327 lines
12 KiB
C++
327 lines
12 KiB
C++
/**
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* \file GeodesicLine.cpp
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* \brief Implementation for GeographicLib::GeodesicLine 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 <GeographicLib/GeodesicLine.hpp>
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#if defined(_MSC_VER)
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// Squelch warnings about mixing enums
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# pragma warning (disable: 5054)
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#endif
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namespace GeographicLib {
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using namespace std;
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void GeodesicLine::LineInit(const Geodesic& g,
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real lat1, real lon1,
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real azi1, real salp1, real calp1,
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unsigned caps) {
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tiny_ = g.tiny_;
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_lat1 = Math::LatFix(lat1);
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_lon1 = lon1;
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_azi1 = azi1;
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_salp1 = salp1;
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_calp1 = calp1;
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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 and unrolling of longitude
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_caps = caps | LATITUDE | AZIMUTH | LONG_UNROLL;
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real cbet1, sbet1;
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Math::sincosd(Math::AngRound(_lat1), sbet1, cbet1); sbet1 *= _f1;
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// Ensure cbet1 = +epsilon at poles
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Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1);
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_dn1 = sqrt(1 + g._ep2 * Math::sq(sbet1));
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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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Math::norm(_ssig1, _csig1); // sig1 in (-pi, pi]
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// Math::norm(_somg1, _comg1); -- don't need to normalize!
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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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_aA1m1 = Geodesic::A1m1f(eps);
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Geodesic::C1f(eps, _cC1a);
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_bB11 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC1a, nC1_);
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real s = sin(_bB11), c = cos(_bB11);
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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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// _bB11 = -SinCosSeries(true, _stau1, _ctau1, _cC1pa, nC1p_);
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}
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if (_caps & CAP_C1p)
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Geodesic::C1pf(eps, _cC1pa);
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if (_caps & CAP_C2) {
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_aA2m1 = Geodesic::A2m1f(eps);
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Geodesic::C2f(eps, _cC2a);
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_bB21 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC2a, nC2_);
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}
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if (_caps & CAP_C3) {
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g.C3f(eps, _cC3a);
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_aA3c = -_f * _salp0 * g.A3f(eps);
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_bB31 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC3a, nC3_-1);
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}
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if (_caps & CAP_C4) {
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g.C4f(eps, _cC4a);
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// Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
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_aA4 = Math::sq(_a) * _calp0 * _salp0 * g._e2;
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_bB41 = Geodesic::SinCosSeries(false, _ssig1, _csig1, _cC4a, nC4_);
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}
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_a13 = _s13 = Math::NaN();
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}
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GeodesicLine::GeodesicLine(const Geodesic& g,
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real lat1, real lon1, real azi1,
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unsigned caps) {
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azi1 = Math::AngNormalize(azi1);
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real salp1, calp1;
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// Guard against underflow in salp0. Also -0 is converted to +0.
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Math::sincosd(Math::AngRound(azi1), salp1, calp1);
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LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
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}
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GeodesicLine::GeodesicLine(const Geodesic& g,
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real lat1, real lon1,
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real azi1, real salp1, real calp1,
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unsigned caps, bool arcmode, real s13_a13) {
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LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
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GenSetDistance(arcmode, s13_a13);
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}
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Math::real GeodesicLine::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) const {
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outmask &= _caps & OUT_MASK;
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if (!( Init() && (arcmode || (_caps & (OUT_MASK & DISTANCE_IN))) ))
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// Uninitialized or impossible distance calculation requested
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return Math::NaN();
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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();
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Math::sincosd(s12_a12, ssig12, csig12);
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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 + _aA1m1)),
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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 = - Geodesic::SinCosSeries(true,
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_stau1 * c + _ctau1 * s,
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_ctau1 * c - _stau1 * s,
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_cC1pa, nC1p_);
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sig12 = tau12 - (B12 - _bB11);
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ssig12 = sin(sig12); csig12 = cos(sig12);
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if (fabs(_f) > 0.01) {
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// Reverted distance series is inaccurate for |f| > 1/100, so correct
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// sig12 with 1 Newton iteration. The following table shows the
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// approximate maximum error for a = WGS_a() and various f relative to
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// GeodesicExact.
