GeodesicLine.cpp

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00001 /**
00002  * \file GeodesicLine.cpp
00003  * \brief Implementation for GeographicLib::GeodesicLine class
00004  *
00005  * Copyright (c) Charles Karney (2009, 2010) <charles@karney.com>
00006  * and licensed under the LGPL.  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 greate 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 "$Id: GeodesicLine.cpp 6921 2010-12-31 14:34:50Z karney $"
00032 
00033 RCSID_DECL(GEOGRAPHICLIB_GEODESICLINE_CPP)
00034 RCSID_DECL(GEOGRAPHICLIB_GEODESICLINE_HPP)
00035 
00036 namespace GeographicLib {
00037 
00038   using namespace std;
00039 
00040   GeodesicLine::GeodesicLine(const Geodesic& g,
00041                              real lat1, real lon1, real azi1,
00042                              unsigned caps) throw()
00043     : _a(g._a)
00044     , _r(g._r)
00045     , _b(g._b)
00046     , _c2(g._c2)
00047     , _f1(g._f1)
00048       // Always allow latitude and azimuth
00049     , _caps(caps | LATITUDE | AZIMUTH)
00050   {
00051     azi1 = Geodesic::AngNormalize(azi1);
00052     // Guard against underflow in salp0
00053     azi1 = Geodesic::AngRound(azi1);
00054     lon1 = Geodesic::AngNormalize(lon1);
00055     _lat1 = lat1;
00056     _lon1 = lon1;
00057     _azi1 = azi1;
00058     // alp1 is in [0, pi]
00059     real alp1 = azi1 * Math::degree<real>();
00060     // Enforce sin(pi) == 0 and cos(pi/2) == 0.  Better to face the ensuing
00061     // problems directly than to skirt them.
00062     _salp1 =     azi1  == -180 ? 0 : sin(alp1);
00063     _calp1 = abs(azi1) ==   90 ? 0 : cos(alp1);
00064     real cbet1, sbet1, phi;
00065     phi = lat1 * Math::degree<real>();
00066     // Ensure cbet1 = +epsilon at poles
00067     sbet1 = _f1 * sin(phi);
00068     cbet1 = abs(lat1) == 90 ? Geodesic::eps2 : cos(phi);
00069     Geodesic::SinCosNorm(sbet1, cbet1);
00070 
00071     // Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
00072     _salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|]
00073     // Alt: calp0 = hypot(sbet1, calp1 * cbet1).  The following
00074     // is slightly better (consider the case salp1 = 0).
00075     _calp0 = Math::hypot(_calp1, _salp1 * sbet1);
00076     // Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
00077     // sig = 0 is nearest northward crossing of equator.
00078     // With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
00079     // With bet1 =  pi/2, alp1 = -pi, sig1 =  pi/2
00080     // With bet1 = -pi/2, alp1 =  0 , sig1 = -pi/2
00081     // Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
00082     // With alp0 in (0, pi/2], quadrants for sig and omg coincide.
