geometry/formulas/andoyer_inverse.hpp

0001 // Boost.Geometry
0002
0003 // Copyright (c) 2018 Adam Wulkiewicz, Lodz, Poland.
0004
0005 // Copyright (c) 2015-2020 Oracle and/or its affiliates.
0006
0007 // Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle
0008
0009 // Use, modification and distribution is subject to the Boost Software License,
0010 // Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
0011 // http://www.boost.org/LICENSE_1_0.txt)
0012
0013 #ifndef BOOST_GEOMETRY_FORMULAS_ANDOYER_INVERSE_HPP
0014 #define BOOST_GEOMETRY_FORMULAS_ANDOYER_INVERSE_HPP
0015
0016
0017 #include <boost/math/constants/constants.hpp>
0018
0019 #include <boost/geometry/core/radius.hpp>
0020
0021 #include <boost/geometry/util/condition.hpp>
0022 #include <boost/geometry/util/math.hpp>
0023
0024 #include <boost/geometry/formulas/differential_quantities.hpp>
0025 #include <boost/geometry/formulas/flattening.hpp>
0026 #include <boost/geometry/formulas/result_inverse.hpp>
0027
0028
0029 namespace boost { namespace geometry { namespace formula
0030 {
0031
0032 /*!
0033 \brief The solution of the inverse problem of geodesics on latlong coordinates,
0034        Forsyth-Andoyer-Lambert type approximation with first order terms.
0035 \author See
0036     - Technical Report: PAUL D. THOMAS, MATHEMATICAL MODELS FOR NAVIGATION SYSTEMS, 1965
0037       http://www.dtic.mil/docs/citations/AD0627893
0038     - Technical Report: PAUL D. THOMAS, SPHEROIDAL GEODESICS, REFERENCE SYSTEMS, AND LOCAL GEOMETRY, 1970
0039       http://www.dtic.mil/docs/citations/AD703541
0040 */
0041 template <
0042     typename CT,
0043     bool EnableDistance,
0044     bool EnableAzimuth,
0045     bool EnableReverseAzimuth = false,
0046     bool EnableReducedLength = false,
0047     bool EnableGeodesicScale = false
0048 >
0049 class andoyer_inverse
0050 {
0051     static const bool CalcQuantities = EnableReducedLength || EnableGeodesicScale;
0052     static const bool CalcAzimuths = EnableAzimuth || EnableReverseAzimuth || CalcQuantities;
0053     static const bool CalcFwdAzimuth = EnableAzimuth || CalcQuantities;
0054     static const bool CalcRevAzimuth = EnableReverseAzimuth || CalcQuantities;
0055
0056 public:
0057     typedef result_inverse<CT> result_type;
0058
0059     template <typename T1, typename T2, typename Spheroid>
0060     static inline result_type apply(T1 const& lon1,
0061                                     T1 const& lat1,
0062                                     T2 const& lon2,
0063                                     T2 const& lat2,
0064                                     Spheroid const& spheroid)
0065     {
0066         result_type result;
0067
0068         // coordinates in radians
0069
0070         if ( math::equals(lon1, lon2) && math::equals(lat1, lat2) )
0071         {
0072             return result;
0073         }
0074
0075         CT const c0 = CT(0);
0076         CT const c1 = CT(1);
0077         CT const pi = math::pi<CT>();
0078         CT const f = formula::flattening<CT>(spheroid);
0079
0080         CT const dlon = lon2 - lon1;
0081         CT const sin_dlon = sin(dlon);
0082         CT const cos_dlon = cos(dlon);
0083         CT const sin_lat1 = sin(lat1);
0084         CT const cos_lat1 = cos(lat1);
0085         CT const sin_lat2 = sin(lat2);
0086         CT const cos_lat2 = cos(lat2);
0087
0088         // H,G,T = infinity if cos_d = 1 or cos_d = -1
0089         // lat1 == +-90 && lat2 == +-90
0090         // lat1 == lat2 && lon1 == lon2
0091         CT cos_d = sin_lat1*sin_lat2 + cos_lat1*cos_lat2*cos_dlon;
0092         // on some platforms cos_d may be outside valid range
0093         if (cos_d < -c1)
0094             cos_d = -c1;
0095         else if (cos_d > c1)
0096             cos_d = c1;
0097
0098         CT const d = acos(cos_d); // [0, pi]
0099         CT const sin_d = sin(d);  // [-1, 1]
0100
0101         if ( BOOST_GEOMETRY_CONDITION(EnableDistance) )
0102         {
0103             CT const K = math::sqr(sin_lat1-sin_lat2);
0104             CT const L = math::sqr(sin_lat1+sin_lat2);
0105             CT const three_sin_d = CT(3) * sin_d;
0106
0107             CT const one_minus_cos_d = c1 - cos_d;
0108             CT const one_plus_cos_d = c1 + cos_d;
0109             // cos_d = 1 means that the points are very close
0110             // cos_d = -1 means that the points are antipodal
0111
0112             CT const H = math::equals(one_minus_cos_d, c0) ?
0113                             c0 :
0114                             (d + three_sin_d) / one_minus_cos_d;
0115             CT const G = math::equals(one_plus_cos_d, c0) ?
0116                             c0 :
0117                             (d - three_sin_d) / one_plus_cos_d;
0118
0119             CT const dd = -(f/CT(4))*(H*K+G*L);
0120
0121             CT const a = CT(get_radius<0>(spheroid));
0122
0123             result.distance = a * (d + dd);
0124         }
0125
0126         if ( BOOST_GEOMETRY_CONDITION(CalcAzimuths) )
0127         {
0128             // sin_d = 0 <=> antipodal points (incl. poles) or very close
0129             if (math::equals(sin_d, c0))
0130             {
0131                 // T = inf
0132                 // dA = inf
0133                 // azimuth = -inf
0134
0135                 // TODO: The following azimuths are inconsistent with distance
0136                 // i.e. according to azimuths below a segment with antipodal endpoints
0137                 // travels through the north pole, however the distance returned above
0138                 // is the length of a segment traveling along the equator.
