/** * \file GeodesicLine.cpp * \brief Implementation for GeographicLib::GeodesicLine class * * Copyright (c) Charles Karney (2009-2022) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ * * This is a reformulation of the geodesic problem. The notation is as * follows: * - at a general point (no suffix or 1 or 2 as suffix) * - phi = latitude * - beta = latitude on auxiliary sphere * - omega = longitude on auxiliary sphere * - lambda = longitude * - alpha = azimuth of great circle * - sigma = arc length along great circle * - s = distance * - tau = scaled distance (= sigma at multiples of pi/2) * - at northwards equator crossing * - beta = phi = 0 * - omega = lambda = 0 * - alpha = alpha0 * - sigma = s = 0 * - a 12 suffix means a difference, e.g., s12 = s2 - s1. * - s and c prefixes mean sin and cos **********************************************************************/ #include #if defined(_MSC_VER) // Squelch warnings about mixing enums # pragma warning (disable: 5054) #endif namespace GeographicLib { using namespace std; void GeodesicLine::LineInit(const Geodesic& g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps) { tiny_ = g.tiny_; _lat1 = Math::LatFix(lat1); _lon1 = lon1; _azi1 = azi1; _salp1 = salp1; _calp1 = calp1; _a = g._a; _f = g._f; _b = g._b; _c2 = g._c2; _f1 = g._f1; // Always allow latitude and azimuth and unrolling of longitude _caps = caps | LATITUDE | AZIMUTH | LONG_UNROLL; real cbet1, sbet1; Math::sincosd(Math::AngRound(_lat1), sbet1, cbet1); sbet1 *= _f1; // Ensure cbet1 = +epsilon at poles Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1); _dn1 = sqrt(1 + g._ep2 * Math::sq(sbet1)); // Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0), _salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|] // Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following // is slightly better (consider the case salp1 = 0). _calp0 = hypot(_calp1, _salp1 * sbet1); // Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1). // sig = 0 is nearest northward crossing of equator. // With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line). // With bet1 = pi/2, alp1 = -pi, sig1 = pi/2 // With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2 // Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1). // With alp0 in (0, pi/2], quadrants for sig and omg coincide. // No atan2(0,0) ambiguity at poles since cbet1 = +epsilon. // With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi. _ssig1 = sbet1; _somg1 = _salp0 * sbet1; _csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1; Math::norm(_ssig1, _csig1); // sig1 in (-pi, pi] // Math::norm(_somg1, _comg1); -- don't need to normalize! _k2 = Math::sq(_calp0) * g._ep2; real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2); if (_caps & CAP_C1) { _aA1m1 = Geodesic::A1m1f(eps); Geodesic::C1f(eps, _cC1a); _bB11 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC1a, nC1_); real s = sin(_bB11), c = cos(_bB11); // tau1 = sig1 + B11 _stau1 = _ssig1 * c + _csig1 * s; _ctau1 = _csig1 * c - _ssig1 * s; // Not necessary because C1pa reverts C1a // _bB11 = -SinCosSeries(true, _stau1, _ctau1, _cC1pa, nC1p_); } if (_caps & CAP_C1p) Geodesic::C1pf(eps, _cC1pa); if (_caps & CAP_C2) { _aA2m1 = Geodesic::A2m1f(eps); Geodesic::C2f(eps, _cC2a); _bB21 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC2a, nC2_); } if (_caps & CAP_C3) { g.C3f(eps, _cC3a); _aA3c = -_f * _salp0 * g.A3f(eps); _bB31 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC3a, nC3_-1); } if (_caps & CAP_C4) { g.C4f(eps, _cC4a); // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) _aA4 = Math::sq(_a) * _calp0 * _salp0 * g._e2; _bB41 = Geodesic::SinCosSeries(false, _ssig1, _csig1, _cC4a, nC4_); } _a13 = _s13 = Math::NaN(); } GeodesicLine::GeodesicLine(const Geodesic& g, real lat1, real lon1, real azi1, unsigned caps) { azi1 = Math::AngNormalize(azi1); real salp1, calp1; // Guard against underflow in salp0. Also -0 is converted to +0. Math::sincosd(Math::AngRound(azi1), salp1, calp1); LineInit(g, lat1, lon1, azi1, salp1, calp1, caps); } GeodesicLine::GeodesicLine(const Geodesic& g, real lat1, real lon1, real azi1, real salp1, real calp1, unsigned caps, bool arcmode, real s13_a13) { LineInit(g, lat1, lon1, azi1, salp1, calp1, caps); GenSetDistance(arcmode, s13_a13); } Math::real GeodesicLine::GenPosition(bool arcmode, real s12_a12, unsigned outmask, real& lat2, real& lon2, real& azi2, real& s12, real& m12, real& M12, real& M21, real& S12) const { outmask &= _caps & OUT_MASK; if (!