boost/geometry/formulas/vincenty_inverse.hpp
// Boost.Geometry // Copyright (c) 2007-2012 Barend Gehrels, Amsterdam, the Netherlands. // Copyright (c) 2018-2023 Adam Wulkiewicz, Lodz, Poland. // This file was modified by Oracle on 2014, 2016, 2017. // Modifications copyright (c) 2014-2017 Oracle and/or its affiliates. // Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle // Use, modification and distribution is subject to the Boost Software License, // Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at // http://www.boost.org/LICENSE_1_0.txt) #ifndef BOOST_GEOMETRY_FORMULAS_VINCENTY_INVERSE_HPP #define BOOST_GEOMETRY_FORMULAS_VINCENTY_INVERSE_HPP #include <boost/math/constants/constants.hpp> #include <boost/geometry/core/radius.hpp> #include <boost/geometry/util/constexpr.hpp> #include <boost/geometry/util/math.hpp> #include <boost/geometry/formulas/differential_quantities.hpp> #include <boost/geometry/formulas/flattening.hpp> #include <boost/geometry/formulas/result_inverse.hpp> #ifndef BOOST_GEOMETRY_DETAIL_VINCENTY_MAX_STEPS #define BOOST_GEOMETRY_DETAIL_VINCENTY_MAX_STEPS 1000 #endif namespace boost { namespace geometry { namespace formula { /*! \brief The solution of the inverse problem of geodesics on latlong coordinates, after Vincenty, 1975 \author See - http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf - http://www.icsm.gov.au/gda/gda-v_2.4.pdf \author Adapted from various implementations to get it close to the original document - http://www.movable-type.co.uk/scripts/LatLongVincenty.html - http://exogen.case.edu/projects/geopy/source/geopy.distance.html - http://futureboy.homeip.net/fsp/colorize.fsp?fileName=navigation.frink */ template < typename CT, bool EnableDistance, bool EnableAzimuth, bool EnableReverseAzimuth = false, bool EnableReducedLength = false, bool EnableGeodesicScale = false > struct vincenty_inverse { static const bool CalcQuantities = EnableReducedLength || EnableGeodesicScale; static const bool CalcAzimuths = EnableAzimuth || EnableReverseAzimuth || CalcQuantities; static const bool CalcFwdAzimuth = EnableAzimuth || CalcQuantities; static const bool CalcRevAzimuth = EnableReverseAzimuth || CalcQuantities; public: typedef result_inverse<CT> result_type; template <typename T1, typename T2, typename Spheroid> static inline result_type apply(T1 const& lon1, T1 const& lat1, T2 const& lon2, T2 const& lat2, Spheroid const& spheroid) { result_type result; if (math::equals(lat1, lat2) && math::equals(lon1, lon2)) { return result; } CT const c0 = 0; CT const c1 = 1; CT const c2 = 2; CT const c3 = 3; CT const c4 = 4; CT const c16 = 16; CT const c_e_12 = CT(1e-12); CT const pi = geometry::math::pi<CT>(); CT const two_pi = c2 * pi; // lambda: difference in longitude on an auxiliary sphere CT L = lon2 - lon1; CT lambda = L; if (L < -pi) L += two_pi; if (L > pi) L -= two_pi; CT const radius_a = CT(get_radius<0>(spheroid)); CT const radius_b = CT(get_radius<2>(spheroid)); CT const f = formula::flattening<CT>(spheroid); // U: reduced latitude, defined by tan U = (1-f) tan phi CT const one_min_f = c1 - f; CT const tan_U1 = one_min_f * tan(lat1); // above (1) CT const tan_U2 = one_min_f * tan(lat2); // above (1) // calculate sin U and cos U using trigonometric identities CT const temp_den_U1 = math::sqrt(c1 + math::sqr(tan_U1)); CT const temp_den_U2 = math::sqrt(c1 + math::sqr(tan_U2)); // cos = 1 / sqrt(1 + tan^2) CT const cos_U1 = c1 / temp_den_U1; CT const cos_U2 = c1 / temp_den_U2; // sin = tan / sqrt(1 + tan^2) // sin = tan * cos CT const sin_U1 = tan_U1 * cos_U1; CT const sin_U2 = tan_U2 * cos_U2; // calculate sin U and cos U directly //CT const U1 = atan(tan_U1); //CT const U2 = atan(tan_U2); //cos_U1 = cos(U1); //cos_U2 = cos(U2); //sin_U1 = tan_U1 * cos_U1; // sin(U1); //sin_U2 = tan_U2 * cos_U2; // sin(U2); CT previous_lambda; CT sin_lambda; CT cos_lambda; CT sin_sigma; CT sin_alpha; CT cos2_alpha; CT cos_2sigma_m; CT cos2_2sigma_m; CT sigma; int counter = 0; // robustness do { previous_lambda = lambda; // (13) sin_lambda = sin(lambda); cos_lambda = cos(lambda); sin_sigma = math::sqrt(math::sqr(cos_U2 * sin_lambda) + math::sqr(cos_U1 * sin_U2 - sin_U1 * cos_U2 * cos_lambda)); // (14) CT cos_sigma = sin_U1 * sin_U2 + cos_U1 * cos_U2 * cos_lambda; // (15) sin_alpha = cos_U1 * cos_U2 * sin_lambda / sin_sigma; // (17) cos2_alpha = c1 - math::sqr(sin_alpha); cos_2sigma_m = math::equals(cos2_alpha, c0) ? c0 : cos_sigma - c2 * sin_U1 * sin_U2 / cos2_alpha; // (18) cos2_2sigma_m = math::sqr(cos_2sigma_m); CT C = f/c16 * cos2_alpha * (c4 + f * (c4 - c3 * cos2_alpha)); // (10) sigma = atan2(sin_sigma, cos_sigma); // (16) lambda = L + (c1 - C) * f * sin_alpha * (sigma + C * sin_sigma * (cos_2sigma_m + C * cos_sigma * (-c1 + c2 * cos2_2sigma_m))); // (11) ++counter; // robustness } while ( geometry::math::abs(previous_lambda - lambda) > c_e_12 && geometry::math::abs(lambda) < pi && counter < BOOST_GEOMETRY_DETAIL_VINCENTY_MAX_STEPS ); // robustness if BOOST_GEOMETRY_CONSTEXPR (EnableDistance) { // Some types cannot divide by doubles CT const c6 = 6; CT const c47 = 47; CT const c74 = 74; CT const c128 = 128; CT const c256 = 256; CT const c175 = 175; CT const c320 = 320; CT const c768 = 768; CT const c1024 = 1024; CT const c4096 = 4096; CT const c16384 = 16384; //CT sqr_u = cos2_alpha * (math::sqr(radius_a) - math::sqr(radius_b)) / math::sqr(radius_b); // above (1) CT sqr_u = cos2_alpha * ( math::sqr(radius_a / radius_b) - c1 ); // above (1) CT A = c1 + sqr_u/c16384 * (c4096 + sqr_u * (-c768 + sqr_u * (c320 - c175 * sqr_u))); // (3) CT B = sqr_u/c1024 * (c256 + sqr_u * ( -c128 + sqr_u * (c74 - c47 * sqr_u))); // (4) CT const cos_sigma = cos(sigma); CT const sin2_sigma = math::sqr(sin_sigma); CT delta_sigma = B * sin_sigma * (cos_2sigma_m + (B/c4) * (cos_sigma* (-c1 + c2 * cos2_2sigma_m) - (B/c6) * cos_2sigma_m * (-c3 + c4 * sin2_sigma) * (-c3 + c4 * cos2_2sigma_m))); // (6) result.distance = radius_b * A * (sigma - delta_sigma); // (19) } if BOOST_GEOMETRY_CONSTEXPR (CalcAzimuths) { if BOOST_GEOMETRY_CONSTEXPR (CalcFwdAzimuth) { result.azimuth = atan2(cos_U2 * sin_lambda, cos_U1 * sin_U2 - sin_U1 * cos_U2 * cos_lambda); // (20) } if BOOST_GEOMETRY_CONSTEXPR (CalcRevAzimuth) { result.reverse_azimuth = atan2(cos_U1 * sin_lambda, -sin_U1 * cos_U2 + cos_U1 * sin_U2 * cos_lambda); // (21) } } if BOOST_GEOMETRY_CONSTEXPR (CalcQuantities) { typedef differential_quantities<CT, EnableReducedLength, EnableGeodesicScale, 2> quantities; quantities::apply(lon1, lat1, lon2, lat2, result.azimuth, result.reverse_azimuth, radius_b, f, result.reduced_length, result.geodesic_scale); } return result; } }; }}} // namespace boost::geometry::formula #endif // BOOST_GEOMETRY_FORMULAS_VINCENTY_INVERSE_HPP