Adding upstream version 14.2.21.upstream/14.2.21upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

17 files changed, 4000 insertions, 0 deletions

diff --git a/src/boost/libs/geometry/test/formulas/Jamfile.v2 b/src/boost/libs/geometry/test/formulas/Jamfile.v2
new file mode 100644
index 00000000..28ca8719
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/Jamfile.v2
@@ -0,0 +1,21 @@
+# Boost.Geometry
+#
+# Copyright (c) 2016-2019, Oracle and/or its affiliates.
+#
+# Contributed and/or modified by Vissarion Fysikopoulos, on behalf of Oracle
+# 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)
+
+test-suite boost-geometry-formulas
+    :
+    [ run inverse.cpp                        : : : : formulas_inverse ]
+    [ run inverse_karney.cpp                 : : : : formulas_inverse_karney ]
+    [ run direct.cpp                         : : : : formulas_direct ]
+    [ run direct_accuracy.cpp                : : : : formulas_direct_accuracy ]
+    [ run direct_meridian.cpp                : : : : formulas_direct_meridian ]
+    [ run intersection.cpp                   : : : : formulas_intersection ]
+    [ run vertex_longitude.cpp               : : : : formulas_vertex_longitude ]
+    ;
diff --git a/src/boost/libs/geometry/test/formulas/direct.cpp b/src/boost/libs/geometry/test/formulas/direct.cpp
new file mode 100644
index 00000000..97fb170f
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/direct.cpp
@@ -0,0 +1,176 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2016-2018 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Vissarion Fysikopoulos, on behalf of Oracle
+// Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle
+// Contributed and/or modified by Adeel Ahmad, as part of Google Summer of Code 2018 program
+
+// 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)
+
+#include "test_formula.hpp"
+#include "direct_cases.hpp"
+#include "direct_cases_antipodal.hpp"
+
+#include <boost/geometry/formulas/vincenty_direct.hpp>
+#include <boost/geometry/formulas/thomas_direct.hpp>
+#include <boost/geometry/formulas/karney_direct.hpp>
+//#include <boost/geometry/formulas/series_expansion_direct.hpp>
+#include <boost/geometry/formulas/spherical.hpp>
+
+#include <boost/geometry/srs/srs.hpp>
+
+#ifdef BOOST_GEOEMTRY_TEST_WITH_GEOGRAPHICLIB
+#include <GeographicLib/Geodesic.hpp>
+#include <GeographicLib/Constants.hpp>
+#endif // BOOST_GEOEMTRY_TEST_WITH_GEOGRAPHICLIB
+
+inline void symmetrize_wrt_origin(expected_result & r)
+{
+    r.lon2 = -r.lon2;
+    r.lat2 = -r.lat2;
+    r.reduced_length = -r.reduced_length;
+}
+
+inline expected_results symmetric_wrt_origin(expected_results r)
+{
+    r.distance = -r.distance;
+    symmetrize_wrt_origin(r.karney);
+    symmetrize_wrt_origin(r.series);
+    symmetrize_wrt_origin(r.spherical);
+    symmetrize_wrt_origin(r.thomas);
+    symmetrize_wrt_origin(r.thomas1st);
+    symmetrize_wrt_origin(r.vincenty);
+    return r;
+}
+
+template <typename Result>
+void check_direct(Result const& result, expected_result const& expected, expected_result const& reference,
+                  double reference_error, bool check_reference_only = false)
+{
+    check_direct_sph(result, expected, reference, reference_error, check_reference_only);
+    check_one(result.reduced_length, expected.reduced_length, reference.reduced_length, reference_error);
+    check_one(result.geodesic_scale, expected.geodesic_scale, reference.geodesic_scale, reference_error);
+}
+
+template <typename Result>
+void check_direct_sph(Result const& result, expected_result const& expected, expected_result const& reference,
+                      double reference_error, bool check_reference_only = false)
+{
+    check_one(result.lon2, expected.lon2, reference.lon2, reference_error, true, check_reference_only);
+    check_one(result.lat2, expected.lat2, reference.lat2, reference_error, true, check_reference_only);
+    check_one(result.reverse_azimuth, expected.reverse_azimuth, reference.reverse_azimuth, reference_error, true, check_reference_only);
+}
+
+void test_all(expected_results const& results)
+{
+    double const d2r = bg::math::d2r<double>();
+    double const r2d = bg::math::r2d<double>();
+
+    double lon1r = results.p1.lon * d2r;
+    double lat1r = results.p1.lat * d2r;
+    double distance = results.distance;
+    double azi12r = results.azimuth12 * d2r;
+
+    double lon1d = results.p1.lon;
+    double lat1d = results.p1.lat;
+    double azi12d = results.azimuth12;
+
+    // WGS84
+    bg::srs::spheroid<double> spheroid(6378137.0, 6356752.3142451793);
+    bg::srs::sphere<double> const sphere;
+
+    bg::formula::result_direct<double> result;
+
+    typedef bg::formula::vincenty_direct<double, true, true, true, true> vi_t;
+    result = vi_t::apply(lon1r, lat1r, distance, azi12r, spheroid);
+    result.lon2 *= r2d;
+    result.lat2 *= r2d;
+    result.reverse_azimuth *= r2d;
+    check_direct(result, results.vincenty, results.karney, 0.00000001);
+
+    typedef bg::formula::thomas_direct<double, true, true, true, true, true> th_t;
+    result = th_t::apply(lon1r, lat1r, distance, azi12r, spheroid);
+    result.lon2 *= r2d;
+    result.lat2 *= r2d;
+    result.reverse_azimuth *= r2d;
+    check_direct(result, results.thomas, results.karney, 0.0000001);
+
+    typedef bg::formula::thomas_direct<double, false, true, true, true, true> th_t1st;
+    result = th_t1st::apply(lon1r, lat1r, distance, azi12r, spheroid);
+    result.lon2 *= r2d;
+    result.lat2 *= r2d;
+    result.reverse_azimuth *= r2d;
+    check_direct(result, results.thomas1st, results.karney, 0.0000001);
+/*
+    typedef bg::formula::series_expansion_direct<double, true, true, true, true, 4> series;
+    result = series::apply(lon1r, lat1r, distance, azi12r, spheroid);
+    result.lon2 *= r2d;
+    result.lat2 *= r2d;
+    result.reverse_azimuth *= r2d;
+    check_direct(result, results.series, results.karney, 0.0000001);
+*/
+    result = bg::formula::spherical_direct<true, true>(lon1r, lat1r, distance,
+                                                       azi12r, sphere);
+    result.lon2 *= r2d;
+    result.lat2 *= r2d;
+    result.reverse_azimuth *= r2d;
+    check_direct_sph(result, results.spherical, results.karney, 0.1);
+
+    typedef bg::formula::karney_direct<double, true, true, true, true, 2> ka_t;
+    result = ka_t::apply(lon1d, lat1d, distance, azi12d, spheroid);
+    check_direct(result, results.karney, results.karney, 0.0000001);
+
+#ifdef BOOST_GEOEMTRY_TEST_WITH_GEOGRAPHICLIB
+    {
+        using namespace GeographicLib;
+        Geodesic geod(Constants::WGS84_a(), Constants::WGS84_f());
+        double foo = 0;
+        geod.Direct(lat1d, lon1d, azi12d, distance,
+                    result.lat2, result.lon2, result.reverse_azimuth,
+                    result.reduced_length, result.geodesic_scale, foo);
+        boost::ignore_unused(foo);
+        check_direct(result, results.karney, results.karney, 0.0000001);
+    }
+#endif
+}
+
+void test_karney_antipodal(expected_results_antipodal const& results)
+{
+    double lon1d = results.p1.lon;
+    double lat1d = results.p1.lat;
+    double distance = results.distance;
+    double azi12d = results.azimuth12;
+
+    // WGS84
+    bg::srs::spheroid<double> spheroid(6378137.0, 6356752.3142451793);
+
+    bg::formula::result_direct<double> result;
+
+    typedef bg::formula::karney_direct<double, true, true, true, true, 8> ka_t;
+    result = ka_t::apply(lon1d, lat1d, distance, azi12d, spheroid);
+    check_direct(result, results.karney, results.karney, 0.0000001, true);
+}
+
+int test_main(int, char*[])
+{
+    for (size_t i = 0; i < expected_size; ++i)
+    {
+        test_all(expected[i]);
+
+        if (expected[i].p1.lon == 0 && expected[i].p1.lat == 0)
+        {
+            test_all(symmetric_wrt_origin(expected[i]));
+        }
+    }
+
+    for (size_t i = 0; i < expected_size_antipodal; ++i)
+    {
+        test_karney_antipodal(expected_antipodal[i]);
+    }
+
+    return 0;
+}
diff --git a/src/boost/libs/geometry/test/formulas/direct_accuracy.cpp b/src/boost/libs/geometry/test/formulas/direct_accuracy.cpp
new file mode 100644
index 00000000..e485caaf
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/direct_accuracy.cpp
@@ -0,0 +1,97 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2019 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Vissarion Fysikopoulos, 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)
+
+#include <geometry_test_common.hpp>
+
+#include <boost/geometry/formulas/karney_direct.hpp>
+#include <boost/geometry/srs/srs.hpp>
+
+#ifdef BOOST_GEOEMTRY_TEST_WITH_GEOGRAPHICLIB
+#include <GeographicLib/Geodesic.hpp>
+#include <GeographicLib/Constants.hpp>
+#endif // BOOST_GEOEMTRY_TEST_WITH_GEOGRAPHICLIB
+
+int test_main(int, char*[])
+{
+
+#ifdef BOOST_GEOEMTRY_TEST_WITH_GEOGRAPHICLIB
+    // accuracy test from https://github.com/boostorg/geometry/issues/560
+    using namespace GeographicLib;
+
+    const long double wgs84_a = 6378137.0L;
+    const long double wgs84_f = 1.0L / 298.257223563L;
+    const long double one_minus_f = 1.0L - wgs84_f;
+    const long double wgs84_b = wgs84_a * one_minus_f;
+
+    const boost::geometry::srs::spheroid<long double> BoostWGS84(wgs84_a, wgs84_b);
+
+    // boost karney_direct function class with azimuth output and SeriesOrder = 6
+    typedef boost::geometry::formula::karney_direct <double, true, true, false, false, 6u>
+            BoostKarneyDirect_6;
+
+    // boost karney_direct function class with azimuth output and SeriesOrder = 8
+    typedef boost::geometry::formula::karney_direct <double, true, true, false, false, 8u>
+            BoostKarneyDirect_8;
+
+    // boost test BOOST_CHECK_CLOSE macro takes a percentage accuracy parameter
+    const double EPSILON = std::numeric_limits<double>::epsilon();
+    const double CALCULATION_TOLERANCE = 100 * EPSILON;
+
+    const Geodesic GeographicLibWGS84(Geodesic::WGS84());
+
+    // Loop around latitudes: 0 to 89 degrees
+    for (int i=0; i < 90; ++i)
+    {
+        // The latitude in degrees.
+        double latitude(1.0 * i);
+
+        // Loop around longitudes: 1 to 179 degrees
+        for (int j=1; j < 180; ++j)
+        {
+            // The longitude in degrees.
+            double longitude(1.0 * j);
+
+            // The Geodesic: distance in metres, start azimuth and finish azimuth in degrees.