( Init() && (arcmode || (_caps & (OUT_MASK & DISTANCE_IN))) )) // Uninitialized or impossible distance calculation requested return Math::NaN(); // Avoid warning about uninitialized B12. real sig12, ssig12, csig12, B12 = 0, AB1 = 0; if (arcmode) { // Interpret s12_a12 as spherical arc length sig12 = s12_a12 * Math::degree(); Math::sincosd(s12_a12, ssig12, csig12); } else { // Interpret s12_a12 as distance real tau12 = s12_a12 / (_b * (1 + _aA1m1)), s = sin(tau12), c = cos(tau12); // tau2 = tau1 + tau12 B12 = - Geodesic::SinCosSeries(true, _stau1 * c + _ctau1 * s, _ctau1 * c - _stau1 * s, _cC1pa, nC1p_); sig12 = tau12 - (B12 - _bB11); ssig12 = sin(sig12); csig12 = cos(sig12); if (fabs(_f) > 0.01) { // Reverted distance series is inaccurate for |f| > 1/100, so correct // sig12 with 1 Newton iteration. The following table shows the // approximate maximum error for a = WGS_a() and various f relative to // GeodesicExact. // erri = the error in the inverse solution (nm) // errd = the error in the direct solution (series only) (nm) // errda = the error in the direct solution // (series + 1 Newton) (nm) // // f erri errd errda // -1/5 12e6 1.2e9 69e6 // -1/10 123e3 12e6 765e3 // -1/20 1110 108e3 7155 // -1/50 18.63 200.9 27.12 // -1/100 18.63 23.78 23.37 // -1/150 18.63 21.05 20.26 // 1/150 22.35 24.73 25.83 // 1/100 22.35 25.03 25.31 // 1/50 29.80 231.9 30.44 // 1/20 5376 146e3 10e3 // 1/10 829e3 22e6 1.5e6 // 1/5 157e6 3.8e9 280e6 real ssig2 = _ssig1 * csig12 + _csig1 * ssig12, csig2 = _csig1 * csig12 - _ssig1 * ssig12; B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_); real serr = (1 + _aA1m1) * (sig12 + (B12 - _bB11)) - s12_a12 / _b; sig12 = sig12 - serr / sqrt(1 + _k2 * Math::sq(ssig2)); ssig12 = sin(sig12); csig12 = cos(sig12); // Update B12 below } } real ssig2, csig2, sbet2, cbet2, salp2, calp2; // sig2 = sig1 + sig12 ssig2 = _ssig1 * csig12 + _csig1 * ssig12; csig2 = _csig1 * csig12 - _ssig1 * ssig12; real dn2 = sqrt(1 + _k2 * Math::sq(ssig2)); if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) { if (arcmode || fabs(_f) > 0.01) B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_); AB1 = (1 + _aA1m1) * (B12 - _bB11); } // sin(bet2) = cos(alp0) * sin(sig2) sbet2 = _calp0 * ssig2; // Alt: cbet2 = hypot(csig2, salp0 * ssig2); cbet2 = hypot(_salp0, _calp0 * csig2); if (cbet2 == 0) // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case cbet2 = csig2 = tiny_; // tan(alp0) = cos(sig2)*tan(alp2) salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize if (outmask & DISTANCE) s12 = arcmode ? _b * ((1 + _aA1m1) * sig12 + AB1) : s12_a12; if (outmask & LONGITUDE) { // tan(omg2) = sin(alp0) * tan(sig2) real somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize E = copysign(real(1), _salp0); // east-going? // omg12 = omg2 - omg1 real omg12 = outmask & LONG_UNROLL ? E * (sig12 - (atan2( ssig2, csig2) - atan2( _ssig1, _csig1)) + (atan2(E * somg2, comg2) - atan2(E * _somg1, _comg1))) : atan2(somg2 * _comg1 - comg2 * _somg1, comg2 * _comg1 + somg2 * _somg1); real lam12 = omg12 + _aA3c * ( sig12 + (Geodesic::SinCosSeries(true, ssig2, csig2, _cC3a, nC3_-1) - _bB31)); real lon12 = lam12 / Math::degree(); lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 : Math::AngNormalize(Math::AngNormalize(_lon1) + Math::AngNormalize(lon12)); } if (outmask & LATITUDE) lat2 = Math::atan2d(sbet2, _f1 * cbet2); if (outmask & AZIMUTH) azi2 = Math::atan2d(salp2, calp2); if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) { real B22 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC2a, nC2_), AB2 = (1 + _aA2m1) * (B22 - _bB21), J12 = (_aA1m1 - _aA2m1) * sig12 + (AB1 - AB2); if (outmask & REDUCEDLENGTH) // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure // accurate cancellation in the case of coincident points. m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2)) - _csig1 * csig2 * J12); if (outmask & GEODESICSCALE) { real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2); M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1; M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2; } } if (outmask & AREA) { real B42 = Geodesic::SinCosSeries(false, ssig2, csig2, _cC4a, nC4_); real salp12, calp12; if (_calp0 == 0 || _salp0 == 0) { // alp12 = alp2 - alp1, used in atan2 so no need to normalize salp12 = salp2 * _calp1 - calp2 * _salp1; calp12 = calp2 * _calp1 + salp2 * _salp1; // We used to include here some patch up code that purported to deal // with nearly meridional geodesics properly. However, this turned out // to be wrong once _salp1 = -0 was allowed (via // Geodesic::InverseLine). In fact, the calculation of {s,c}alp12 // was already correct (following the IEEE rules for handling signed // zeros). So the patch up code was unnecessary (as well as // dangerous). } else { // tan(alp) = tan(alp0) * sec(sig) // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1) // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2) // If csig12 > 0, write // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1) // else // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1 // No need to normalize salp12 = _calp0 * _salp0 * (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 : ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1)); calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2; } S12 = _c2 * atan2(salp12, calp12) + _aA4 * (B42 - _bB41); } return arcmode ? s12_a12 : sig12 / Math::degree(); } void GeodesicLine::SetDistance(real s13) { _s13 = s13; real t; // This will set _a13 to NaN if the GeodesicLine doesn't have the // DISTANCE_IN capability. _a13 = GenPosition(false, _s13, 0u, t, t, t, t, t, t, t, t); } void GeodesicLine::SetArc(real a13) { _a13 = a13; // In case the GeodesicLine doesn't have the DISTANCE capability. _s13 = Math::NaN(); real t; GenPosition(true, _a13, DISTANCE, t, t, t, _s13, t, t, t, t); } void GeodesicLine::GenSetDistance(bool arcmode, real s13_a13) { arcmode ? SetArc(s13_a13) : SetDistance(s13_a13); } } // namespace GeographicLib