 *
Math::sq(cbet1)) / _f1),
218 dn2 = (_f >= 0 ? sqrt(1 + _ep2 *
Math::sq(sbet2)) :
219 sqrt(1 - _e2 *
Math::sq(cbet2)) / _f1);
223 slam12 = abs(lon12) == 180 ? 0 : sin(lam12),
227 real a12, sig12, calp1, salp1, calp2 = 0, salp2 = 0;
229 bool meridian = lat1 == -90 || slam12 == 0;
236 calp1 = clam12; salp1 = slam12;
237 calp2 = 1; salp2 = 0;
241 ssig1 = sbet1, csig1 = calp1 * cbet1,
242 ssig2 = sbet2, csig2 = calp2 * cbet2;
245 sig12 = atan2(max(csig1 * ssig2 - ssig1 * csig2, real(0)),
246 csig1 * csig2 + ssig1 * ssig2);
249 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
250 cbet1, cbet2, s12x, m12x, dummy,
260 if (sig12 < 1 || m12x >= 0) {
276 calp1 = calp2 = 0; salp1 = salp2 = 1;
278 sig12 = omg12 = lam12 / _f1;
279 m12x = _b * sin(sig12);
281 M12 = M21 = cos(sig12);
284 }
else if (!meridian) {
291 sig12 = InverseStart(E, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
293 salp1, calp1, salp2, calp2, dnm);
297 s12x = sig12 * _b * dnm;
298 m12x =
Math::sq(dnm) * _b * sin(sig12 / dnm);
300 M12 = M21 = cos(sig12 / dnm);
302 omg12 = lam12 / (_f1 * dnm);
318 real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0;
321 real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
322 for (
bool tripn =
false, tripb =
false;
347 real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
348 salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
349 E, omg12, numit < maxit1_, dv) - lam12;
352 if (tripb || !(abs(v) >= (tripn ? 8 : 2) * tol0_))
break;
354 if (v > 0 && (numit > maxit1_ || calp1/salp1 > calp1b/salp1b))
355 { salp1b = salp1; calp1b = calp1; }
356 else if (v < 0 && (numit > maxit1_ || calp1/salp1 < calp1a/salp1a))
357 { salp1a = salp1; calp1a = calp1; }
358 if (numit < maxit1_ && dv > 0) {
362 sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
363 nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
364 if (nsalp1 > 0 && abs(dalp1) <
Math::pi()) {
365 calp1 = calp1 * cdalp1 - salp1 * sdalp1;
371 tripn = abs(v) <= 16 * tol0_;
383 salp1 = (salp1a + salp1b)/2;
384 calp1 = (calp1a + calp1b)/2;
387 tripb = (abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
388 abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
392 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
393 cbet1, cbet2, s12x, m12x, dummy,
408 if (outmask &
AREA) {
411 salp0 = salp1 * cbet1,
414 if (calp0 != 0 && salp0 != 0) {
417 ssig1 = sbet1, csig1 = calp1 * cbet1,
418 ssig2 = sbet2, csig2 = calp2 * cbet2,
420 eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
422 A4 =
Math::sq(_a) * calp0 * salp0 * _e2;
428 B41 = CosSeries(ssig1, csig1, C4a, nC4_),
429 B42 = CosSeries(ssig2, csig2, C4a, nC4_);
430 S12 = A4 * (B42 - B41);
437 sbet2 - sbet1 < real(1.75)) {
442 somg12 = sin(omg12), domg12 = 1 + cos(omg12),
443 dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
444 alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
445 domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
449 salp12 = salp2 * calp1 - calp2 * salp1,
450 calp12 = calp2 * calp1 + salp2 * salp1;
455 if (salp12 == 0 && calp12 < 0) {
456 salp12 = tiny_ * calp1;
459 alp12 = atan2(salp12, calp12);
462 S12 *= swapp * lonsign * latsign;
475 salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
476 salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
489 real ssig1, real csig1, real dn1,
490 real ssig2, real csig2, real dn2,
491 real cbet1, real cbet2,
492 real& s12b, real& m12b, real& m0,
493 bool scalep, real& M12, real& M21)
const {
501 (sig12 + E.
deltaD(ssig2, csig2, dn2) - E.
deltaD(ssig1, csig1, dn1));
505 m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) - csig1 * csig2 * J12;
508 (sig12 + E.
deltaE(ssig2, csig2, dn2) - E.
deltaE(ssig1, csig1, dn1));
510 real csig12 = csig1 * csig2 + ssig1 * ssig2;
511 real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
512 M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
513 M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
517 Math::real GeodesicExact::Astroid(real x, real y) {
525 if ( !(q == 0 && r <= 0) ) {
534 disc = S * (S + 2 * r3);
541 T3 += T3 < 0 ? -sqrt(disc) : sqrt(disc);
545 u += T + (T ? r2 / T : 0);
548 real ang = atan2(sqrt(-disc), -(S + r3));
551 u += 2 * r * cos(ang / 3);
556 uv = u < 0 ? q / (v - u) : u + v,
557 w = (uv - q) / (2 * v);
560 k = uv / (sqrt(uv +
Math::sq(w)) + w);
570 real sbet1, real cbet1, real dn1,
571 real sbet2, real cbet2, real dn2,
573 real& salp1, real& calp1,
575 real& salp2, real& calp2,
585 sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
586 cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
587 #if defined(__GNUC__) && __GNUC__ == 4 && \ 588 (__GNUC_MINOR__ < 6 || defined(__MINGW32__)) 602 real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
604 bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
605 cbet2 * lam12 < real(0.5);
608 real sbetm2 =
Math::sq(sbet1 + sbet2);
611 sbetm2 /= sbetm2 +
Math::sq(cbet1 + cbet2);
612 dnm = sqrt(1 + _ep2 * sbetm2);
615 real somg12 = sin(omg12), comg12 = cos(omg12);
617 salp1 = cbet2 * somg12;
618 calp1 = comg12 >= 0 ?