00083     // No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
00084     // With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
00085     _ssig1 = sbet1; _somg1 = _salp0 * sbet1;
00086     _csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1;
00087     Geodesic::SinCosNorm(_ssig1, _csig1); // sig1 in (-pi, pi]
00088     Geodesic::SinCosNorm(_somg1, _comg1);
00089 
00090     _k2 = sq(_calp0) * g._ep2;
00091     real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2);
00092 
00093     if (_caps & CAP_C1) {
00094       _A1m1 =  Geodesic::A1m1f(eps);
00095       Geodesic::C1f(eps, _C1a);
00096       _B11 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C1a, nC1);
00097       real s = sin(_B11), c = cos(_B11);
00098       // tau1 = sig1 + B11
00099       _stau1 = _ssig1 * c + _csig1 * s;
00100       _ctau1 = _csig1 * c - _ssig1 * s;
00101       // Not necessary because C1pa reverts C1a
00102       //    _B11 = -SinCosSeries(true, _stau1, _ctau1, _C1pa, nC1p);
00103     }
00104 
00105     if (_caps & CAP_C1p)
00106       Geodesic::C1pf(eps, _C1pa);
00107 
00108     if (_caps & CAP_C2) {
00109       _A2m1 =  Geodesic::A2m1f(eps);
00110       Geodesic::C2f(eps, _C2a);
00111       _B21 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C2a, nC2);
00112     }
00113 
00114     if (_caps & CAP_C3) {
00115       g.C3f(eps, _C3a);
00116       _A3c = -g._f * _salp0 * g.A3f(eps);
00117       _B31 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _C3a, nC3-1);
00118     }
00119 
00120     if (_caps & CAP_C4) {
00121       g.C4f(_k2, _C4a);
00122       // Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
00123       _A4 = sq(g._a) * _calp0 * _salp0 * g._e2;
00124       _B41 = Geodesic::SinCosSeries(false, _ssig1, _csig1, _C4a, nC4);
00125     }
00126   }
00127 
00128   Math::real GeodesicLine::GenPosition(bool arcmode, real s12_a12,
00129                                        unsigned outmask,
00130                                        real& lat2, real& lon2, real& azi2,
00131                                        real& s12, real& m12,
00132                                        real& M12, real& M21,
00133                                        real& S12)
00134   const throw() {
00135     outmask &= _caps & OUT_ALL;
00136     if (!( Init() && (arcmode || (_caps & DISTANCE_IN & OUT_ALL)) ))
00137       // Uninitialized or impossible distance calculation requested
00138       return Math::NaN();
00139 
00140     // Avoid warning about uninitialized B12.
00141     real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
00142     if (arcmode) {
00143       // Interpret s12_a12 as spherical arc length
00144       sig12 = s12_a12 * Math::degree<real>();
00145       real s12a = abs(s12_a12);
00146       s12a -= 180 * floor(s12a / 180);
00147       ssig12 = s12a ==  0 ? 0 : sin(sig12);
00148       csig12 = s12a == 90 ? 0 : cos(sig12);
00149     } else {
00150       // Interpret s12_a12 as distance
00151       real
00152         tau12 = s12_a12 / (_b * (1 + _A1m1)),
00153         s = sin(tau12),
00154         c = cos(tau12);
00155       // tau2 = tau1 + tau12
00156       B12 = - Geodesic::SinCosSeries(true, _stau1 * c + _ctau1 * s,
00157                                      _ctau1 * c - _stau1 * s,
00158                                      _C1pa, nC1p);
00159       sig12 = tau12 - (B12 - _B11);
00160       ssig12 = sin(sig12);
00161       csig12 = cos(sig12);
00162     }
00163 
00164     real omg12, lam12, lon12;
00165     real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2;
00166     // sig2 = sig1 + sig12
00167     ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
00168     csig2 = _csig1 * csig12 - _ssig1 * ssig12;
00169     if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
00170       if (arcmode)
00171         B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _C1a, nC1);
00172       AB1 = (1 + _A1m1) * (B12 - _B11);
00173     }
00174     // sin(bet2) = cos(alp0) * sin(sig2)
00175     sbet2 = _calp0 * ssig2;
00176     // Alt: cbet2 = hypot(csig2, salp0 * ssig2);
00177     cbet2 = Math::hypot(_salp0, _calp0 * csig2);
00178     if (cbet2 == 0)
00179       // I.e., salp0 = 0, csig2 = 0.  Break the degeneracy in this case
00180       cbet2 = csig2 = Geodesic::eps2;
00181     // tan(omg2) = sin(alp0) * tan(sig2)
00182     somg2 = _salp0 * ssig2; comg2 = csig2;  // No need to normalize
00183     // tan(alp0) = cos(sig2)*tan(alp2)
00184     salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
00185     // omg12 = omg2 - omg1
00186     omg12 = atan2(somg2 * _comg1 - comg2 * _somg1,
00187                   comg2 * _comg1 + somg2 * _somg1);
00188 
00189     if (outmask & DISTANCE)
00190       s12 = arcmode ? _b * ((1 + _A1m1) * sig12 + AB1) : s12_a12;
00191 
00192     if (outmask & LONGITUDE) {
00193       lam12 = omg12 + _A3c *
00194         ( sig12 + (Geodesic::SinCosSeries(true, ssig2, csig2, _C3a, nC3-1)
00195                    - _B31));
00196       lon12 = lam12 / Math::degree<real>();
00197       // Can't use AngNormalize because longitude might have wrapped multiple
00198       // times.