+            double distance_m, azimuth, azi2;
+            GeographicLibWGS84.Inverse(0.0, 0.0, latitude, longitude, distance_m, azimuth, azi2);
+
+            // The GeographicLib position and azimuth at the distance in metres
+            double lat2k, lon2k, azi2k;
+            GeographicLibWGS84.Direct(0.0, 0.0, azimuth, distance_m, lat2k, lon2k, azi2k);
+            BOOST_CHECK_CLOSE(latitude, lat2k, 140 * CALCULATION_TOLERANCE);
+            BOOST_CHECK_CLOSE(longitude, lon2k, 120 * CALCULATION_TOLERANCE);
+
+            // The boost karney_direct order 6 position at the azimuth and distance in metres.
+            boost::geometry::formula::result_direct<double> results_6
+                    = BoostKarneyDirect_6::apply(0.0, 0.0, distance_m, azimuth, BoostWGS84);
+            BOOST_CHECK_CLOSE(azi2, results_6.reverse_azimuth, 140 * CALCULATION_TOLERANCE);
+            BOOST_CHECK_CLOSE(latitude, results_6.lat2, 220 * CALCULATION_TOLERANCE);
+
+            /******** Test below only passes with >= 10172000 * CALCULATION_TOLERANCE !! ********/
+            BOOST_CHECK_CLOSE(longitude, results_6.lon2, 10171000 * CALCULATION_TOLERANCE);
+            /*****************************************************************************/
+
+            // The boost karney_direct order 8 position at the azimuth and distance in metres.
+            boost::geometry::formula::result_direct<double> results_8
+                    = BoostKarneyDirect_8::apply(0.0, 0.0, distance_m, azimuth, BoostWGS84);
+            BOOST_CHECK_CLOSE(azi2, results_8.reverse_azimuth, 140 * CALCULATION_TOLERANCE);
+            BOOST_CHECK_CLOSE(latitude, results_8.lat2, 220 * CALCULATION_TOLERANCE);
+
+            /******** Test below only passes with >= 10174000 * CALCULATION_TOLERANCE !! ********/
+            BOOST_CHECK_CLOSE(longitude, results_8.lon2, 10173000 * CALCULATION_TOLERANCE);
+            /*****************************************************************************/
+        }
+    }
+#endif // BOOST_GEOEMTRY_TEST_WITH_GEOGRAPHICLIB
+
+    return 0;
+}
+
diff --git a/src/boost/libs/geometry/test/formulas/direct_cases.hpp b/src/boost/libs/geometry/test/formulas/direct_cases.hpp
new file mode 100644
index 00000000..1ac436e6
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/direct_cases.hpp
@@ -0,0 +1,700 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2016-2018 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Vissarion Fysikopoulos, on behalf of Oracle
+// 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_TEST_DIRECT_CASES_HPP
+#define BOOST_GEOMETRY_TEST_DIRECT_CASES_HPP
+
+struct coordinates
+{
+    double lon;
+    double lat;
+};
+
+struct expected_result
+{
+    double lon2;
+    double lat2;
+    double reverse_azimuth;
+    double reduced_length;
+    double geodesic_scale;
+};
+
+struct expected_results
+{
+    coordinates p1;
+    double distance;
+    double azimuth12;
+    expected_result karney;
+    expected_result vincenty;
+    expected_result thomas;
+    expected_result thomas1st;
+    expected_result series;
+    expected_result spherical;
+};
+
+expected_results expected[] =
+{
+
+        {
+            { 0, 0 }, 250000, 0,
+            { 0.00000000000000000000, 2.26091191238511868278, 0.00000000000000000000, 249935.55905595037620514631, 0.99922674639115516282 },
+            { 0.00000000000000000000, 2.26091191236402178077, 0.00000000000000000000, 249935.55905951990280300379, 0.99922674639116959572 },
+            { 0.00000000000000000000, 2.26091189386641744363, 0.00000000000000000000, 249935.55701571033569052815, 0.99922674640382092015 },
+            { 0.00000000000000000000, 2.26091190837723177154, 0.00000000000000000000, 249935.55863258230965584517, 0.99922674639389641449 },
+            { 0.00000000000000000000, 2.26091191238508226746, 0.00000000000000000000, 249935.55905602785060182214, 0.99922674639115527384 },
+            { 0.00000000000000000000, 2.24830091939884413321, 0.00000000000000000000, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 0, 0 }, 250000, 45,
+            { 1.58842150169031337548, 1.59850419267109766785, 45.02216068943542381930, 249935.55885449703782796860, 0.99922674504834751996 },
+            { 1.58842150168977558344, 1.59850419267017707092, 45.02216068943540250302, 249935.55886070139240473509, 0.99922674504834840814 },
+            { 1.58842149958854261804, 1.59850419056543535667, 45.02216068937701010100, 249935.55853169565671123564, 0.99922674505038500126 },
+            { 1.58842150075770360829, 1.59850419406187715943, 45.02216068947401339528, 249935.55908163820276968181, 0.99922674504700181863 },
+            { 1.58842150169031270934, 1.59850419267109744581, 45.02216068943543092473, 249935.55885453775408677757, 0.99922674504834751996 },
+            { 1.59019688120571478507, 1.58958477520588337129, 45.02206163500609648054, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 0, 0 }, 250000, 90,
+            { 2.24578821029880382198, 0.00000000000000000000, 90.00000000000000000000, 249935.55865304186590947211, 0.99922674370552955203 },
+            { 2.24578821029880382198, 0.00000000000000013841, 90.00000000000000000000, 249935.55865304186590947211, 0.99922674370552955203 },
+            { 2.24578821029880382198, 0.00000000000000013841, 90.00000000000000000000, 249935.55865304186590947211, 0.99922674370552955203 },
+            { 2.24578821029880382198, 0.00000000000000013841, 90.00000000000000000000, 249935.55865304186590947211, 0.99922674370552955203 },
+            { 2.24578821029880382198, 0.00000000000000013841, 90.00000000000000000000, 249935.55865304186590947211, 0.99922674370552955203 },
+            { 2.24830091939884413321, 0.00000000000000013763, 90.00000000000000000000, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 0, 0 }, 250000, 135,
+            { 1.58842150169031337548, -1.59850419267109766785, 134.97783931056457618070, 249935.55885449703782796860, 0.99922674504834751996 },
+            { 1.58842150168977580549, -1.59850419267017707092, 134.97783931056460460240, 249935.55886070139240473509, 0.99922674504834840814 },
+            { 1.58842149958854261804, -1.59850419056543602281, 134.97783931062298279357, 249935.55853169583133421838, 0.99922674505038500126 },
+            { 1.58842150075770360829, -1.59850419406187760352, 134.97783931052597949929, 249935.55908163831918500364, 0.99922674504700181863 },
+            { 1.58842150169033402562, -1.59850419267108945220, 134.97783931056457618070, 249935.55885453653172589839, 0.99922674504834751996 },
+            { 1.59019688120571345280, -1.58958477520588448151, 134.97793836499391773032, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 0, 0 }, 250000, 180,
+            { 0.00000000000000000000, -2.26091191238511868278, -180.00000000000000000000, 249935.55905595037620514631, 0.99922674639115516282 },
+            { 0.00000000000000027517, -2.26091191236402178077, 180.00000000000000000000, 249935.55905951990280300379, 0.99922674639116959572 },
+            { 0.00000000000000000000, -2.26091189386641744363, 180.00000000000000000000, 249935.55701571033569052815, 0.99922674640382092015 },
+            { 0.00000000000000000000, -2.26091190837723177154, 180.00000000000000000000, 249935.55863258230965584517, 0.99922674639389641449 },
+            { 0.00000000000000000000, -2.26091191238511468597, 180.00000000000000000000, 249935.55905603148858062923, 0.99922674639115527384 },
+            { 0.00000000000000000000, -2.24830091939884635366, 180.00000000000000000000, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 0, 0 }, 250000, -45,
+            { -1.58842150169031337548, 1.59850419267109766785, -45.02216068943542381930, 249935.55885449703782796860, 0.99922674504834751996 },
+            { -1.58842150168977558344, 1.59850419267017707092, -45.02216068943540250302, 249935.55886070139240473509, 0.99922674504834840814 },
+            { -1.58842149958854261804, 1.59850419056543535667, -45.02216068937701010100, 249935.55853169565671123564, 0.99922674505038500126 },
+            { -1.58842150075770360829, 1.59850419406187715943, -45.02216068947401339528, 249935.55908163820276968181, 0.99922674504700181863 },
+            { -1.58842150169031270934, 1.59850419267109744581, -45.02216068943543092473, 249935.55885453775408677757, 0.99922674504834751996 },
+            { -1.59019688120571478507, 1.58958477520588337129, -45.02206163500609648054, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 0, 0 }, 250000, -90,
+            { -2.24578821029880382198, 0.00000000000000000000, -90.00000000000000000000, 249935.55865304186590947211, 0.99922674370552955203 },
+            { -2.24578821029880382198, 0.00000000000000013841, -90.00000000000000000000, 249935.55865304186590947211, 0.99922674370552955203 },
+            { -2.24578821029880382198, 0.00000000000000013841, -90.00000000000000000000, 249935.55865304186590947211, 0.99922674370552955203 },
+            { -2.24578821029880382198, 0.00000000000000013841, -90.00000000000000000000, 249935.55865304186590947211, 0.99922674370552955203 },
+            { -2.24578821029880382198, 0.00000000000000013841, -90.00000000000000000000, 249935.55865304186590947211, 0.99922674370552955203 },
+            { -2.24830091939884413321, 0.00000000000000013763, -90.00000000000000000000, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 0, 0 }, 250000, -135,
+            { -1.58842150169031337548, -1.59850419267109766785, -134.97783931056457618070, 249935.55885449703782796860, 0.99922674504834751996 },
+            { -1.58842150168977580549, -1.59850419267017707092, -134.97783931056460460240, 249935.55886070139240473509, 0.99922674504834840814 },
+            { -1.58842149958854261804, -1.59850419056543602281, -134.97783931062298279357, 249935.55853169583133421838, 0.99922674505038500126 },
+            { -1.58842150075770360829, -1.59850419406187760352, -134.97783931052597949929, 249935.55908163831918500364, 0.99922674504700181863 },
+            { -1.58842150169033402562, -1.59850419267108945220, -134.97783931056457618070, 249935.55885453653172589839, 0.99922674504834751996 },
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+        },{
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+            { 3.24612766651409989649, -0.99922666842209439952, 89.96080977150054991398, 249935.55891569345840252936, 0.99922674685767398639 },
+            { 3.24612766651347328661, -0.99922666842209439952, 89.96080977150054991398, 249935.55891513984533958137, 0.99922674686822765544 },
+            { 3.24612766651409989649, -0.99922666842208529570, 89.96080977150032254031, 249935.55891660030465573072, 0.99922674686821844059 },
+            { 3.24612766651409812013, -0.99922666841944873806, 89.96080977143354573400, 249935.55934300241642631590, 0.99922674685495815883 },
+            { 3.24612766651407813612, -0.99922666842209184601, 89.96080977150054991398, 249935.55891566007630899549, 0.99922674685767387537 },
+            { 3.24864304714222429382, -0.99923012194947846698, 89.96076583819413485799, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 1, -1 }, 250000, 135,
+            { 2.58943065821228035617, -2.59810442478829362045, 134.95009296248679220298, 249935.55953696221695281565, 0.99922675155892082266 },
