271 lines
10 KiB
C++
271 lines
10 KiB
C++
/**
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* \file GeodesicLine30.cpp
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* \brief Implementation for GeographicLib::GeodesicLine30 class
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*
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* Copyright (c) Charles Karney (2009-2022) <charles@karney.com> and licensed
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* under the MIT/X11 License. For more information, see
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* https://geographiclib.sourceforge.io/
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*
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* This is a reformulation of the geodesic problem. The notation is as
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* follows:
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* - at a general point (no suffix or 1 or 2 as suffix)
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* - phi = latitude
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* - beta = latitude on auxiliary sphere
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* - omega = longitude on auxiliary sphere
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* - lambda = longitude
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* - alpha = azimuth of great circle
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* - sigma = arc length along great circle
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* - s = distance
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* - tau = scaled distance (= sigma at multiples of pi/2)
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* - at northwards equator crossing
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* - beta = phi = 0
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* - omega = lambda = 0
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* - alpha = alpha0
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* - sigma = s = 0
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* - a 12 suffix means a difference, e.g., s12 = s2 - s1.
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* - s and c prefixes mean sin and cos
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**********************************************************************/
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#include "GeodesicLine30.hpp"
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namespace GeographicLib {
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using namespace std;
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template<typename real>
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GeodesicLine30<real>::GeodesicLine30(const Geodesic30<real>& g,
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real lat1, real lon1, real azi1,
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unsigned caps)
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: _a(g._a)
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, _f(g._f)
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, _b(g._b)
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, _c2(g._c2)
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, _f1(g._f1)
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// Always allow latitude and azimuth
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, _caps(caps | LATITUDE | AZIMUTH)
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{
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azi1 = Math::AngNormalize(azi1);
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// Guard against underflow in salp0
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azi1 = Geodesic30<real>::AngRound(azi1);
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lon1 = Math::AngNormalize(lon1);
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_lat1 = lat1;
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_lon1 = lon1;
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_azi1 = azi1;
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// alp1 is in [0, pi]
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real alp1 = azi1 * Math::degree<real>();
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// Enforce sin(pi) == 0 and cos(pi/2) == 0. Better to face the ensuing
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// problems directly than to skirt them.
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_salp1 = azi1 == -180 ? 0 : sin(alp1);
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_calp1 = abs(azi1) == 90 ? 0 : cos(alp1);
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real cbet1, sbet1, phi;
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phi = lat1 * Math::degree<real>();
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// Ensure cbet1 = +epsilon at poles
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sbet1 = _f1 * sin(phi);
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cbet1 = abs(lat1) == 90 ? Geodesic30<real>::tiny_ : cos(phi);
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Geodesic30<real>::SinCosNorm(sbet1, cbet1);
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// Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
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_salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|]
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// Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
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// is slightly better (consider the case salp1 = 0).
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_calp0 = hypot(_calp1, _salp1 * sbet1);
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// Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
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// sig = 0 is nearest northward crossing of equator.
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// With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
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// With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
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// With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
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// Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
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// With alp0 in (0, pi/2], quadrants for sig and omg coincide.
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// No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
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// With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
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_ssig1 = sbet1; _somg1 = _salp0 * sbet1;
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_csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1;
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Geodesic30<real>::SinCosNorm(_ssig1, _csig1); // sig1 in (-pi, pi]
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Geodesic30<real>::SinCosNorm(_somg1, _comg1);
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_k2 = Math::sq(_calp0) * g._ep2;
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real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2);
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if (_caps & CAP_C1) {
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_A1m1 = Geodesic30<real>::A1m1f(eps);
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Geodesic30<real>::C1f(eps, _C1a);
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_B11 = Geodesic30<real>::SinCosSeries(true, _ssig1, _csig1, _C1a, nC1_);
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real s = sin(_B11), c = cos(_B11);
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// tau1 = sig1 + B11
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_stau1 = _ssig1 * c + _csig1 * s;
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_ctau1 = _csig1 * c - _ssig1 * s;
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// Not necessary because C1pa reverts C1a
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// _B11 = -SinCosSeries(true, _stau1, _ctau1, _C1pa, nC1p_);
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}
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if (_caps & CAP_C1p)
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Geodesic30<real>::C1pf(eps, _C1pa);
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if (_caps & CAP_C2) {
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_A2m1 = Geodesic30<real>::A2m1f(eps);
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Geodesic30<real>::C2f(eps, _C2a);
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_B21 = Geodesic30<real>::SinCosSeries(true, _ssig1, _csig1, _C2a, nC2_);
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}
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if (_caps & CAP_C3) {
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g.C3f(eps, _C3a);
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_A3c = -_f * _salp0 * g.A3f(eps);
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_B31 = Geodesic30<real>::SinCosSeries(true, _ssig1, _csig1,
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_C3a, nC3_-1);
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}
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if (_caps & CAP_C4) {
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g.C4f(_k2, _C4a);
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// Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
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_A4 = Math::sq(_a) * _calp0 * _salp0 * g._e2;
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_B41 = Geodesic30<real>::SinCosSeries(false, _ssig1, _csig1, _C4a, nC4_);
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}
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}
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template<typename real>
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real GeodesicLine30<real>::GenPosition(bool arcmode, real s12_a12,
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unsigned outmask,
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real& lat2, real& lon2, real& azi2,
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real& s12, real& m12,
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real& M12, real& M21,
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real& S12)
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const {
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outmask &= _caps & OUT_ALL;
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if (!( Init() && (arcmode || (_caps & DISTANCE_IN & OUT_ALL)) ))
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// Uninitialized or impossible distance calculation requested
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return Math::NaN<real>();
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// Avoid warning about uninitialized B12.