00199       lon12 = lon12 - 360 * floor(lon12/360 + real(0.5));
00200       lon2 = Geodesic::AngNormalize(_lon1 + lon12);
00201     }
00202 
00203     if (outmask & LATITUDE)
00204       lat2 = atan2(sbet2, _f1 * cbet2) / Math::degree<real>();
00205 
00206     if (outmask & AZIMUTH)
00207       // minus signs give range [-180, 180). 0- converts -0 to +0.
00208       azi2 = 0 - atan2(-salp2, calp2) / Math::degree<real>();
00209 
00210     if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
00211       real
00212         ssig1sq = sq(_ssig1),
00213         ssig2sq = sq( ssig2),
00214         w1 = sqrt(1 + _k2 * ssig1sq),
00215         w2 = sqrt(1 + _k2 * ssig2sq),
00216         B22 = Geodesic::SinCosSeries(true, ssig2, csig2, _C2a, nC2),
00217         AB2 = (1 + _A2m1) * (B22 - _B21),
00218         J12 = (_A1m1 - _A2m1) * sig12 + (AB1 - AB2);
00219       if (outmask & REDUCEDLENGTH)
00220         // Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
00221         // accurate cancellation in the case of coincident points.
00222         m12 = _b * ((w2 * (_csig1 * ssig2) - w1 * (_ssig1 * csig2))
00223                   - _csig1 * csig2 * J12);
00224       if (outmask & GEODESICSCALE) {
00225         M12 = csig12 + (_k2 * (ssig2sq - ssig1sq) *  ssig2 / (w1 + w2)
00226                         - csig2 * J12) * _ssig1 / w1;
00227         M21 = csig12 - (_k2 * (ssig2sq - ssig1sq) * _ssig1 / (w1 + w2)
00228                         - _csig1 * J12) * ssig2 / w2;
00229       }
00230     }
00231 
00232     if (outmask & AREA) {
00233       real
00234         B42 = Geodesic::SinCosSeries(false, ssig2, csig2, _C4a, nC4);
00235       real salp12, calp12;
00236       if (_calp0 == 0 || _salp0 == 0) {
00237         // alp12 = alp2 - alp1, used in atan2 so no need to normalized
00238         salp12 = salp2 * _calp1 - calp2 * _salp1;
00239         calp12 = calp2 * _calp1 + salp2 * _salp1;
00240         // The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
00241         // salp12 = -0 and alp12 = -180.  However this depends on the sign being
00242         // attached to 0 correctly.  The following ensures the correct behavior.
00243         if (salp12 == 0 && calp12 < 0) {
00244           salp12 = Geodesic::eps2 * _calp1;
00245           calp12 = -1;
00246         }
00247       } else {
00248         // tan(alp) = tan(alp0) * sec(sig)
00249         // tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
00250         // = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
00251         // If csig12 > 0, write
00252         //   csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
00253         // else
00254         //   csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
00255         // No need to normalize
00256         salp12 = _calp0 * _salp0 *
00257           (csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
00258            ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
00259         calp12 = sq(_salp0) + sq(_calp0) * _csig1 * csig2;
00260       }
00261       S12 = _c2 * atan2(salp12, calp12) + _A4 * (B42 - _B41);
00262     }
00263 
00264     return arcmode ? s12_a12 : sig12 /  Math::degree<real>();
00265   }
00266 } // namespace GeographicLib
00267