+            { 2.58943065821171281016, -2.59810442478739078709, 134.95009296248684904640, 249935.55953448649961501360, 0.99922675162094776180 },
+            { 2.58943065613921019619, -2.59810442273856878614, 134.95009296257936171060, 249935.55921406237757764757, 0.99922675162293117523 },
+            { 2.58943065729835319999, -2.59810443328209572655, 134.95009296210329807764, 249935.56090466637397184968, 0.99922675155085038945 },
+            { 2.58943065821226792167, -2.59810442478831138402, 134.95009296248679220298, 249935.55953712956397794187, 0.99922675155892148879 },
+            { 2.59120965314638285193, -2.58919942720575324557, 134.95016080642784572774, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 1, -1 }, 250000, 180,
+            { 1.00000000000000000000, -3.26088938011404927408, -180.00000000000000000000, 249935.55991226507467217743, 0.99922675429255258805 },
+            { 1.00000000000000022204, -3.26088938009322193423, 180.00000000000000000000, 249935.55988221566076390445, 0.99922675435769237051 },
+            { 1.00000000000000000000, -3.26088936183515265554, 180.00000000000000000000, 249935.55786483851261436939, 0.99922675437017982603 },
+            { 1.00000000000000000000, -3.26088938879229361945, 180.00000000000000000000, 249935.56092769638053141534, 0.99922675428677187881 },
+            { 1.00000000000000000000, -3.26088938011400975014, 180.00000000000000000000, 249935.55991249711951240897, 0.99922675429255336521 },
+            { 0.99999999999998867573, -3.24830091939884901819, 180.00000000000000000000, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 1, -1 }, 250000, -45,
+            { -0.58789352027561725400, 0.59889382252477618707, -44.99444098733987829064, 249935.55869720416376367211, 0.99922674484119022864 },
+            { -0.58789352027508279264, 0.59889382252384637528, -44.99444098733987118521, 249935.55869934445945546031, 0.99922674484277662632 },
+            { -0.58789351816042800802, 0.59889382039736038799, -44.99444098731778751699, 249935.55836710281437262893, 0.99922674484483320345 },
+            { -0.58789351933467837164, 0.59889382227058296504, -44.99444098733723507166, 249935.55866038752719759941, 0.99922674484142481877 },
+            { -0.58789352027561669889, 0.59889382252477563195, -44.99444098733987829064, 249935.55869721624185331166, 0.99922674484119011762 },
+            { -0.58966906957485232255, 0.58996975014221642031, -44.99431096332273227745, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 1, -1 }, 250000, -90,
+            { -1.24612766651409989649, -0.99922666842209439952, -89.96080977150054991398, 249935.55891569345840252936, 0.99922674685767398639 },
+            { -1.24612766651347328661, -0.99922666842209439952, -89.96080977150054991398, 249935.55891513984533958137, 0.99922674686822765544 },
+            { -1.24612766651409945240, -0.99922666842208529570, -89.96080977150032254031, 249935.55891660030465573072, 0.99922674686821844059 },
+            { -1.24612766651409834218, -0.99922666841944873806, -89.96080977143354573400, 249935.55934300241642631590, 0.99922674685495815883 },
+            { -1.24612766651407835816, -0.99922666842209184601, -89.96080977150054991398, 249935.55891566007630899549, 0.99922674685767387537 },
+            { -1.24864304714222140724, -0.99923012194947846698, -89.96076583819413485799, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 1, -1 }, 250000, -135,
+            { -0.58943065821228057821, -2.59810442478829362045, -134.95009296248679220298, 249935.55953696221695281565, 0.99922675155892082266 },
+            { -0.58943065821171258811, -2.59810442478739078709, -134.95009296248684904640, 249935.55953448649961501360, 0.99922675162094776180 },
+            { -0.58943065613921030721, -2.59810442273856878614, -134.95009296257936171060, 249935.55921406237757764757, 0.99922675162293117523 },
+            { -0.58943065729835331101, -2.59810443328209572655, -134.95009296210329807764, 249935.56090466637397184968, 0.99922675155085038945 },
+            { -0.58943065821226814371, -2.59810442478831138402, -134.95009296248679220298, 249935.55953712956397794187, 0.99922675155892148879 },
+            { -0.59120965314640527843, -2.58919942720575324557, -134.95016080642784572774, 0.00000000000000000000, 1.00000000000000000000 }
+        },{
+            { 1, -1 }, 250000, -180,
+            { 1.00000000000000000000, -3.26088938011404927408, -180.00000000000000000000, 249935.55991226507467217743, 0.99922675429255258805 },
+            { 0.99999999999999977796, -3.26088938009322193423, -180.00000000000000000000, 249935.55988221566076390445, 0.99922675435769237051 },
+            { 1.00000000000000000000, -3.26088936183515265554, -180.00000000000000000000, 249935.55786483851261436939, 0.99922675437017982603 },
+            { 1.00000000000000000000, -3.26088938879229361945, -180.00000000000000000000, 249935.56092769638053141534, 0.99922675428677187881 },
+            { 1.00000000000000000000, -3.26088938011400975014, -180.00000000000000000000, 249935.55991249711951240897, 0.99922675429255336521 },
+            { 0.99999999999998867573, -3.24830091939884901819, -180.00000000000000000000, 0.00000000000000000000, 1.00000000000000000000 }
+        }
+
+};
+
+size_t const expected_size = sizeof(expected) / sizeof(expected_results);
+#endif // BOOST_GEOMETRY_TEST_DIRECT_CASES_HPP
diff --git a/src/boost/libs/geometry/test/formulas/direct_cases_antipodal.hpp b/src/boost/libs/geometry/test/formulas/direct_cases_antipodal.hpp
new file mode 100644
index 00000000..c6ba8524
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/direct_cases_antipodal.hpp
@@ -0,0 +1,343 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2018 Adeel Ahmad, Islamabad, Pakistan.
+
+// Contributed and/or modified by Adeel Ahmad, as part of Google Summer of Code 2018 program.
+
+// 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_TEST_DIRECT_CASES_ANTIPODAL_HPP
+#define BOOST_GEOMETRY_TEST_DIRECT_CASES_ANTIPODAL_HPP
+
+#include "direct_cases.hpp"
+
+struct expected_results_antipodal
+{
+    coordinates p1;
+    double distance;
+    double azimuth12;
+    expected_result karney;
+};
+
+/*
+ These values are collected from GeodTest which is associated with GeographicLib:
+     https://zenodo.org/record/32156
+
+ The conversion to C++ array format is done using this Python script:
+     https://github.com/adl1995/boost-geometry-extra/blob/master/geographicLib-dataset-parse.py
+
+ Geodesic scale (M12) is absent from the GeodTest dataset, so it is manually generated
+ using GeographicLib through this C++ script:
+     https://github.com/adl1995/boost-geometry-extra/blob/master/geographicLib-direct-antipodal.cpp
+*/
+expected_results_antipodal expected_antipodal[] =
+{
+    {
+        { 0, 31.394417440639 }, 19980218.4055399, 34.266322930672,
+        { 179.615601631202912322, -31.275540610835465807, 145.782701113414306756, 49490.8807994496209, -0.996116451012525883079717914370121434 }
+    },{
+        { 0, 29.788792273749 }, 19887224.5407334, 74.302205994192,
+        { 178.569451327813675741, -29.558013672069422725, 106.156240654579267308, 97043.7545600593058, -0.998624031147844926081802441331092268 }
+    },{
+        { 0, 46.471843094141 }, 19944337.8863917, 63.693680310665,
+        { 179.083144618009561276, -46.284166405924629853, 116.699978859005570535, 53139.140576552365, -0.997597309645591900917338534782174975 }
+    },{
+        { 0, 63.016506345929 }, 20000925.7533636, 153.393656073038,
+        { 179.862869954071637855, -63.02943882703369735, 26.619056019474552953, 12713.9284725111772, -1.00381317792143387457315384381217882 }
+    },{
+        { 0, 19.796231412719 }, 19956338.1330537, 28.272934411318,
+        { 179.546498474461283862, -19.470586923091672503, 151.789094611690988249, 87191.1749625132931, -0.997015409027664833985227232915349305 }
+    },{
+        { 0, 6.373459459035 }, 19946581.6983394, 56.859050230583,
+        { 179.240009269347556917, -6.204887833274217382, 123.169200847008284851, 53958.8698005263939, -0.999349049081101004077254401636309922 }
+    },{
+        { 0, 66.380766469414 }, 19986277.7696849, 38.646950203356,
+        { 179.632633596894388233, -66.27177494016956425, 141.550919825824399405, 22198.215635049214, -0.996949176054954366854587988200364634 }
+    },{
+        { 0, 16.483421185231 }, 19962737.9842573, 163.431254767325,
+        { 179.731567273052604726, -16.818424446748042212, 16.598399455529231288, 95318.4104529881431, -1.00272210232979741562076014815829694 }
+    },{
+        { 0, 4.215702155486 }, 19932517.393764, 65.543168480886,
+        { 179.093771177769992874, -4.051917290690976764, 114.482669479963380006, 55205.4553703842317, -0.999655858425056553784315838129259646 }
+    },{
+        { 0, 40.71372085907 }, 19951133.3595356, 143.672151631634,
+        { 179.404612926861498984, -41.047052242159400671, 36.54002600969304553, 70931.1530155553621, -1.00414169574077272173440178448799998 }
+    },{
+        { 0, 15.465481491654 }, 19877383.8879911, 36.289185640976,
+        { 179.020726605204181801, -14.622355549425900341, 143.875673907461159912, 156419.0806764376957, -0.997639074397169589580869342171354219 }
+    },{
+        { 0, 17.586197343531 }, 19982280.4639115, 157.929615091529,
+        { 179.722490735835379144, -17.731394230364437075, 22.089021105298661023, 69727.5357849255557, -1.00280451698301242835498214844847098 }
+    },{
+        { 0, 5.7442768247 }, 19902873.7431814, 116.146983678305,
+        { 178.85894724576868462, -6.039853564481335581, 63.91482549951374061, 87149.6188944111673, -1.00039332893096744037109147029696032 }
+    },{
+        { 0, 32.002904282111 }, 19967670.3104795, 163.052160078191,
+        { 179.744925422107715439, -32.297934520693132807, 17.004175883388454943, 78311.3164829640582, -1.00449903445302446414189034840092063 }
+    },{
+        { 0, 55.902716926362 }, 19970525.337607, 98.927641063414,
+        { 179.300685189522463007, -55.934320218634018206, 81.374264168520557301, 23554.0093185709067, -1.00072788779083454713259015989024192 }
+    },{
+        { 0, 22.69939784398 }, 19959286.1903172, 74.253870776761,
+        { 179.294173474584020749, -22.654875407651067149, 105.811588890213155275, 22369.7179951557679, -0.998972181419003457669703038845909759 }
+    },{
+        { 0, 41.312328471121 }, 19962690.5721867, 11.277616109847,
+        { 179.817186837717804928, -40.954523601529804886, 168.784288786443902199, 77252.6121237260201, -0.994825151471527391322524636052548885 }
+    },{
+        { 0, 27.927415327453 }, 19961296.8828333, 23.166421459647,
+        { 179.636508875679110143, -27.607314264234172721, 156.905194492817275222, 83096.5801709291101, -0.995959692767656723511038308060960844 }
+    },{
+        { 0, 41.567228741451 }, 19944253.4454809, 176.66609526064,
+        { 179.931812964300204608, -42.103039532074194347, 3.361859685835349219, 96859.08180779197, -1.00513607140487626345759508694754913 }
+    },{
+        { 0, 37.384208978567 }, 19928705.5911445, 39.072534864532,
+        { 179.225180174670992261, -36.916085670712060029, 141.212743814390850106, 92667.7834060578402, -0.995955516859159284415170532156480476 }
+    },{
+        { 0, 59.011868682852 }, 19970442.3788306, 44.970301291063,
+        { 179.424923485514312807, -58.82705468054708336, 135.333817989802309531, 38071.1136293083857, -0.996658942892707400140750451100757346 }
