我需要一个函数来计算一对WGS 84位置之间的距离，以达到高度准确性，我计划使用boost geometry中的geographic函数。
boost geometry Design Rational州：
有Andoyer方法，快速而精确，还有Vincenty方法，更慢，更精确..
但是，在使用boost::geometry::distance和Andoyer策略测试Vincenty函数时，我得到了以下结果：
WGS 84 values (metres)
Semimajor axis: 6378137.000000
Flattening: 0.003353
Semiminor axis: 6356752.314245
Semimajor distance: 20037508.342789
Semiminor distance: 19970326.371123
Boost geometry near poles
Andoyer function:
Semimajor distance: 20037508.151445
Semiminor distance: 20003917.164970
Vincenty function:
Semimajor distance: **19970326.180419**
Semiminor distance: 20003931.266635
Boost geometry at poles
Andoyer function:
Semimajor distance: 0.000000
Semiminor distance: 0.000000
Vincenty function:
Semimajor distance: **19970326.371122**
Semiminor distance: 20003931.458623
沿着Semimajor轴（即赤道附近）的Vincenty距离小于北极和南极之间的Semiminor轴周围的距离。这不是正确的。
Semiminor和Andoyer距离看起来合理。除非点位于地球的另一侧，否则当boost Andoyer函数返回零时！
问题在于：Vincenty算法，boost geometry实现，还是我的测试代码？
测试代码：
/// boost geometry WGS84 distance issue
// Note: M_PI is not part of the C or C++ standards, _USE_MATH_DEFINES enables it
#define _USE_MATH_DEFINES
#include <boost/geometry.hpp>
#include <cmath>
#include <iostream>
#include <ios>
// WGS 84 parameters from: Eurocontrol WGS 84 Implementation Manual
// Version 2.4 Chapter 3, page 14
/// The Semimajor axis measured in metres.
/// This is the radius at the equator.
constexpr double a = 6378137.0;
/// Flattening, a ratio.
/// This is the flattening of the ellipse at the poles
constexpr double f = 1.0/298.257223563;
/// The Semiminor axis measured in metres.
/// This is the radius at the poles.
/// Note: this is derived from the Semimajor axis and the flattening.
/// See WGS 84 Implementation Manual equation B-2, page 69.
constexpr double b = a * (1.0 - f);
int main(int /*argc*/, char ** /*argv*/)
{
std::cout.setf(std::ios::fixed);
std::cout << "WGS 84 values (metres)\n";
std::cout << "\tSemimajor axis:\t\t" << a << "\n";
std::cout << "\tFlattening:\t\t" << f << "\n";
std::cout << "\tSemiminor axis:\t\t" << b << "\n\n";
std::cout << "\tSemimajor distance:\t" << M_PI * a << "\n";
std::cout << "\tSemiminor distance:\t" << M_PI * b << "\n";
std::cout << std::endl;
// Min value for delta. 0.000000014 causes Andoyer to fail.
const double DELTA(0.000000015);
// For boost::geometry:
typedef boost::geometry::cs::geographic<boost::geometry::radian> Wgs84Coords;
typedef boost::geometry::model::point<double, 2, Wgs84Coords> GeographicPoint;
// Note boost points are Long & Lat NOT Lat & Long
GeographicPoint near_north_pole (0.0, M_PI_2 - DELTA);
GeographicPoint near_south_pole (0.0, -M_PI_2 + DELTA);
GeographicPoint near_equator_east ( M_PI_2 - DELTA, 0.0);
GeographicPoint near_equator_west (-M_PI_2 + DELTA, 0.0);
// Note: the default boost geometry spheroid is WGS84
// #include <boost/geometry/core/srs.hpp>
typedef boost::geometry::srs::spheroid<double> SpheroidType;
SpheroidType spheriod;
//#include <boost/geometry/strategies/geographic/distance_andoyer.hpp>
typedef boost::geometry::strategy::distance::andoyer<SpheroidType>
AndoyerStrategy;
AndoyerStrategy andoyer(spheriod);
std::cout << "Boost geometry near poles\n";
std::cout << "Andoyer function:\n";
double andoyer_major(boost::geometry::distance(near_equator_east, near_equator_west, andoyer));
std::cout << "\tSemimajor distance:\t" << andoyer_major << "\n";
double andoyer_minor(boost::geometry::distance(near_north_pole, near_south_pole, andoyer));
std::cout << "\tSemiminor distance:\t" << andoyer_minor << "\n";
//#include <boost/geometry/strategies/geographic/distance_vincenty.hpp>
typedef boost::geometry::strategy::distance::vincenty<SpheroidType>
VincentyStrategy;
VincentyStrategy vincenty(spheriod);
std::cout << "Vincenty function:\n";
double vincenty_major(boost::geometry::distance(near_equator_east, near_equator_west, vincenty));
std::cout << "\tSemimajor distance:\t" << vincenty_major << "\n";
double vincenty_minor(boost::geometry::distance(near_north_pole, near_south_pole, vincenty));
std::cout << "\tSemiminor distance:\t" << vincenty_minor << "\n\n";
// Note boost points are Long & Lat NOT Lat & Long
GeographicPoint north_pole (0.0, M_PI_2);
GeographicPoint south_pole (0.0, -M_PI_2);
GeographicPoint equator_east ( M_PI_2, 0.0);
GeographicPoint equator_west (-M_PI_2, 0.0);
std::cout << "Boost geometry at poles\n";
std::cout << "Andoyer function:\n";
andoyer_major = boost::geometry::distance(equator_east, equator_west, andoyer);
std::cout << "\tSemimajor distance:\t" << andoyer_major << "\n";
