GeographicLib
1.21
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00001 /** 00002 * \file GeodesicLine.cpp 00003 * \brief Implementation for GeographicLib::GeodesicLine class 00004 * 00005 * Copyright (c) Charles Karney (2009-2011) <charles@karney.com> and licensed 00006 * under the MIT/X11 License. For more information, see 00007 * http://geographiclib.sourceforge.net/ 00008 * 00009 * This is a reformulation of the geodesic problem. The notation is as 00010 * follows: 00011 * - at a general point (no suffix or 1 or 2 as suffix) 00012 * - phi = latitude 00013 * - beta = latitude on auxiliary sphere 00014 * - omega = longitude on auxiliary sphere 00015 * - lambda = longitude 00016 * - alpha = azimuth of great circle 00017 * - sigma = arc length along great circle 00018 * - s = distance 00019 * - tau = scaled distance (= sigma at multiples of pi/2) 00020 * - at northwards equator crossing 00021 * - beta = phi = 0 00022 * - omega = lambda = 0 00023 * - alpha = alpha0 00024 * - sigma = s = 0 00025 * - a 12 suffix means a difference, e.g., s12 = s2 - s1. 00026 * - s and c prefixes mean sin and cos 00027 **********************************************************************/ 00028 00029 #include <GeographicLib/GeodesicLine.hpp> 00030 00031 #define GEOGRAPHICLIB_GEODESICLINE_CPP \ 00032 "$Id: d95fea8e73fd86fdc558e5b0397a97241cfe40c2 $" 00033 00034 RCSID_DECL(GEOGRAPHICLIB_GEODESICLINE_CPP) 00035 RCSID_DECL(GEOGRAPHICLIB_GEODESICLINE_HPP) 00036 00037 namespace GeographicLib { 00038 00039 using namespace std; 00040 00041 GeodesicLine::GeodesicLine(const Geodesic& g, 00042 real lat1, real lon1, real azi1, 00043 unsigned caps) throw() 00044 : _a(g._a) 00045 , _f(g._f) 00046 , _b(g._b) 00047 , _c2(g._c2) 00048 , _f1(g._f1) 00049 // Always allow latitude and azimuth 00050 , _caps(caps | LATITUDE | AZIMUTH) 00051 { 00052 azi1 = Geodesic::AngNormalize(azi1); 00053 // Guard against underflow in salp0 00054 azi1 = Geodesic::AngRound(azi1); 00055 lon1 = Geodesic::AngNormalize(lon1); 00056 _lat1 = lat1; 00057 _lon1 = lon1; 00058 _azi1 = azi1; 00059 // alp1 is in [0, pi] 00060 real alp1 = azi1 * Math::degree<real>(); 00061 // Enforce sin(pi) == 0 and cos(pi/2) == 0. Better to face the ensuing 00062 // problems directly than to skirt them. 00063 _salp1 = azi1 == -180 ? 0 : sin(alp1); 00064 _calp1 = abs(azi1) == 90 ? 0 : cos(alp1); 00065 real cbet1, sbet1, phi; 00066 phi = lat1 * Math::degree<real>(); 00067 // Ensure cbet1 = +epsilon at poles 00068 sbet1 = _f1 * sin(phi); 00069 cbet1 = abs(lat1) == 90 ? Geodesic::tiny_ : cos(phi); 00070 Geodesic::SinCosNorm(sbet1, cbet1); 00071 00072 // Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0), 00073 _salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|] 00074 // Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following 00075 // is slightly better (consider the case salp1 = 0). 00076 _calp0 = Math::hypot(_calp1, _salp1 * sbet1); 00077 // Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1). 00078 // sig = 0 is nearest northward crossing of equator. 00079 // With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line). 00080 // With bet1 = pi/2, alp1 = -pi, sig1 = pi/2 00081 // With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2 00082 // Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1). 00083 // With alp0 in (0, pi/2], quadrants for sig and omg coincide. 00084 // No atan2(0,0) ambiguity at poles since cbet1 = +epsilon. 