+    },{
+        { 0, 35.515406087737 }, 19948918.9139751, 28.528972431952,
+        { 179.50369572149476218, -35.119747127350258822, 151.622257906284404073, 84564.0387217601751, -0.995562861799169418475230486365035176 }
+    },{
+        { 0, 58.170252463184 }, 19961407.0813807, 128.021116291844,
+        { 179.254737571455023977, -58.372261836268550805, 52.399129705193347143, 43715.3070711393309, -1.00285273713280753682397516968194395 }
+    },{
+        { 0, 34.012183807959 }, 19970955.843065, 168.944519134772,
+        { 179.83713352180447672, -34.29640782899529639, 11.093048811826875835, 76493.5814538538151, -1.0047652354558671561335359001532197 }
+    },{
+        { 0, 45.510762948553 }, 19940248.3450143, 99.886784003837,
+        { 178.981682578823726535, -45.582753595227824235, 80.542330522982505877, 48555.1946627894972, -1.00083807750906350619857221317943186 }
+    },{
+        { 0, 4.19841765451 }, 19970496.5132933, 89.561550657928,
+        { 179.398024428225540172, -4.198416896099783242, 90.438456568689151881, 14.8790480103109, -0.999994104810285944218151144013972953 }
+    },{
+        { 0, 40.890119148103 }, 19926563.5817492, 165.437641169967,
+        { 179.6557148951668192, -41.553556264538302258, 14.713597527941311478, 111805.7305227545923, -1.00492294933406567380984597548376769 }
+    },{
+        { 0, 28.096672787686 }, 19883901.8482359, 115.174366374632,
+        { 178.606868012231657724, -28.472055035513955205, 65.257367020445564176, 107880.4353518862363, -1.00170803073331593502359737613005564 }
+    },{
+        { 0, 6.50572154271 }, 19917276.4101551, 79.069492719523,
+        { 178.926013840891647541, -6.411745140559297675, 100.985091481519557845, 57073.3242952680707, -0.999736666933808471036115861352300271 }
+    },{
+        { 0, .468835109567 }, 19849380.7342734, 80.234636214474,
+        { 178.325942223692180692, -.281751687044281805, 99.77243368342786593, 123845.4568822078908, -0.999801437209140719808431185811059549 }
+    },{
+        { 0, 1.682746325049 }, 19890026.0274781, 10.076182752451,
+        { 179.717131561406935483, -.677647430701204515, 169.927471515299313238, 177917.2104306563981, -0.999538055691262194990542866435134783 }
+    },{
+        { 0, 10.711305126218 }, 19962987.2134077, 7.528253696796,
+        { 179.874050163405229937, -10.349315378531556046, 172.480576051850009046, 104175.1095378254456, -0.998071853755238880268052525934763253 }
+    },{
+        { 0, 53.374321544652 }, 19980478.1457438, 23.324715976877,
+        { 179.729445806011012057, -53.196257519024042184, 156.777734080146664812, 41907.8869272231053, -0.995333596277707566279957518418086693 }
+    },{
+        { 0, 39.680221664519 }, 19956191.7841809, 7.075406493429,
+        { 179.87506206720154785, -39.256187213040660911, 172.967624741991546131, 86943.8110669895148, -0.994801087909667924868983845954062417 }
+    },{
+        { 0, 1.377666714083 }, 19925401.4931301, 95.29199069739,
+        { 178.994542525209058878, -1.415358715570225495, 84.7178724483824156, 45800.9140624827059, -0.99999803170512457928253979844157584 }
+    },{
+        { 0, 48.751426624188 }, 19988599.1160495, 40.252328570137,
+        { 179.661697715070846977, -48.688146707479475147, 139.808452951157199824, 26322.3790862461568, -0.995999245724129789181233718409202993 }
+    },{
+        { 0, 59.443039048494 }, 19969935.9534732, 93.052184108221,
+        { 179.247605418616998285, -59.454371825393424121, 87.331416513795326158, 25342.4691896499534, -1.00020727848897084122370415570912883 }
+    },{
+        { 0, 4.122408476235 }, 19938291.6332293, 167.73479753304,
+        { 179.749430572914989772, -4.689124208743755363, 12.274635577599782826, 127855.6475863583497, -1.00068600902837667732114823593292385 }
+    },{
+        { 0, 46.422470082432 }, 19931980.7029341, 86.67365350297,
+        { 178.857408435141563774, -46.390934261324541952, 93.852683224054943377, 56114.680046867064, -0.999607096116300386512421027873642743 }
+    },{
+        { 0, 32.614423729024 }, 19926887.3785175, 24.943814520557,
+        { 179.460593512880455451, -32.01874745886238612, 155.229917137448282531, 112355.3319340873104, -0.995562150676871926435751447570510209 }
+    },{
+        { 0, 3.242895277973 }, 19964490.4789049, 30.247458779683,
+        { 179.556428318080663113, -3.001106476068264917, 149.760178923092147784, 80929.0418317066044, -0.999474184270344845337774586369050667 }
+    },{
+        { 0, 6.29069210113 }, 19877160.8505733, 94.34299459284,
+        { 178.556859259685624933, -6.354208910915346725, 85.750059038253282986, 94127.1566760840083, -0.999976397350904933070125935046235099 }
+    },{
+        { 0, 18.232086569498 }, 19927978.7462175, 164.41905055334,
+        { 179.658073278238477245, -18.87394850776853555, 15.640779355822506503, 129771.1882449660559, -1.00293460439063886191490837518358603 }
+    },{
+        { 0, 12.049849333181 }, 19908004.4552909, 9.418096768309,
+        { 179.761046682699610657, -11.201990279782499264, 170.610608272305604585, 157761.5040571466343, -0.997761474497510958414636661473196 }
+    },{
+        { 0, 40.289465276136 }, 19985674.936106, 143.092606818963,
+        { 179.644208494155329095, -40.370034926441385999, 36.958610382613096419, 36200.8933724688593, -1.00414965876091266672176516294712201 }
+    },{
+        { 0, 2.197784650379 }, 19910509.7517973, 1.542117609437,
+        { 179.961199531084784854, -1.353440827124394777, 178.458582198505846426, 160403.6285079348996, -0.999488724639301051588802238256903365 }
+    },{
+        { 0, 1.966575272177 }, 19875595.6267266, 170.112968791865,
+        { 179.699817324905962184, -3.101125282483752618, 9.89572776349855838, 192355.7206665719908, -1.00015463589804554089823795948177576 }
+    },{
+        { 0, 25.078832492684 }, 19887997.7953866, 77.264585323781,
+        { 178.600804840925824646, -24.897833702325682511, 103.101167809583406892, 92442.9124509225839, -0.998981189838600847075156252685701475 }
+    },{
+        { 0, 31.740361941314 }, 19972325.3556069, 143.930820896999,
+        { 179.553485210731879874, -31.909206787477701871, 36.145242998351638503, 54883.4113710054145, -1.00379461628115951299378139083273709 }
+    },{
+        { 0, .05479250563 }, 19858049.4780499, 41.349430623518,
+        { 178.822647462220726609, .836079031223269324, 138.645259065012502544, 169078.442370111714, -0.9997793696948588104689292777038645 }
+    },{
+        { 0, 36.685139871608 }, 19968965.6773632, 89.167975517493,
+        { 179.366667224014334712, -36.6833040833258687, 90.921025521408327068, 13327.2156799476918, -0.999916537946348604748436628142371774 }
+    },{
+        { 0, 3.451199399671 }, 19938203.3838544, 91.541212417048,
+        { 179.107509334399258305, -3.459003521120242021, 88.476282464773035164, 32316.1747698810781, -1.00000397484395819880376166111091152 }
+    },{
+        { 0, 27.692898794247 }, 19883493.6699045, 88.406440883665,
+        { 178.512356615673144314, -27.666009301228316555, 92.036345087713397961, 94128.7880896190836, -0.999736458322951659916100197733612731 }
+    },{
+        { 0, 17.363238291869 }, 19980749.7638027, 39.697196316589,
+        { 179.567921315455829491, -17.288872648596950413, 140.321938237586060826, 46975.9359427664379, -0.997687691981715030209443284547887743 }
+    },{
+        { 0, 37.006775102539 }, 19949309.9180043, 116.455543532607,
+        { 179.191103068859169842, -37.156365616364686838, 63.771817992036617793, 45856.1961421018701, -1.00221962858918423044940482213860378 }
+    },{
+        { 0, 45.572883540957 }, 19940027.8586414, 137.627256708444,
+        { 179.224707765088686272, -45.94675931323086696, 42.723991162977357301, 74208.4359612889496, -1.00380887786447159371050474874209613 }
+    },{
+        { 0, 43.63393981955 }, 19931045.2914508, 91.203625101465,
+        { 178.878236417027994157, -43.642335115130514773, 89.268780774643462256, 55253.5406349861764, -1.00002974153150514524668324156664312 }
+    },{
+        { 0, 38.4995307019 }, 19918391.2222193, 141.232864609445,
+        { 179.143856004445269342, -39.042223438550921467, 39.117947060740562295, 102217.2563106863077, -1.00388164115732947401227193040540442 }
+    },{
+        { 0, 27.55015339382 }, 19986004.7358853, 137.025135713548,
+        { 179.596220103573824099, -27.587412128122249651, 42.992898351962011956, 33938.7346646670654, -1.00316044390281167153489150223322213 }
+    },{
+        { 0, 1.54507498314 }, 19978593.3191777, 36.816106412092,
+        { 179.567115633151308577, -1.448861185025252004, 143.185763012309022403, 56320.5800276739168, -0.999770499462467210349814195069484413 }
+    },{
+        { 0, 45.217063644222 }, 19987042.0782465, 18.114645812265,
+        { 179.807382581661125, -45.086424050571516283, 161.928120141429818658, 45544.2915061261936, -0.994974179414854997816064496873877943 }
+    },{
+        { 0, 13.473522450751 }, 19987364.078382, 156.839609002403,
+        { 179.726941062277208626, -13.570372758027936877, 23.170293747820406391, 65329.9068132034472, -1.00219093189506569530067281448282301 }
+    },{
+        { 0, 6.287741997374 }, 19912159.8245954, 132.954797451112,
+        { 179.071252372259552052, -6.743450924917895817, 47.100789519677419746, 104772.4027498097375, -1.00071252411103017720961361192166805 }
+    },{
+        { 0, 7.639709001531 }, 19976374.3699535, 29.731916588299,
+        { 179.616156296978583335, -7.48702643786017917, 150.279582966919438164, 69224.6591757209539, -0.998789792086741234911073661351110786 }
+    },{
+        { 0, 5.893688050348 }, 19886907.2520668, 14.653438882877,
+        { 179.586212000450856399, -4.888408917114795625, 165.371181401863458848, 177183.5330818593022, -0.998794647031120752522781458537792787 }
+    },{
+        { 0, 61.997076235476 }, 19976288.2901729, 149.562797049254,
+        { 179.605779116829636081, -62.19593758437129915, 30.65850204223272625, 36696.2853801462176, -1.00373071432437144245852778112748638 }
+    },{
+        { 0, 50.507637741656 }, 19979542.5263293, 171.564028344478,
+        { 179.893569206021038536, -50.721890799900161112, 8.4746613464253591, 50644.5234828162697, -1.00508881632281776852266830246662721 }
+    },{
+        { 0, 7.484475238477 }, 19867425.2906303, 57.020570370985,
+        { 178.638400003000590878, -6.926155588124333461, 123.087267812322270238, 132929.2775641349633, -0.999097042677338120775232255255104974 }
+    },{
+        { 0, 56.851165323215 }, 19988235.9960515, 112.345749045605,
+        { 179.587046628550073045, -56.875248360744638525, 67.744017057185404441, 9971.0934553515518, -1.00182859249871403228837607457535341 }
+    },{
+        { 0, 10.692273150738 }, 19893210.3050033, 102.824601316946,
+        { 178.709520715733071393, -10.851727623036704339, 77.308514969817191459, 83032.7122948051111, -1.00034345584508432835946223349310458 }
+    },{
+        { 0, 46.694739303788 }, 19975447.9283188, 174.663684259477,
+        { 179.926838145841924189, -46.948618153686522669, 5.361568174833475454, 59614.5876209460645, -1.00520484875201732144489596976200119 }
+    },{
+        { 0, 15.804386137005 }, 19855850.8800526, 74.932089158884,
+        { 178.367587635209819128, -15.522042847777054984, 105.357235560913450667, 123350.4326645237628, -0.999091578546475345135036150168161839 }