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real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
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if (arcmode) {
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// Interpret s12_a12 as spherical arc length
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sig12 = s12_a12 * Math::degree<real>();
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real s12a = abs(s12_a12);
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s12a -= 180 * floor(s12a / 180);
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ssig12 = s12a == 0 ? 0 : sin(sig12);
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csig12 = s12a == 90 ? 0 : cos(sig12);
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} else {
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// Interpret s12_a12 as distance
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real
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tau12 = s12_a12 / (_b * (1 + _A1m1)),
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s = sin(tau12),
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c = cos(tau12);
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// tau2 = tau1 + tau12
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B12 = - Geodesic30<real>::SinCosSeries(true, _stau1 * c + _ctau1 * s,
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_ctau1 * c - _stau1 * s,
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_C1pa, nC1p_);
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sig12 = tau12 - (B12 - _B11);
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ssig12 = sin(sig12);
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csig12 = cos(sig12);
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}
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real omg12, lam12, lon12;
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real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2;
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// sig2 = sig1 + sig12
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ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
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csig2 = _csig1 * csig12 - _ssig1 * ssig12;
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if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
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if (arcmode)
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B12 = Geodesic30<real>::SinCosSeries(true, ssig2, csig2, _C1a, nC1_);
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AB1 = (1 + _A1m1) * (B12 - _B11);
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}
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// sin(bet2) = cos(alp0) * sin(sig2)
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sbet2 = _calp0 * ssig2;
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// Alt: cbet2 = hypot(csig2, salp0 * ssig2);
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cbet2 = hypot(_salp0, _calp0 * csig2);
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if (cbet2 == 0)
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// I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
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cbet2 = csig2 = Geodesic30<real>::tiny_;
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// tan(omg2) = sin(alp0) * tan(sig2)
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somg2 = _salp0 * ssig2; comg2 = csig2; // No need to normalize
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// tan(alp0) = cos(sig2)*tan(alp2)
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salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
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// omg12 = omg2 - omg1
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omg12 = atan2(somg2 * _comg1 - comg2 * _somg1,
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comg2 * _comg1 + somg2 * _somg1);
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if (outmask & DISTANCE)
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s12 = arcmode ? _b * ((1 + _A1m1) * sig12 + AB1) : s12_a12;
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if (outmask & LONGITUDE) {
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lam12 = omg12 + _A3c *
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( sig12 +
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(Geodesic30<real>::SinCosSeries(true, ssig2, csig2, _C3a, nC3_-1)
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- _B31));
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lon12 = lam12 / Math::degree<real>();
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lon12 = Math::AngNormalize(lon12);
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lon2 = Math::AngNormalize(_lon1 + lon12);
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}
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if (outmask & LATITUDE)
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lat2 = atan2(sbet2, _f1 * cbet2) / Math::degree<real>();
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if (outmask & AZIMUTH)
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// minus signs give range [-180, 180). 0- converts -0 to +0.
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azi2 = 0 - atan2(-salp2, calp2) / Math::degree<real>();
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if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
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real
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ssig1sq = Math::sq(_ssig1),
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ssig2sq = Math::sq( ssig2),
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w1 = sqrt(1 + _k2 * ssig1sq),
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w2 = sqrt(1 + _k2 * ssig2sq),
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B22 = Geodesic30<real>::SinCosSeries(true, ssig2, csig2, _C2a, nC2_),
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AB2 = (1 + _A2m1) * (B22 - _B21),
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J12 = (_A1m1 - _A2m1) * sig12 + (AB1 - AB2);
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if (outmask & REDUCEDLENGTH)
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// Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
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// accurate cancellation in the case of coincident points.
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m12 = _b * ((w2 * (_csig1 * ssig2) - w1 * (_ssig1 * csig2))
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- _csig1 * csig2 * J12);
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if (outmask & GEODESICSCALE) {
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M12 = csig12 + (_k2 * (ssig2sq - ssig1sq) * ssig2 / (w1 + w2)
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- csig2 * J12) * _ssig1 / w1;
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M21 = csig12 - (_k2 * (ssig2sq - ssig1sq) * _ssig1 / (w1 + w2)
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- _csig1 * J12) * ssig2 / w2;
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}
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}
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if (outmask & AREA) {
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real
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B42 = Geodesic30<real>::SinCosSeries(false, ssig2, csig2, _C4a, nC4_);
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real salp12, calp12;
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if (_calp0 == 0 || _salp0 == 0) {
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// alp12 = alp2 - alp1, used in atan2 so no need to normalize
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salp12 = salp2 * _calp1 - calp2 * _salp1;
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calp12 = calp2 * _calp1 + salp2 * _salp1;
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// The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
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// salp12 = -0 and alp12 = -180. However this depends on the sign
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// being attached to 0 correctly. The following ensures the correct
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// behavior.
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if (salp12 == 0 && calp12 < 0) {
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salp12 = Geodesic30<real>::tiny_ * _calp1;
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calp12 = -1;
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}
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} else {
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// tan(alp) = tan(alp0) * sec(sig)
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// tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
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// = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
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// If csig12 > 0, write
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// csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
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// else
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// csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
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// No need to normalize
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salp12 = _calp0 * _salp0 *
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(csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
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ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
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calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2;
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}
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S12 = _c2 * atan2(salp12, calp12) + _A4 * (B42 - _B41);
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}
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return arcmode ? s12_a12 : sig12 / Math::degree<real>();
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}
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template class GeodesicLine30<double>;
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#if GEOGRAPHICLIB_HAVE_LONG_DOUBLE
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template class GeodesicLine30<long double>;
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#endif
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} // namespace GeographicLib
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