619 sbet12 + cbet2 * sbet1 *
Math::sq(somg12) / (1 + comg12) :
620 sbet12a - cbet2 * sbet1 *
Math::sq(somg12) / (1 - comg12);
624 csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
626 if (shortline && ssig12 < _etol2) {
628 salp2 = cbet1 * somg12;
629 calp2 = sbet12 - cbet1 * sbet2 *
630 (comg12 >= 0 ?
Math::sq(somg12) / (1 + comg12) : 1 - comg12);
633 sig12 = atan2(ssig12, csig12);
634 }
else if (abs(_n) > real(0.1) ||
641 real y, lamscale, betscale;
650 E.
Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
651 lamscale = _e2/_f1 * cbet1 * 2 * E.
H();
653 betscale = lamscale * cbet1;
655 x = (lam12 -
Math::pi()) / lamscale;
656 y = sbet12a / betscale;
660 cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
661 bet12a = atan2(sbet12a, cbet12a);
662 real m12b, m0, dummy;
666 sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
667 cbet1, cbet2, dummy, m12b, m0,
false,
669 x = -1 + m12b / (cbet1 * cbet2 * m0 *
Math::pi());
670 betscale = x < -real(0.01) ? sbet12a / x :
672 lamscale = betscale / cbet1;
673 y = (lam12 -
Math::pi()) / lamscale;
676 if (y > -tol1_ && x > -1 - xthresh_) {
680 salp1 = min(real(1), -real(x)); calp1 = - sqrt(1 -
Math::sq(salp1));
682 calp1 = max(real(x > -tol1_ ? 0 : -1), real(x));
720 real k = Astroid(x, y);
722 omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
723 somg12 = sin(omg12a); comg12 = -cos(omg12a);
725 salp1 = cbet2 * somg12;
726 calp1 = sbet12a - cbet2 * sbet1 *
Math::sq(somg12) / (1 - comg12);
733 salp1 = 1; calp1 = 0;
738 Math::real GeodesicExact::Lambda12(real sbet1, real cbet1, real dn1,
739 real sbet2, real cbet2, real dn2,
740 real salp1, real calp1,
741 real& salp2, real& calp2,
743 real& ssig1, real& csig1,
744 real& ssig2, real& csig2,
747 bool diffp, real& dlam12)
const 750 if (sbet1 == 0 && calp1 == 0)
757 salp0 = salp1 * cbet1,
760 real somg1, comg1, somg2, comg2, cchi1, cchi2, lam12;
763 ssig1 = sbet1; somg1 = salp0 * sbet1;
764 csig1 = comg1 = calp1 * cbet1;
766 cchi1 = _f1 * dn1 * comg1;
775 salp2 = cbet2 != cbet1 ? salp0 / cbet2 : salp1;
780 calp2 = cbet2 != cbet1 || abs(sbet2) != -sbet1 ?
783 (cbet2 - cbet1) * (cbet1 + cbet2) :
784 (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
788 ssig2 = sbet2; somg2 = salp0 * sbet2;
789 csig2 = comg2 = calp2 * cbet2;
791 cchi2 = _f1 * dn2 * comg2;
797 sig12 = atan2(max(csig1 * ssig2 - ssig1 * csig2, real(0)),
798 csig1 * csig2 + ssig1 * ssig2);
801 omg12 = atan2(max(comg1 * somg2 - somg1 * comg2, real(0)),
802 comg1 * comg2 + somg1 * somg2);
804 E.
Reset(-k2, -_ep2, 1 + k2, 1 + _ep2);
805 real chi12 = atan2(max(cchi1 * somg2 - somg1 * cchi2, real(0)),
806 cchi1 * cchi2 + somg1 * somg2);
808 _e2/_f1 * salp0 * E.
H() / (
Math::pi() / 2) *
809 (sig12 + E.
deltaH(ssig2, csig2, dn2) - E.
deltaH(ssig1, csig1, dn1) );
813 dlam12 = - 2 * _f1 * dn1 / sbet1;
816 Lengths(E, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
817 cbet1, cbet2, dummy, dlam12, dummy,
818 false, dummy, dummy);
819 dlam12 *= _f1 / (calp2 * cbet2);
826 void GeodesicExact::C4f(real eps, real c[])
const {
831 for (
int l = 0; l < nC4_; ++l) {
832 int m = nC4_ - l - 1;
static T AngNormalize(T x)
void Reset(real k2=0, real alpha2=0)
static bool isfinite(T x)
Mathematical functions needed by GeographicLib.
Elliptic integrals and functions.
static void norm(T &x, T &y)
#define GEOGRAPHICLIB_VOLATILE
Math::real GenInverse(real lat1, real lon1, real lat2, real lon2, unsigned outmask, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21, real &S12) const
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.
static T AngDiff(T x, T y)
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.
Exception handling for GeographicLib.
friend class GeodesicLineExact
Math::real deltaD(real sn, real cn, real dn) const
#define GEOGRAPHICLIB_PANIC
static const GeodesicExact & WGS84()