+    },{
+        { 0, 4.371450175299 }, 19979071.1035552, 164.163592252794,
+        { 179.780887420199549421, -4.566109732313098407, 15.840695025950408814, 84137.2115482558728, -1.00076323969894742660358133434783667 }
+    },{
+        { 0, 30.894388279688 }, 19968681.8321577, 77.35154610481,
+        { 179.375426183521944524, -30.871308884744172663, 102.709506078439532936, 14048.0277985734058, -0.998975176336422854284080585784977302 }
+    },{
+        { 0, 9.541166838639 }, 19848553.7844137, 118.441353539081,
+        { 178.432934555386452839, -10.09982228112793472, 61.736686215549403663, 144831.1911566651614, -1.00060548620110489892454097571317106 }
+    },{
+        { 0, 8.489292700054 }, 19995477.1669578, 171.963952699866,
+        { 179.906698338023119097, -8.559237750032113623, 8.037517851139094467, 72192.60793572974, -1.00152068486306466965629624610301107 }
+    },{
+        { 0, 19.562401114224 }, 19893208.1788508, 126.362762598128,
+        { 178.838724116996037606, -20.05038360490599475, 53.875560227496658204, 112181.7524188837615, -1.00185202668802775249901060306001455 }
+    },{
+        { 0, 42.260350252749 }, 19942715.0054774, 170.703419847646,
+        { 179.807860448877064601, -42.79985897702184353, 9.377654670896439828, 96336.3477142010769, -1.00508642406443549077721399953588843 }
+    },{
+        { 0, 24.511403144656 }, 19924809.5184876, 102.913211410163,
+        { 178.957598444862223515, -24.616808725039883945, 77.297538210434837096, 55403.453072179318, -1.0008408309188838725134473861544393 }
+    },{
+        { 0, 20.844284170708 }, 19909084.6340808, 44.172784008084,
+        { 179.069258863637226633, -20.321320573298341477, 136.01627115731728436, 111009.0987238994608, -0.997389183621778974142557672166731209 }
+    },{
+        { 0, 2.426010809098 }, 19840940.6924189, 94.315194952561,
+        { 178.236397468862000784, -2.513715200833756776, 85.734896842737189557, 130002.6104886615638, -0.999825249844991659209370027383556589 }
+    },{
+        { 0, 6.600682554664 }, 19878412.28273, 168.167678684515,
+        { 179.646475458013797028, -7.699164822656561787, 11.861035812918738552, 187426.3958525886692, -1.00098284856064978498579876031726599 }
+    },{
+        { 0, 23.372339802326 }, 19899498.4582543, 161.197647943542,
+        { 179.499422665106094027, -24.239465200482591299, 18.932355367478826536, 151863.2545535951091, -1.00347666868431395492677893344080076 }
+    },{
+        { 0, 16.194668264095 }, 19874825.6683239, 148.942349959054,
+        { 179.115193814080201851, -17.129419031459576897, 31.225656401221968078, 166033.3161394594622, -1.00222032222233647935638600756647065 }
+    },{
+        { 0, 1.528726471528 }, 19897803.9939987, 69.212891442493,
+        { 178.791047180477802091, -1.282203000582034597, 110.802928803578167132, 85252.8333849204133, -0.999827144228156883265512533398577943 }
+    },{
+        { 0, 6.297249676078 }, 19864042.0495193, 56.274639904925,
+        { 178.623258703845895437, -5.709470001196540278, 123.817184177744186806, 137475.1283083659258, -0.999190450178399580671850799262756482 }
+    },{
+        { 0, 17.393540327984 }, 19962624.6302607, 107.855062015266,
+        { 179.330156510680163326, -17.431100690958209424, 72.181322855288535245, 19320.5501845044839, -1.00091841779689127989172447996679693 }
+    },{
+        { 0, 46.284685151236 }, 19990422.3478916, 14.758013867151,
+        { 179.852534804091121255, -46.176234945675219984, 165.271681964991897184, 42614.1796365710104, -0.994894592261839960656288894824683666 }
+    },{
+        { 0, 14.924320176299 }, 19891861.8615337, 31.446544793174,
+        { 179.195663739713760883, -14.125476432252858442, 148.678916887199611191, 149419.6596309045804, -0.997620142585332936313591289945179597 }
+    },{
+        { 0, 23.668824656069 }, 19938736.4442268, 148.091483667618,
+        { 179.409875478773990359, -24.107855233601412399, 32.02919257641173958, 97771.7687385830819, -1.00323262872000595891108787327539176 }
+    },{
+        { 0, 46.986276695896 }, 19968596.0414782, 174.796708941456,
+        { 179.92040916864362177, -47.301644191214905832, 5.234240076649939638, 66113.7417494369769, -1.00519095452608087093437916337279603 }
+    },{
+        { 0, 65.946144289524 }, 19993734.5109736, 25.375428509648,
+        { 179.808282612725835525, -65.871840130833632868, 154.703163938350061652, 18355.2254271672769, -0.996436935914610577569305860379245132 }
+    },{
+        { 0, 10.950298933293 }, 19975919.5586889, 28.779018914489,
+        { 179.624609619829763098, -10.787771536605316781, 151.238005588662201946, 70291.1998404303581, -0.998272071834115148902810688014142215 }
+    },{
+        { 0, 13.609869340778 }, 19913213.8514358, 129.616021271129,
+        { 179.035623147420893383, -14.023624108675206222, 50.506400999466711623, 97596.7664002074776, -1.00146664642314031645753402699483559 }
+    },{
+        { 0, 48.701427557433 }, 19972955.2699173, 102.875149183407,
+        { 179.385565054218238481, -48.735316652259656533, 77.294384444682547869, 18461.7742226227697, -1.00114676855429074464609584538266063 }
+    },{
+        { 0, 31.519172055785 }, 19952318.3772514, 26.247105619999,
+        { 179.555251675378549409, -31.140142027808697534, 153.865822276646938125, 86354.7117605101002, -0.995739948399825047786748655198607594 }
+    },{
+        { 0, 31.863784754278 }, 19993324.8682601, 29.572313410211,
+        { 179.722489476483407524, -31.826935359797657785, 150.440607907359037187, 41427.6181613499234, -0.995888009001147267440501309465616941 }
+    },{
+        { 0, 76.434608546092 }, 19997750.023578, 167.428385412814,
+        { 179.918287057674124459, -76.48787937532808951, 12.621032110142724567, 9619.5267710862108, -1.00233963893091582164629471662919968 }
+    },{
+        { 0, 73.114273316483 }, 19992866.6147806, 78.154765899661,
+        { 179.576736605988553624, -73.098788070892914568, 102.085693546950923465, 8580.6475692800946, -0.999384143308475469957841141876997426 }
+    },{
+        { 0, 1.125639056292 }, 19852573.5442848, 67.184842289382,
+        { 178.426819580880619395, -.694775021853292564, 112.831314850896246589, 132932.8743502563937, -0.999732957962833457266071945923613384 }
+    }
+};
+
+size_t const expected_size_antipodal = sizeof(expected_antipodal) / sizeof(expected_results_antipodal);
+
+#endif // BOOST_GEOMETRY_TEST_DIRECT_CASES_ANTIPODAL_HPP
diff --git a/src/boost/libs/geometry/test/formulas/direct_meridian.cpp b/src/boost/libs/geometry/test/formulas/direct_meridian.cpp
new file mode 100644
index 00000000..aa94413b
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/direct_meridian.cpp
@@ -0,0 +1,115 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2018 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Vissarion Fysikopoulos, 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)
+
+#define BOOST_GEOMETRY_NORMALIZE_LATITUDE
+
+#include <sstream>
+
+#include "test_formula.hpp"
+#include "direct_meridian_cases.hpp"
+
+#include <boost/geometry/srs/srs.hpp>
+#include <boost/geometry/formulas/vincenty_direct.hpp>
+#include <boost/geometry/formulas/meridian_direct.hpp>
+
+template <typename Result, typename Expected, typename Reference>
+void check_meridian_direct(Result& result,
+                           Expected const& expected,
+                           Reference& reference,
+                           double reference_error)
+{
+    boost::geometry::math::normalize_spheroidal_coordinates
+        <
+            boost::geometry::radian,
+            double
+        >(result.lon2, result.lat2);
+
+    boost::geometry::math::normalize_spheroidal_coordinates
+        <
+            boost::geometry::radian,
+            double
+        >(reference.lon2, reference.lat2);
+
+    std::stringstream ss;
+    ss << "(" << result.lon2 * bg::math::r2d<double>()
+       << " " << result.lat2 * bg::math::r2d<double>() << ")";
+
+    check_one("lon:" + ss.str(), result.lon2, expected.lon, reference.lon2,
+              reference_error);
+    check_one("lat:" + ss.str(), result.lat2, expected.lat, reference.lat2,
+              reference_error);
+    check_one("rev_az:" + ss.str(), result.reverse_azimuth,
+              result.reverse_azimuth, reference.reverse_azimuth, reference_error);
+    check_one("red len:" + ss.str(), result.reduced_length, result.reduced_length,
+              reference.reduced_length, 0.01);
+    check_one("geo scale:" + ss.str(), result.geodesic_scale, result.geodesic_scale,
+              reference.geodesic_scale, 0.01);
+
+}
+
+void test_all(expected_results const& results)
+{
+    double const d2r = bg::math::d2r<double>();
+
+    double lon1_rad = results.p1.lon * d2r;
+    double lat1_rad = results.p1.lat * d2r;
+    coordinates expected_point;
+    expected_point.lon = results.p2.lon * d2r;
+    expected_point.lat = results.p2.lat * d2r;
+    double distance = results.distance;
+    bool direction = results.direction;
+
+    // WGS84
+    bg::srs::spheroid<double> spheroid(6378137.0, 6356752.3142451793);
+
+    bg::formula::result_direct<double> vincenty_result;
+    bg::formula::result_direct<double> meridian_result;
+
+    typedef bg::formula::vincenty_direct<double, true, true, true, true> vi_t;
+    double vincenty_azimuth = direction ? 0.0 : bg::math::pi<double>();
+    vincenty_result = vi_t::apply(lon1_rad, lat1_rad, distance, vincenty_azimuth, spheroid);
+
+    {
+        typedef bg::formula::meridian_direct<double, true, true, true, true, 1> eli;
+        meridian_result = eli::apply(lon1_rad, lat1_rad, distance, direction, spheroid);
+        check_meridian_direct(meridian_result, expected_point, vincenty_result, 0.001);
+    }
+    {
+        typedef bg::formula::meridian_direct<double, true, true, true, true, 2> eli;
+        meridian_result = eli::apply(lon1_rad, lat1_rad, distance, direction, spheroid);
+        check_meridian_direct(meridian_result, expected_point, vincenty_result, 0.00001);
+    }
+    {
+        typedef bg::formula::meridian_direct<double, true, true, true, true, 3> eli;
+        meridian_result = eli::apply(lon1_rad, lat1_rad, distance, direction, spheroid);
+        check_meridian_direct(meridian_result, expected_point, vincenty_result, 0.00000001);
+    }
+    {
+        typedef bg::formula::meridian_direct<double, true, true, true, true, 4> eli;
+        meridian_result = eli::apply(lon1_rad, lat1_rad, distance, direction, spheroid);
+        check_meridian_direct(meridian_result, expected_point, vincenty_result, 0.00000001);
+    }
+    {
+        typedef bg::formula::meridian_direct<double, true, true, true, true, 5> eli;
+        meridian_result = eli::apply(lon1_rad, lat1_rad, distance, direction, spheroid);
+        check_meridian_direct(meridian_result, expected_point, vincenty_result, 0.00000000001);
+    }
+}
+
+int test_main(int, char*[])
+{
+    for (size_t i = 0; i < expected_size; ++i)
+    {
+        test_all(expected[i]);
+    }
+
+    return 0;
+}
diff --git a/src/boost/libs/geometry/test/formulas/direct_meridian_cases.hpp b/src/boost/libs/geometry/test/formulas/direct_meridian_cases.hpp
new file mode 100644
index 00000000..b02c42b2
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/direct_meridian_cases.hpp