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// erri = the error in the inverse solution (nm)
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// errd = the error in the direct solution (series only) (nm)
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// errda = the error in the direct solution
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// (series + 1 Newton) (nm)
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//
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// f erri errd errda
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// -1/5 12e6 1.2e9 69e6
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// -1/10 123e3 12e6 765e3
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// -1/20 1110 108e3 7155
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// -1/50 18.63 200.9 27.12
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// -1/100 18.63 23.78 23.37
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// -1/150 18.63 21.05 20.26
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// 1/150 22.35 24.73 25.83
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// 1/100 22.35 25.03 25.31
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// 1/50 29.80 231.9 30.44
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// 1/20 5376 146e3 10e3
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// 1/10 829e3 22e6 1.5e6
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// 1/5 157e6 3.8e9 280e6
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real
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ssig2 = _ssig1 * csig12 + _csig1 * ssig12,
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csig2 = _csig1 * csig12 - _ssig1 * ssig12;
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B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_);
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real serr = (1 + _aA1m1) * (sig12 + (B12 - _bB11)) - s12_a12 / _b;
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sig12 = sig12 - serr / sqrt(1 + _k2 * Math::sq(ssig2));
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ssig12 = sin(sig12); csig12 = cos(sig12);
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// Update B12 below
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}
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}
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real ssig2, csig2, sbet2, cbet2, 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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real dn2 = sqrt(1 + _k2 * Math::sq(ssig2));
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if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
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if (arcmode || fabs(_f) > 0.01)
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B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_);
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AB1 = (1 + _aA1m1) * (B12 - _bB11);
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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 = tiny_;
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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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if (outmask & DISTANCE)
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s12 = arcmode ? _b * ((1 + _aA1m1) * sig12 + AB1) : s12_a12;
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if (outmask & LONGITUDE) {
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// tan(omg2) = sin(alp0) * tan(sig2)
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real somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize
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E = copysign(real(1), _salp0); // east-going?
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// omg12 = omg2 - omg1
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real omg12 = outmask & LONG_UNROLL
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? E * (sig12
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- (atan2( ssig2, csig2) - atan2( _ssig1, _csig1))
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+ (atan2(E * somg2, comg2) - atan2(E * _somg1, _comg1)))
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: atan2(somg2 * _comg1 - comg2 * _somg1,
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comg2 * _comg1 + somg2 * _somg1);
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real lam12 = omg12 + _aA3c *
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( sig12 + (Geodesic::SinCosSeries(true, ssig2, csig2, _cC3a, nC3_-1)
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- _bB31));
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real lon12 = lam12 / Math::degree();
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lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 :
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Math::AngNormalize(Math::AngNormalize(_lon1) +
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Math::AngNormalize(lon12));
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}
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if (outmask & LATITUDE)
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lat2 = Math::atan2d(sbet2, _f1 * cbet2);
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if (outmask & AZIMUTH)
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azi2 = Math::atan2d(salp2, calp2);
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if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
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real
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B22 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC2a, nC2_),
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AB2 = (1 + _aA2m1) * (B22 - _bB21),
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J12 = (_aA1m1 - _aA2m1) * 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 * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
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- _csig1 * csig2 * J12);
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if (outmask & GEODESICSCALE) {
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real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
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M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
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M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2;
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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 = Geodesic::SinCosSeries(false, ssig2, csig2, _cC4a, 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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// We used to include here some patch up code that purported to deal
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// with nearly meridional geodesics properly. However, this turned out
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// to be wrong once _salp1 = -0 was allowed (via
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// Geodesic::InverseLine). In fact, the calculation of {s,c}alp12
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// was already correct (following the IEEE rules for handling signed
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// zeros). So the patch up code was unnecessary (as well as
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// dangerous).
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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) + _aA4 * (B42 - _bB41);
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}
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return arcmode ? s12_a12 : sig12 / Math::degree();
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}
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void GeodesicLine::SetDistance(real s13) {
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_s13 = s13;
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real t;
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// This will set _a13 to NaN if the GeodesicLine doesn't have the
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// DISTANCE_IN capability.
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_a13 = GenPosition(false, _s13, 0u, t, t, t, t, t, t, t, t);
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}
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void GeodesicLine::SetArc(real a13) {
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_a13 = a13;
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// In case the GeodesicLine doesn't have the DISTANCE capability.
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_s13 = Math::NaN();
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real t;
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GenPosition(true, _a13, DISTANCE, t, t, t, _s13, t, t, t, t);
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}
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void GeodesicLine::GenSetDistance(bool arcmode, real s13_a13) {
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arcmode ? SetArc(s13_a13) : SetDistance(s13_a13);
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}
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} // namespace GeographicLib
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