00085 // With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi. 00086 _ssig1 = sbet1; _somg1 = _salp0 * sbet1; 00087 _csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1; 00088 Geodesic::SinCosNorm(_ssig1, _csig1); // sig1 in (-pi, pi] 00089 Geodesic::SinCosNorm(_somg1, _comg1); 00090 00091 _k2 = Math::sq(_calp0) * g._ep2; 00092 real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2); 00093 00094 if (_caps & CAP_C1) { 00095 _A1m1 = Geodesic::A1m1f(eps); 00096 Geodesic::C1f(eps, _C1a); 00097 _B11 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C1a, nC1_); 00098 real s = sin(_B11), c = cos(_B11); 00099 // tau1 = sig1 + B11 00100 _stau1 = _ssig1 * c + _csig1 * s; 00101 _ctau1 = _csig1 * c - _ssig1 * s; 00102 // Not necessary because C1pa reverts C1a 00103 // _B11 = -SinCosSeries(true, _stau1, _ctau1, _C1pa, nC1p_); 00104 } 00105 00106 if (_caps & CAP_C1p) 00107 Geodesic::C1pf(eps, _C1pa); 00108 00109 if (_caps & CAP_C2) { 00110 _A2m1 = Geodesic::A2m1f(eps); 00111 Geodesic::C2f(eps, _C2a); 00112 _B21 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C2a, nC2_); 00113 } 00114 00115 if (_caps & CAP_C3) { 00116 g.C3f(eps, _C3a); 00117 _A3c = -_f * _salp0 * g.A3f(eps); 00118 _B31 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C3a, nC3_-1); 00119 } 00120 00121 if (_caps & CAP_C4) { 00122 g.C4f(_k2, _C4a); 00123 // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) 00124 _A4 = Math::sq(_a) * _calp0 * _salp0 * g._e2; 00125 _B41 = Geodesic::SinCosSeries(false, _ssig1, _csig1, _C4a, nC4_); 00126 } 00127 } 00128 00129 Math::real GeodesicLine::GenPosition(bool arcmode, real s12_a12, 00130 unsigned outmask, 00131 real& lat2, real& lon2, real& azi2, 00132 real& s12, real& m12, 00133 real& M12, real& M21, 00134 real& S12) 00135 const throw() { 00136 outmask &= _caps & OUT_ALL; 00137 if (!( Init() && (arcmode || (_caps & DISTANCE_IN & OUT_ALL)) )) 00138 // Uninitialized or impossible distance calculation requested 00139 return Math::NaN<real>(); 00140 00141 // Avoid warning about uninitialized B12. 00142 real sig12, ssig12, csig12, B12 = 0, AB1 = 0; 00143 if (arcmode) { 00144 // Interpret s12_a12 as spherical arc length 00145 sig12 = s12_a12 * Math::degree<real>(); 00146 real s12a = abs(s12_a12); 00147 s12a -= 180 * floor(s12a / 180); 00148 ssig12 = s12a == 0 ? 0 : sin(sig12); 00149 csig12 = s12a == 90 ? 0 : cos(sig12); 00150 } else { 00151 // Interpret s12_a12 as distance 00152 real 00153 tau12 = s12_a12 / (_b * (1 + _A1m1)), 00154 s = sin(tau12), 00155 c = cos(tau12); 00156 // tau2 = tau1 + tau12 00157 B12 = - Geodesic::SinCosSeries(true, _stau1 * c + _ctau1 * s, 00158 _ctau1 * c - _stau1 * s, 00159 _C1pa, nC1p_); 00160 sig12 = tau12 - (B12 - _B11); 00161 ssig12 = sin(sig12); 00162 csig12 = cos(sig12); 00163 } 00164 00165 real omg12, lam12, lon12; 00166 real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2; 00167 // sig2 = sig1 + sig12 00168 ssig2 = _ssig1 * csig12 + _csig1 * ssig12; 00169 csig2 = _csig1 * csig12 - _ssig1 * ssig12; 00170 if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) { 00171 if (arcmode) 00172 B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _C1a, nC1_); 00173 AB1 = (1 + _A1m1) * (B12 - _B11); 00174 } 00175 // sin(bet2) = cos(alp0) * sin(sig2) 00176 sbet2 = _calp0 * ssig2; 00177 // Alt: cbet2 = hypot(csig2, salp0 * ssig2); 00178 cbet2 = Math::hypot(_salp0, _calp0 * csig2); 00179 if (cbet2 == 0) 00180 // I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case 00181 cbet2 = csig2 = Geodesic::tiny_; 00182 // tan(omg2) = sin(alp0) * tan(sig2) 00183 somg2 = _salp0 * ssig2; comg2 = csig2; // No need to normalize 00184 // tan(alp0) = cos(sig2)*tan(alp2) 00185 salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize 00186 // omg12 = omg2 - omg1 00187 omg12 = atan2(somg2 * _comg1 - comg2 * _somg1, 00188 comg2 * _comg1 + somg2 * _somg1); 00189 00190 if (outmask & DISTANCE) 00191 s12 = arcmode ? _b * ((1 + _A1m1) * sig12 + AB1) : s12_a12; 00192 00193 if (outmask & LONGITUDE) { 00194 lam12 = omg12 + _A3c * 00195 ( sig12 + (Geodesic::SinCosSeries(true, ssig2, csig2, _C3a, nC3_-1) 00196 - _B31)); 00197 lon12 = lam12 / Math::degree<real>(); 00198 // Can't use AngNormalize because longitude might have wrapped multiple 00199 // times. 00200 lon12 = lon12 - 360 * floor(lon12/360 + real(0.5)); 00201 lon2 = Geodesic::AngNormalize(_lon1 + lon12); 00202 } 00203 00204 if (outmask & LATITUDE) 00205 lat2 = atan2(sbet2, _f1 * cbet2) / Math::degree<real>(); 00206 00207 if (outmask & AZIMUTH) 00208 // minus signs give range [-180, 180). 0- converts -0 to +0. 00209 azi2 = 0 - atan2(-salp2, calp2) / Math::degree<real>(); 00210 00211 if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) { 00212 real 00213 ssig1sq = Math::sq(_ssig1), 00214 ssig2sq = Math::sq( ssig2), 00215 w1 = sqrt(1 + _k2 * ssig1sq), 00216 w2 = sqrt(1 + _k2 * ssig2sq), 00217 B22 = Geodesic::SinCosSeries(true, ssig2, csig2, _C2a, nC2_), 00218 AB2 = (1 + _A2m1) * (B22 - _B21), 00219 J12 = (_A1m1 - _A2m1) * sig12 + (AB1 - AB2); 00220 if (outmask & REDUCEDLENGTH) 00221 // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure 00222 // accurate cancellation in the case of coincident points. 00223 m12 = _b * ((w2 * (_csig1 * ssig2) - w1 * (_ssig1 * csig2)) 00224 - _csig1 * csig2 * J12); 00225 if (outmask & GEODESICSCALE) { 00226 M12 = csig12 + (_k2 * (ssig2sq - ssig1sq) * ssig2 / (w1 + w2) 00227 - csig2 * J12) * _ssig1 / w1; 00228 M21 = csig12 - (_k2 * (ssig2sq - ssig1sq) * _ssig1 / (w1 + w2) 00229 - _csig1 * J12) * ssig2 / w2; 00230 } 00231 } 00232 00233 if (outmask & AREA) { 00234 real 00235 B42 = Geodesic::SinCosSeries(false, ssig2, csig2, _C4a, nC4_); 00236 real salp12, calp12; 00237 if (_calp0 == 0 || _salp0 == 0) { 00238 // alp12 = alp2 - alp1, used in atan2 so no need to normalized 00239 salp12 = salp2 * _calp1 - calp2 * _salp1; 00240 calp12 = calp2 * _calp1 + salp2 * _salp1; 00241 // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz 00242 // salp12 = -0 and alp12 = -180. However this depends on the sign being 00243 // attached to 0 correctly. The following ensures the correct behavior. 00244 if (salp12 == 0 && calp12 < 0) { 00245 salp12 = Geodesic::tiny_ * _calp1; 00246 calp12 = -1; 00247 } 00248 } else { 00249 // tan(alp) = tan(alp0) * sec(sig) 00250 // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1) 00251 // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2) 00252 // If csig12 > 0, write 00253 // csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1) 00254 // else 00255 // csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1 00256 // No need to normalize 00257 salp12 = _calp0 * _salp0 * 00258 (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 : 00259 ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1)); 00260 calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2; 00261 } 00262 S12 = _c2 * atan2(salp12, calp12) + _A4 * (B42 - _B41); 00263 } 00264 00265 return arcmode ? s12_a12 : sig12 / Math::degree<real>(); 00266 } 00267 00268 } // namespace GeographicLib