@@ -0,0 +1,57 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2018 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Vissarion Fysikopoulos, 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_TEST_DIRECT_MERIDIAN_CASES_HPP
+#define BOOST_GEOMETRY_TEST_DIRECT_MERIDIAN_CASES_HPP
+
+struct coordinates
+{
+    double lon;
+    double lat;
+};
+
+struct expected_results
+{
+    coordinates p1;
+    double distance;
+    bool direction;//north=true, south=false
+    coordinates p2;
+};
+
+expected_results expected[] =
+{
+    { {0, 0}, 100000, true, {0, 0.90435537782} },
+    { {0, 0}, 200000, true, {0, 1.8087062846} },
+    { {0, 0}, 300000, true, {0, 2.7130880778385226826} },
+    { {0, 0}, 400000, true, {0, 3.6174296804714929365} },
+    { {0, 0}, 500000, true, {0, 4.521753233678117212} },
+    { {0, 0}, 600000, true, {0, 5.4260542596891729872} },
+    { {0, 0}, 700000, true, {0, 6.3303283037366426811} },
+    { {0, 0}, 800000, true, {0, 7.234570938597034484} },
+    { {0, 0}, 900000, true, {0, 8.1386639105161684427} },
+    { {0, 0}, 1000000, true, {0, 9.0428195432854963087} },
+    { {0, 0}, 10000000, true, {0, 89.982400761362015373} },
+    { {0, 0}, 10001965, true, {0, 90} },
+    { {0, 0}, 11000000, true, {180, 81.063836359} },
+    { {10, 0}, 11000000, true, {-170, 81.063836359} },
+    { {0, 1}, 100000, true, {0, 1.9043634563000368942} },
+    { {0, 10}, 100000, true, {0, 10.904070448} },
+    { {0, 80}, 100000, true, {0, 80.895552833} },
+    { {0, 0}, 40010000, true, {0, 0.01932683869} },//wrap the spheroid once
+    { {0, 10}, 100000, false, {0, 9.0956502718} },
+    { {0, 10}, 2000000, false, {0, -8.0858305929} },
+    { {0, 0}, 11000000, false, {180, -81.063836359} },
+    { {0, 0}, 40010000, false, {0, -0.01932683869} } //wrap the spheroid once
+};
+
+size_t const expected_size = sizeof(expected) / sizeof(expected_results);
+
+#endif // BOOST_GEOMETRY_TEST_DIRECT_MERIDIAN_CASES_HPP
diff --git a/src/boost/libs/geometry/test/formulas/intersection.cpp b/src/boost/libs/geometry/test/formulas/intersection.cpp
new file mode 100644
index 00000000..4ab53aa2
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/intersection.cpp
@@ -0,0 +1,198 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2016-2019 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Vissarion Fysikopoulos, on behalf of Oracle
+// 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)
+
+#include "test_formula.hpp"
+#include "intersection_cases.hpp"
+
+#include <boost/geometry/formulas/andoyer_inverse.hpp>
+#include <boost/geometry/formulas/geographic.hpp>
+#include <boost/geometry/formulas/gnomonic_intersection.hpp>
+#include <boost/geometry/formulas/sjoberg_intersection.hpp>
+#include <boost/geometry/formulas/thomas_direct.hpp>
+#include <boost/geometry/formulas/thomas_inverse.hpp>
+#include <boost/geometry/formulas/vincenty_direct.hpp>
+#include <boost/geometry/formulas/vincenty_inverse.hpp>
+
+#include <boost/geometry/strategies/geographic/parameters.hpp>
+
+#include <boost/geometry/srs/spheroid.hpp>
+
+void check_result(expected_result const& result, expected_result const& expected,
+                  expected_result const& reference, double reference_error,
+                  bool check_reference_only)
+{
+    //BOOST_CHECK_MESSAGE((false), "(" << result.lon << " " << result.lat << ") vs (" << expected.lon << " " << expected.lat << ")");
+    check_one(result.lon, expected.lon, reference.lon, reference_error, false, check_reference_only);
+    check_one(result.lat, expected.lat, reference.lat, reference_error, false, check_reference_only);
+}
+
+void test_formulas(expected_results const& results, bool check_reference_only)
+{
+    // reference result
+    if (results.sjoberg_vincenty.lon == ND)
+    {
+        return;
+    }
+
+    double const d2r = bg::math::d2r<double>();
+    double const r2d = bg::math::r2d<double>();
+
+    double lona1r = results.p1.lon * d2r;
+    double lata1r = results.p1.lat * d2r;
+    double lona2r = results.p2.lon * d2r;
+    double lata2r = results.p2.lat * d2r;
+    double lonb1r = results.q1.lon * d2r;
+    double latb1r = results.q1.lat * d2r;
+    double lonb2r = results.q2.lon * d2r;
+    double latb2r = results.q2.lat * d2r;
+
+    expected_result result;
+
+    // WGS84
+    bg::srs::spheroid<double> spheroid(6378137.0, 6356752.3142451793);
+
+    if (results.gnomonic_vincenty.lon != ND)
+    {
+        bg::formula::gnomonic_intersection<double, bg::formula::vincenty_inverse, bg::formula::vincenty_direct>
+            ::apply(lona1r, lata1r, lona2r, lata2r, lonb1r, latb1r, lonb2r, latb2r, result.lon, result.lat, spheroid);
+        result.lon *= r2d;
+        result.lat *= r2d;
+        check_result(result, results.gnomonic_vincenty, results.sjoberg_vincenty, 0.00000001, check_reference_only);
+    }
+
+    if (results.gnomonic_thomas.lon != ND)
+    {
+        bg::formula::gnomonic_intersection<double, bg::strategy::thomas::inverse, bg::strategy::thomas::direct>
+                ::apply(lona1r, lata1r, lona2r, lata2r, lonb1r, latb1r, lonb2r, latb2r, result.lon, result.lat, spheroid);
+        result.lon *= r2d;
+        result.lat *= r2d;
+        check_result(result, results.gnomonic_thomas, results.sjoberg_vincenty, 0.0000001, check_reference_only);
+    }
+
+    if (results.sjoberg_vincenty.lon != ND)
+    {
+        bg::formula::sjoberg_intersection<double, bg::formula::vincenty_inverse, 4>
+            ::apply(lona1r, lata1r, lona2r, lata2r, lonb1r, latb1r, lonb2r, latb2r, result.lon, result.lat, spheroid);
+        result.lon *= r2d;
+        result.lat *= r2d;
+        check_result(result, results.sjoberg_vincenty, results.sjoberg_vincenty, 0.00000001, check_reference_only);
+    }
+
+    if (results.sjoberg_thomas.lon != ND)
+    {
+        bg::formula::sjoberg_intersection<double, bg::formula::thomas_inverse, 2>
+            ::apply(lona1r, lata1r, lona2r, lata2r, lonb1r, latb1r, lonb2r, latb2r, result.lon, result.lat, spheroid);
+        result.lon *= r2d;
+        result.lat *= r2d;
+        check_result(result, results.sjoberg_thomas, results.sjoberg_vincenty, 0.0000001, check_reference_only);
+    }
+
+    if (results.sjoberg_andoyer.lon != ND)
+    {
+        bg::formula::sjoberg_intersection<double, bg::formula::andoyer_inverse, 1>
+            ::apply(lona1r, lata1r, lona2r, lata2r, lonb1r, latb1r, lonb2r, latb2r, result.lon, result.lat, spheroid);
+        result.lon *= r2d;
+        result.lat *= r2d;
+        check_result(result, results.sjoberg_andoyer, results.sjoberg_vincenty, 0.0001, check_reference_only);
+    }
+
+    if (results.great_elliptic.lon != ND)
+    {
+        typedef bg::model::point<double, 2, bg::cs::geographic<bg::degree> > point_geo;
+        typedef bg::model::point<double, 3, bg::cs::cartesian> point_3d;
+        point_geo a1(results.p1.lon, results.p1.lat);
+        point_geo a2(results.p2.lon, results.p2.lat);
+        point_geo b1(results.q1.lon, results.q1.lat);
+        point_geo b2(results.q2.lon, results.q2.lat);
+        point_3d a1v = bg::formula::geo_to_cart3d<point_3d>(a1, spheroid);
+        point_3d a2v = bg::formula::geo_to_cart3d<point_3d>(a2, spheroid);
+        point_3d b1v = bg::formula::geo_to_cart3d<point_3d>(b1, spheroid);
+        point_3d b2v = bg::formula::geo_to_cart3d<point_3d>(b2, spheroid);
+        point_3d resv(0, 0, 0);
+        point_geo res(0, 0);
+        bg::formula::great_elliptic_intersection(a1v, a2v, b1v, b2v, resv, spheroid);
+        res = bg::formula::cart3d_to_geo<point_geo>(resv, spheroid);
+        result.lon = bg::get<0>(res);
+        result.lat = bg::get<1>(res);
+        check_result(result, results.great_elliptic, results.sjoberg_vincenty, 0.01, check_reference_only);
+    }
+}
+
+void test_4_input_combinations(expected_results const& results, bool check_reference_only)
+{
+    test_formulas(results, check_reference_only);
+
+#ifdef BOOST_GEOMETRY_TEST_GEO_INTERSECTION_TEST_SIMILAR
+    {
+        expected_results results_alt = results;
+        std::swap(results_alt.p1, results_alt.p2);
+        test_formulas(results_alt, true);
+    }
+    {
+        expected_results results_alt = results;
+        std::swap(results_alt.q1, results_alt.q2);
+        test_formulas(results_alt, true);
+    }
+    {
+        expected_results results_alt = results;
+        std::swap(results_alt.p1, results_alt.p2);
+        std::swap(results_alt.q1, results_alt.q2);
+        test_formulas(results_alt, true);
+    }
+#endif
+}
+
+void test_all(expected_results const& results)
+{
+    test_4_input_combinations(results, false);
+
+#ifdef BOOST_GEOMETRY_TEST_GEO_INTERSECTION_TEST_SIMILAR
+    expected_results results_alt = results;
+    results_alt.p1.lat *= -1;
+    results_alt.p2.lat *= -1;
+    results_alt.q1.lat *= -1;
+    results_alt.q2.lat *= -1;
+    results_alt.gnomonic_vincenty.lat *= -1;
+    results_alt.gnomonic_thomas.lat *= -1;
+    results_alt.sjoberg_vincenty.lat *= -1;
+    results_alt.sjoberg_thomas.lat *= -1;
+    results_alt.sjoberg_andoyer.lat *= -1;
+    results_alt.great_elliptic.lat *= -1;
+    test_4_input_combinations(results_alt, true);
+#endif
+}
+
+void test_bugs()
+{
+    // https://github.com/boostorg/geometry/issues/612
+    {
+        double lon, lat;
+        bg::formula::sjoberg_intersection<double, bg::formula::andoyer_inverse, 1>
+            ::apply(-0.0872665, -0.0872665, -0.0872665, 0.0872665,
+                    0.0, 1.57e-07, -0.392699, 1.57e-07,
+                    lon, lat, bg::srs::spheroid<double>());
+        check_one("issue 612", lon, -0.087266500535674751);
+        check_one("issue 612", lat, 1.5892499139622920e-07);
+    }
+}
+
+int test_main(int, char*[])
+{
+    for (size_t i = 0; i < expected_size; ++i)
+    {
+        test_all(expected[i]);
+    }
+
+    test_bugs();
+
+    return 0;
+}
diff --git a/src/boost/libs/geometry/test/formulas/intersection_cases.hpp b/src/boost/libs/geometry/test/formulas/intersection_cases.hpp
new file mode 100644
index 00000000..8fdcdb52
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/intersection_cases.hpp
@@ -0,0 +1,335 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2016 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_TEST_INTERSECTION_CASES_HPP
+#define BOOST_GEOMETRY_TEST_INTERSECTION_CASES_HPP
+
+struct coordinates
+{
+    double lon;
+    double lat;
+};
+
+struct expected_result
+{
+    double lon;
+    double lat;
+};
+
+struct expected_results
+{
+    coordinates p1;
+    coordinates p2;
+    coordinates q1;
+    coordinates q2;
+    expected_result gnomonic_vincenty;
+    expected_result gnomonic_thomas;
+    expected_result sjoberg_vincenty;
+    expected_result sjoberg_thomas;
+    expected_result sjoberg_andoyer;
+    expected_result great_elliptic;
+};
+
+#define ND 1000
+
+expected_results expected[] =
+{
+    {
+        { -1, -1 },{ 1, 1 },
+        { -1, 1 },{ 1, -1 },
+        { 0.0000000000000000, 0.0000000000000000 },
+        { 0.0000000000000000, 0.0000000000000000 },
+        { 0.0000000000626537, -0.0000000000000000 },
+        { -0.0000000187861002, -0.0000000000000000 },
+        { -0.0000055778585615, -0.0000000000000000 },
+        { -0.0000000000000000, -0.0000000000000000 }
+    },{
+        // TODO: Newton method in Sjoberg formula is not used in this case
+        //    due to f1 - f2 == 0. The other method is used instead.