andoyer_minor = boost::geometry::distance(north_pole, south_pole, andoyer);
std::cout << "\tSemiminor distance:\t" << andoyer_minor << "\n";
std::cout << "Vincenty function:\n";
vincenty_major = boost::geometry::distance(equator_east, equator_west, vincenty);
std::cout << "\tSemimajor distance:\t" << vincenty_major << "\n";
vincenty_minor = boost::geometry::distance(north_pole, south_pole, vincenty);
std::cout << "\tSemiminor distance:\t" << vincenty_minor << "\n";
return 0;
}
答案 0 :(得分：3)
我按照@ jwd630的建议检查了geographiclib 结果如下：
WGS 84 values (metres)
Semimajor distance: 20037508.342789
Semiminor distance: 19970326.371123
GeographicLib near antipodal
Semimajor distance: 20003931.458625
Semiminor distance: 20003931.455275
GeographicLib antipodal
Semimajor distance: 20003931.458625
Semiminor distance: 20003931.458625
GeographicLib verify
JFK to LHR distance: 5551759.400319
即。它提供与Vincenty相同的距离，用于两极之间的半距离（到5dp），并计算赤道上对映点的相同距离。
这是正确的，因为赤道上的对映点之间的最短距离是通过其中一个极点而不是赤道周围，因为默认的增强Andoyer算法计算。
所以@jwd630的答案是正确的，在三种算法中，geographiclib是唯一一个计算整个WGS84大地水准面的正确距离的算法。
这是测试代码：
/// GeographicLib WGS84 distance
// Note: M_PI is not part of the C or C++ standards, _USE_MATH_DEFINES enables it
#define _USE_MATH_DEFINES
#include <GeographicLib/Geodesic.hpp>
#include <cmath>
#include <iostream>
#include <ios>
// WGS 84 parameters from: Eurocontrol WGS 84 Implementation Manual
// Version 2.4 Chapter 3, page 14
/// The Semimajor axis measured in metres.
/// This is the radius at the equator.
constexpr double a = 6378137.0;
/// Flattening, a ratio.
/// This is the flattening of the ellipse at the poles
constexpr double f = 1.0/298.257223563;
/// The Semiminor axis measured in metres.
/// This is the radius at the poles.
/// Note: this is derived from the Semimajor axis and the flattening.
/// See WGS 84 Implementation Manual equation B-2, page 69.
constexpr double b = a * (1.0 - f);
int main(int /*argc*/, char ** /*argv*/)
{
const GeographicLib::Geodesic& geod(GeographicLib::Geodesic::WGS84());
std::cout.setf(std::ios::fixed);
std::cout << "WGS 84 values (metres)\n";
std::cout << "\tSemimajor axis:\t\t" << a << "\n";
std::cout << "\tFlattening:\t\t" << f << "\n";
std::cout << "\tSemiminor axis:\t\t" << b << "\n\n";
std::cout << "\tSemimajor distance:\t" << M_PI * a << "\n";
std::cout << "\tSemiminor distance:\t" << M_PI * b << "\n";
std::cout << std::endl;
// Min value for delta. 0.000000014 causes boost Andoyer to fail.
const double DELTA(0.000000015);
std::pair<double, double> near_equator_east (0.0, 90.0 - DELTA);
std::pair<double, double> near_equator_west (0.0, -90.0 + DELTA);
std::cout << "GeographicLib near antipodal\n";
double distance_metres(0.0);
geod.Inverse(near_equator_east.first, near_equator_east.second,
near_equator_west.first, near_equator_west.second, distance_metres);
std::cout << "\tSemimajor distance:\t" << distance_metres << "\n";
std::pair<double, double> near_north_pole (90.0 - DELTA, 0.0);
std::pair<double, double> near_south_pole (-90.0 + DELTA, 0.0);
geod.Inverse(near_north_pole.first, near_north_pole.second,
near_south_pole.first, near_south_pole.second, distance_metres);
std::cout << "\tSemiminor distance:\t" << distance_metres << "\n\n";
std::pair<double, double> equator_east (0.0, 90.0);
std::pair<double, double> equator_west (0.0, -90.0);
std::cout << "GeographicLib antipodal\n";
geod.Inverse(equator_east.first, equator_east.second,
equator_west.first, equator_west.second, distance_metres);
std::cout << "\tSemimajor distance:\t" << distance_metres << "\n";
std::pair<double, double> north_pole (90.0, 0.0);
std::pair<double, double> south_pole (-90.0, 0.0);
geod.Inverse(north_pole.first, north_pole.second,
south_pole.first, south_pole.second, distance_metres);
std::cout << "\tSemiminor distance:\t" << distance_metres << "\n\n";
std::pair<double, double> JFK (40.6, -73.8);
std::pair<double, double> LHR (51.6, -0.5);
std::cout << "GeographicLib verify distance\n";
geod.Inverse(JFK.first, JFK.second,
LHR.first, LHR.second, distance_metres);
std::cout << "\tJFK to LHR distance:\t" << distance_metres << std::endl;
return 0;
}
在他的论文Algorithms for geodesics中，
Charles F. F. Karney表示，＆＃34; Vincenty的方法无法收敛于近乎对映的点＃34;
这可以回答我原来的问题，即Vincenty算法不适合对映点。
注意：我已经提出boost票#11817来描述问题所在
Andoyer算法为对映点返回零，并向boost发送拉取请求并修复了它。
然而，对错误距离的唯一正确修复是使用正确的算法，即：geographiclib
非常感谢Charles F. F. Karney（@cffk）礼貌地指出我的愚蠢错误！
答案 1 :(得分：2)
作为另一种选择，查看Charles F. F. Karney的geographiclib。正如文档所说：“重点是返回准确的结果，误差接近四舍五入（约5-15纳米）。”