0139                 // Furthermore, this special case handling is only done in andoyer
0140                 // formula.
0141                 // The most correct way of fixing it is to handle antipodal regions
0142                 // correctly and consistently across all formulas.
0143
0144                 // points very close
0145                 if (cos_d >= c0)
0146                 {
0147                     result.azimuth = c0;
0148                     result.reverse_azimuth = c0;
0149                 }
0150                 // antipodal points
0151                 else
0152                 {
0153                     // Set azimuth to 0 unless the first endpoint is the north pole
0154                     if (! math::equals(sin_lat1, c1))
0155                     {
0156                         result.azimuth = c0;
0157                         result.reverse_azimuth = pi;
0158                     }
0159                     else
0160                     {
0161                         result.azimuth = pi;
0162                         result.reverse_azimuth = c0;
0163                     }
0164                 }
0165             }
0166             else
0167             {
0168                 CT const c2 = CT(2);
0169
0170                 CT A = c0;
0171                 CT U = c0;
0172                 if (math::equals(cos_lat2, c0))
0173                 {
0174                     if (sin_lat2 < c0)
0175                     {
0176                         A = pi;
0177                     }
0178                 }
0179                 else
0180                 {
0181                     CT const tan_lat2 = sin_lat2/cos_lat2;
0182                     CT const M = cos_lat1*tan_lat2-sin_lat1*cos_dlon;
0183                     A = atan2(sin_dlon, M);
0184                     CT const sin_2A = sin(c2*A);
0185                     U = (f/ c2)*math::sqr(cos_lat1)*sin_2A;
0186                 }
0187
0188                 CT B = c0;
0189                 CT V = c0;
0190                 if (math::equals(cos_lat1, c0))
0191                 {
0192                     if (sin_lat1 < c0)
0193                     {
0194                         B = pi;
0195                     }
0196                 }
0197                 else
0198                 {
0199                     CT const tan_lat1 = sin_lat1/cos_lat1;
0200                     CT const N = cos_lat2*tan_lat1-sin_lat2*cos_dlon;
0201                     B = atan2(sin_dlon, N);
0202                     CT const sin_2B = sin(c2*B);
0203                     V = (f/ c2)*math::sqr(cos_lat2)*sin_2B;
0204                 }
0205
0206                 CT const T = d / sin_d;
0207
0208                 // even with sin_d == 0 checked above if the second point
0209                 // is somewhere in the antipodal area T may still be great
0210                 // therefore dA and dB may be great and the resulting azimuths
0211                 // may be some more or less arbitrary angles
0212
0213                 if (BOOST_GEOMETRY_CONDITION(CalcFwdAzimuth))
0214                 {
0215                     CT const dA = V*T - U;
0216                     result.azimuth = A - dA;
0217                     normalize_azimuth(result.azimuth, A, dA);
0218                 }
0219
0220                 if (BOOST_GEOMETRY_CONDITION(CalcRevAzimuth))
0221                 {
0222                     CT const dB = -U*T + V;
0223                     if (B >= 0)
0224                         result.reverse_azimuth = pi - B - dB;
0225                     else
0226                         result.reverse_azimuth = -pi - B - dB;
0227                     normalize_azimuth(result.reverse_azimuth, B, dB);
0228                 }
0229             }
0230         }
0231
0232         if (BOOST_GEOMETRY_CONDITION(CalcQuantities))
0233         {
0234             CT const b = CT(get_radius<2>(spheroid));
0235
0236             typedef differential_quantities<CT, EnableReducedLength, EnableGeodesicScale, 1> quantities;
0237             quantities::apply(dlon, sin_lat1, cos_lat1, sin_lat2, cos_lat2,
0238                               result.azimuth, result.reverse_azimuth,
0239                               b, f,
0240                               result.reduced_length, result.geodesic_scale);
0241         }
0242
0243         return result;
0244     }
0245
0246 private:
0247     static inline void normalize_azimuth(CT & azimuth, CT const& A, CT const& dA)
0248     {
0249         CT const c0 = 0;
0250
0251         if (A >= c0) // A indicates Eastern hemisphere
0252         {
0253             if (dA >= c0) // A altered towards 0
0254             {
0255                 if (azimuth < c0)
0256                 {
0257                     azimuth = c0;
0258                 }
0259             }
0260             else // dA < 0, A altered towards pi
0261             {
0262                 CT const pi = math::pi<CT>();
0263                 if (azimuth > pi)
0264                 {
0265                     azimuth = pi;
0266                 }
0267             }
0268         }
0269         else // A indicates Western hemisphere
0270         {
0271             if (dA <= c0) // A altered towards 0
0272             {
0273                 if (azimuth > c0)
0274                 {
0275                     azimuth = c0;
0276                 }
0277             }
0278             else // dA > 0, A altered towards -pi
0279             {
0280                 CT const minus_pi = -math::pi<CT>();
0281                 if (azimuth < minus_pi)
0282                 {
0283                     azimuth = minus_pi;
0284                 }
0285             }
0286         }
0287     }
0288 };
0289
0290 }}} // namespace boost::geometry::formula
0291
0292
0293 #endif // BOOST_GEOMETRY_FORMULAS_ANDOYER_INVERSE_HPP