+        //    Hence different results. Look into this.
+        { 1, 1 },{ -1, -1 },
+        { -1, 1 },{ 1, -1 },
+        { 0.0000000000000000, 0.0000000000000000 },
+        { 0.0000000000000000, 0.0000000000000000 },
+        { 0.0000000000000000, 0.0000000000626632 },
+        { -0.0000000000000006, -0.0000000187889745 },
+        { -0.0000000000000001, -0.0000055787431353 },
+        { -0.0000000000000000, 0.0000000000000000 }
+    },{
+        // TODO: Newton method in Sjoberg formula is not used in this case
+        //    due to f1 - f2 == 0. The other method is used instead.
+        //    Hence different results. Look into this.
+        { -1, -1 },{ 1, 1 },
+        { 1, -1 },{ -1, 1 },
+        { 0.0000000000000000, 0.0000000000000000 },
+        { 0.0000000000000000, 0.0000000000000000 },
+        { -0.0000000000000000, -0.0000000000626632 },
+        { -0.0000000000000004, 0.0000000187889746 },
+        { 0.0000000000000001, 0.0000055787431353 },
+        { 0.0000000000000000, 0.0000000000000000 }
+    },{
+        { 1, 1 },{ -1, -1 },
+        { 1, -1 },{ -1, 1 },
+        { 0.0000000000000000, 0.0000000000000000 },
+        { 0.0000000000000000, 0.0000000000000000 },
+        { -0.0000000000626537, -0.0000000000000000 },
+        { 0.0000000187860994, 0.0000000000000001 },
+        { 0.0000055778585615, -0.0000000000000000 },
+        { -0.0000000000000000, 0.0000000000000000 }
+    },{
+        { 0, 0 },{ 1, 1 },
+        { 0, 1 },{ 1, 0 },
+        { 0.5000000000000000, 0.5000573755188470 },
+        { 0.5000000000000000, 0.5000573755109839 },
+        { 0.5000000000313266, 0.5000573755188389 },
+        { 0.4999999906069524, 0.5000573755152582 },
+        { 0.4999972102164753, 0.5000573755151276 },
+        { 0.4999999999999999, 0.5000571197534014 }
+    },{
+        { 1, 1 },{ 0, 0 },
+        { 0, 1 },{ 1, 0 },
+        { 0.5000000000000000, 0.5000573755188470 },
+        { 0.5000000000000000, 0.5000573755109839 },
+        { 0.5000000000000000, 0.5000573755501669 },
+        { 0.4999999999999996, 0.5000573661218464 },
+        { 0.4999999999999999, 0.5000545856093679 },
+        { 0.4999999999999999, 0.5000571197534014 }
+    },{
+        { 0, 0 },{ 1, 1 },
+        { 1, 0 },{ 0, 1 },
+        { 0.5000000000000000, 0.5000573755188470 },
+        { 0.5000000000000000, 0.5000573755109839 },
+        { 0.4999999999999999, 0.5000573754875109 },
+        { 0.4999999999999999, 0.5000573849086647 },
+        { 0.5000000000000000, 0.5000601654208935 },
+        { 0.4999999999999999, 0.5000571197534014 }
+    },{
+        { 1, 1 },{ 0, 0 },
+        { 1, 0 },{ 0, 1 },
+        { 0.5000000000000000, 0.5000573755188470 },
+        { 0.5000000000000000, 0.5000573755109839 },
+        { 0.4999999999686731, 0.5000573755188389 },
+        { 0.5000000093930521, 0.5000573755152582 },
+        { 0.5000027897835244, 0.5000573755151276 },
+        { 0.4999999999999999, 0.5000571197534014 }
+    },{
+        { 1, 1 }, {2, 2},
+        {-2, -1}, {-1, -1},
+        { ND, ND },
+        { ND, ND },
+        { -0.9981650042162076, -0.999999718164758 },
+        { ND, ND },
+        { ND, ND },
+        { -0.9981731758085121, -0.9999997213000095 }
+    },{
+        {-25, -31}, {3, 44},
+        {-66, -14}, {-1, -1},
+        { ND, ND },
+        { ND, ND },
+        { -15.83269406235058, -4.87746450262433 },
+        { ND, ND },
+        { ND, ND },
+        { -15.82846301029918, -4.869650527718342 }
+    },{
+        {-47, -8}, {-1, -4},
+        {-40, -5}, {-5, -5},
+        { ND, ND },
+        { ND, ND },
+        { -10.28209064194141, -5.124101297536392 },
+        { ND, ND },
+        { ND, ND },
+        { -10.29663941896089, -5.123527178300465 }
+    }, {
+        {-43, -8}, {3, -4},
+        {-43, -5}, {3, -7},
+        { ND, ND },
+        { ND, ND },
+        { -19.95519754153282, -6.521540589691206 },
+        { ND, ND },
+        { ND, ND },
+        { -19.95535387979888, -6.517861579906892 }
+    }, {
+        {-1, -17}, {-5, 3},
+        {-40, -5}, {-1, -5},
+        { ND, ND },
+        { ND, ND },
+        { -3.424525475838061, -5.070550173747042 },
+        { ND, ND },
+        { ND, ND },
+        { -3.424493504575876, -5.070060631936228 }
+    }, {
+        {-29, -4}, {-1, -5},
+        {-40, -5}, {-1, -5},
+        { ND, ND },
+        { ND, ND },
+        { -0.9999999978280006, -5.00000000000684 },
+        { ND, ND },
+        { ND, ND },
+        { -0.9999999999999919, -5 }
+    }, {
+        {-4, -6}, {-40, -5},
+        {-40, -5}, {-1, -5},
+        { ND, ND },
+        { ND, ND },
+        { -40.00000000470697, -4.999999999854269 },
+        { ND, ND },
+        { ND, ND },
+        { -40.00000000000003, -4.999999999999998 }
+    }, {
+        {5, -4}, {-25, -5},
+        {-40, -5}, {-1, -5},
+        { ND, ND },
+        { ND, ND },
+        { -40.04870859732954, -4.998490166905039 },
+        { ND, ND },
+        { ND, ND },
+        { -39.87674154148858, -5.003778171392734 }
+    }, {
+        {-44, -1}, {38, -7},
+        {-54, -10}, {27, 3},
+        { ND, ND },
+        { ND, ND },
+        { -12.49994722584679, -4.492062263169279 },
+        { ND, ND },
+        { ND, ND },
+        { -12.49786552878283, -4.48224349980724 }
+    }, {
+        {-29, -5}, {10, -5},
+        {-40, -5}, {-1, -5},
+        { ND, ND },
+        { ND, ND },
+        { -14.99999999880286, -5.280237387890117 },
+        { ND, ND },
+        { ND, ND },
+        { -14.99999999999999, -5.278284829442087 }
+    }, {
+        {-29, -5}, {7, -5},
+        {-40, -5}, {-1, -5},
+        { ND, ND },
+        { ND, ND },
+        { -12.68814198534678, -5.255404335144863 },
+        { ND, ND },
+        { ND, ND },
+        { -12.68883814704657, -5.253634733357015 }
+    }, {
+        {-29, -5}, {2, -5},
+        {-40, -5}, {-1, -5},
+        { ND, ND },
+        { ND, ND },
+        { -6.793546728871506, -5.153444544719724 },
+        { ND, ND },
+        { ND, ND },
+        { -6.794933162619669, -5.152409434095275 }
+    }, {
+        {-29, -5}, {2, -5},
+        {-30, -5}, {-1, -5},
+        { ND, ND },
+        { ND, ND },
+        { -22.13104411461423, -5.130155316882458 },
+        { ND, ND },
+        { ND, ND },
+        { -22.13016591567375, -5.129277412801533 }
+    }, {
+        {-29, -5}, {2, -5},
+        {-32, -5}, {-1, -5},
+        { ND, ND },
+        { ND, ND },
+        { -14.99999999908104, -5.187213913248173 },
+        { ND, ND },
+        { ND, ND },
+        { -15.00000000000009, -5.185931965560752 }
+    }, {
+        {-92, -24}, {44, 19},
+        {-78, -5}, {50, -5},
+        { ND, ND },
+        { ND, ND },
+        { -31.65127861442256, -10.84411293410938 },
+        { ND, ND },
+        { ND, ND },
+        { -31.65294079308247, -10.76785432928444 }
+    }, {
+        {-93, -15}, {22, 3},
+        {-78, -5}, {50, -5},
+        { ND, ND },
+        { ND, ND },
+        { -32.33383382062637, -10.80295061031259 },
+        { ND, ND },
+        { ND, ND },
+        { -32.26069578187354, -10.73176059393484 }
+    }, {
+        {-93, -15}, {28, 3},
+        {-78, -5}, {50, -5},
+        { ND, ND },
+        { ND, ND },
+        { -25.71257598566304, -11.13752154787724 },
+        { ND, ND },
+        { ND, ND },
+        { -25.67761999514295, -11.05881610957402 }
+    }, {
+        {-54, -21}, {20, 17},
+        {-59, -5}, {13, -5},
+        { ND, ND },
+        { ND, ND },
+        { -22.96397490169162, -6.181426780426698 },
+        { ND, ND },
+        { ND, ND },
+        { -22.95999443662035, -6.172089364736989 }
+    }, {
+        {-31, -10}, {-31, -2},
+        {-37, -10}, {-25, -2},
+        { ND, ND },
+        { ND, ND },
+        { -31.00000000000001, -6.062729839503469 },
+        { ND, ND },
+        { ND, ND },
+        { -31, -6.062411484514648 }
+    }, {
+        {-26, -10}, {-26, 13},
+        {-37, -4}, {-18, -4},
+        { ND, ND },
+        { ND, ND },
+        { -25.99999999999969, -4.05441766735837 },
+        { ND, ND },
+        { ND, ND },
+        { -26, -4.05405101510235 }
+    }, {
+        {-26, -10}, {154, 78},
+        {-46, 41}, {-13, 36},
+        { ND, ND },
+        { ND, ND },
+        { -26, 39.19492836828103 },
+        { ND, ND },
+        { ND, ND },
+        { -26, 39.19004133747198 }
+    }, {
+        {-26, -10}, {154, 68},
+        {-85, 79}, {22, 80},
+        { ND, ND },
+        { ND, ND },
+        { -26.00000000000013, 83.7164889918963 },
+        { ND, ND },
+        { ND, ND },
+        { -26.00000000000002, 83.71603758698208 }
+    }, {
+        {-25, 55}, {155, 68},
+        {178, 79}, {124, 80},
+        { ND, ND },
+        { ND, ND },
+        { 155, 80.55982670305147 },
+        { ND, ND },
+        { ND, ND },
+        { 155, 80.55961176607357 }
+    }
+};
+
+size_t const expected_size = sizeof(expected) / sizeof(expected_results);
+
+#endif // BOOST_GEOMETRY_TEST_INTERSECTION_CASES_HPP
diff --git a/src/boost/libs/geometry/test/formulas/inverse.cpp b/src/boost/libs/geometry/test/formulas/inverse.cpp
new file mode 100644
index 00000000..cab67820
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/inverse.cpp
@@ -0,0 +1,93 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2016-2019 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle
+
+// Copyright (c) 2018 Adeel Ahmad, Islamabad, Pakistan.
+
+// Contributed and/or modified by Adeel Ahmad, as part of Google Summer of Code 2018 program
+
+// 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)
+
+
+#include <sstream>
+
+#include "test_formula.hpp"
+#include "inverse_cases.hpp"
+
+#include <boost/geometry/formulas/vincenty_inverse.hpp>
+#include <boost/geometry/formulas/thomas_inverse.hpp>
+#include <boost/geometry/formulas/andoyer_inverse.hpp>
+
+#include <boost/geometry/srs/spheroid.hpp>
+
+template <typename Result>
+void check_inverse(std::string const& name,
+                   Result const& results,
+                   bg::formula::result_inverse<double> const& result,
+                   expected_result const& expected,
+                   expected_result const& reference,
+                   double reference_error)
+{
+    std::stringstream ss;
+    ss << "(" << results.p1.lon << " " << results.p1.lat << ")->(" << results.p2.lon << " " << results.p2.lat << ")";
+
+    check_one(name + "_d  " + ss.str(),
+              result.distance, expected.distance, reference.distance, reference_error);
+    check_one(name + "_a  " + ss.str(),
+              result.azimuth, expected.azimuth, reference.azimuth, reference_error, true);
+    check_one(name + "_ra " + ss.str(),
+              result.reverse_azimuth, expected.reverse_azimuth, reference.reverse_azimuth, reference_error, true);
+    check_one(name + "_rl " + ss.str(),
+              result.reduced_length, expected.reduced_length, reference.reduced_length, reference_error);
+    check_one(name + "_gs " + ss.str(),
+              result.geodesic_scale, expected.geodesic_scale, reference.geodesic_scale, reference_error);
+}
+
+void test_all(expected_results const& results)
+{
+    double const d2r = bg::math::d2r<double>();
+    double const r2d = bg::math::r2d<double>();
+
+    double lon1r = results.p1.lon * d2r;
+    double lat1r = results.p1.lat * d2r;
+    double lon2r = results.p2.lon * d2r;
+    double lat2r = results.p2.lat * d2r;
+
+    // WGS84
+    bg::srs::spheroid<double> spheroid(6378137.0, 6356752.3142451793);
+
+    bg::formula::result_inverse<double> result_v, result_t, result_a;
+
+    typedef bg::formula::vincenty_inverse<double, true, true, true, true, true> vi_t;
+    result_v = vi_t::apply(lon1r, lat1r, lon2r, lat2r, spheroid);
+    result_v.azimuth *= r2d;
+    result_v.reverse_azimuth *= r2d;
+    check_inverse("vincenty", results, result_v, results.vincenty, results.reference, 0.0000001);
+
+    typedef bg::formula::thomas_inverse<double, true, true, true, true, true> th_t;
+    result_t = th_t::apply(lon1r, lat1r, lon2r, lat2r, spheroid);
+    result_t.azimuth *= r2d;
+    result_t.reverse_azimuth *= r2d;
+    check_inverse("thomas", results, result_t, results.thomas, results.reference, 0.00001);
+
+    typedef bg::formula::andoyer_inverse<double, true, true, true, true, true> an_t;
+    result_a = an_t::apply(lon1r, lat1r, lon2r, lat2r, spheroid);
+    result_a.azimuth *= r2d;
+    result_a.reverse_azimuth *= r2d;
+    check_inverse("andoyer", results, result_a, results.andoyer, results.reference, 0.001);
+}
+
+int test_main(int, char*[])
+{
+    for (size_t i = 0; i < expected_size; ++i)
+    {
+        test_all(expected[i]);
+    }
+
+    return 0;
+}
diff --git a/src/boost/libs/geometry/test/formulas/inverse_cases.hpp b/src/boost/libs/geometry/test/formulas/inverse_cases.hpp
new file mode 100644
index 00000000..b3b868d5
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/inverse_cases.hpp
@@ -0,0 +1,605 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2016-2017 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_TEST_INVERSE_CASES_HPP
+#define BOOST_GEOMETRY_TEST_INVERSE_CASES_HPP
+
+struct coordinates
+{
+    double lon;
+    double lat;
+};
+
+struct expected_result
+{
+    double distance;
+    double azimuth;
+    double reverse_azimuth;
+    double reduced_length;
+    double geodesic_scale;
+};
+
+struct expected_results
+{
+    coordinates p1;
+    coordinates p2;
+    expected_result reference; // karney or vincenty
+    expected_result vincenty;
+    expected_result thomas;
+    expected_result andoyer;
+
+};
+
+expected_results expected[] =
+{
+    {
+        { 0, 0 },{ 2, 0 },
+        { 222638.98158654713188298047, 90.00000000000000000000, 90.00000000000000000000, 222593.46652303214068524539, 0.99938672191946420487 },
+        { 222638.98158645280636847019, 90.00000000000000000000, 90.00000000000000000000, 222593.46652303214068524539, 0.99938672191946420487 },
+        { 222638.98158654122380539775, 90.00000000000000000000, 90.00000000000000000000, 222593.46652303214068524539, 0.99938672191946420487 },
+        { 222638.98158654125290922821, 90.00000000000000000000, 90.00000000000000000000, 222593.46652303214068524539, 0.99938672191946420487 }
+    },{
+        { 0, 0 },{ 2, 2 },
+        { 313775.70942918420769274235, 45.17488858648466987233, 45.20980230803674970730, 313648.30600607168162241578, 0.99878199603375228399 },
+        { 313775.70929574419278651476, 45.17488858646600391467, 45.20980230801805532792, 313648.30601466231746599078, 0.99878199603375328319 },
+        { 313775.70869052573107182980, 45.17488750916628958976, 45.20980122940459722258, 313648.30008120078127831221, 0.99878199607986462016 },
+        { 313772.17715740547282621264, 45.17456700769592004008, 45.20948072916769433505, 313646.54128309490624815226, 0.99878200978130526888 }
+    },{
+        { 0, 0 },{ 0, 2 },
+        { 221149.45337213488528504968, 0.00000000000000000000, 0.00000000000000000000, 221104.84592454897938296199, 0.99939490135383823866 },
+        { 221149.45318342180689796805, 0.00000000000000000000, 0.00000000000000000000, 221104.84593068063259124756, 0.99939490135383812763 },
+        { 221149.45337045041378587484, 0.00000000000000000000, 0.00000000000000000000, 221104.84593068063259124756, 0.99939490135383812763 },
+        { 221146.95820782837108708918, 0.00000000000000002350, 0.00000000000000002350, 221104.84893554644077084959, 0.99939490135383812763 }
+    },{
+        { 0, 0 },{ -2, 2 },
+        { 313775.70942918420769274235, -45.17488858648466987233, -45.20980230803674970730, 313648.30600607168162241578, 0.99878199603375228399 },
+        { 313775.70929574419278651476, -45.17488858646600391467, -45.20980230801805532792, 313648.30601466231746599078, 0.99878199603375328319 },
+        { 313775.70869052573107182980, -45.17488750916629669518, -45.20980122940459722258, 313648.30008120078127831221, 0.99878199607986462016 },
+        { 313772.17715740547282621264, -45.17456700769592004008, -45.20948072916767301876, 313646.54128309484804049134, 0.99878200978130526888 }
+    },{
+        { 0, 0 },{ -2, 0 },
+        { 222638.98158654713188298047, -90.00000000000000000000, -90.00000000000000000000, 222593.46652303214068524539, 0.99938672191946420487 },
+        { 222638.98158645280636847019, -90.00000000000000000000, -90.00000000000000000000, 222593.46652303214068524539, 0.99938672191946420487 },
+        { 222638.98158654122380539775, -90.00000000000000000000, -90.00000000000000000000, 222593.46652303214068524539, 0.99938672191946420487 },
+        { 222638.98158654125290922821, -90.00000000000000000000, -90.00000000000000000000, 222593.46652303214068524539, 0.99938672191946420487 }
+    },{
+        { 0, 0 },{ -2, -2 },
+        { 313775.70942918420769274235, -134.82511141351534433852, -134.79019769196324318727, 313648.30600607168162241578, 0.99878199603375228399 },
+        { 313775.70929574419278651476, -134.82511141353401740162, -134.79019769198197309379, 313648.30601466225925832987, 0.99878199603375317217 },
+        { 313775.70869052573107182980, -134.82511249083370330482, -134.79019877059539567199, 313648.30008120083948597312, 0.99878199607986462016 },
+        { 313772.17715740547282621264, -134.82543299230408706535, -134.79051927083230566495, 313646.54128309502266347408, 0.99878200978130537990 }
+    },{
+        { 0, 0 },{ 0, -2 },
+        { 221149.45337213488528504968, -180.00000000000000000000, -180.00000000000000000000, 221104.84592454897938296199, 0.99939490135383823866 },
+        { 221149.45318342180689796805, 180.00000000000000000000, 180.00000000000000000000, 221104.84593068063259124756, 0.99939490135383812763 },
+        { 221149.45337045041378587484, 180.00000000000000000000, 180.00000000000000000000, 221104.84593068063259124756, 0.99939490135383812763 },
+        { 221146.95820782837108708918, 180.00000000000000000000, 180.00000000000000000000, 221104.84893554644077084959, 0.99939490135383812763 }
+    },{
+        { 0, 0 },{ 2, -2 },
+        { 313775.70942918420769274235, 134.82511141351534433852, 134.79019769196324318727, 313648.30600607168162241578, 0.99878199603375228399 },
+        { 313775.70929574419278651476, 134.82511141353401740162, 134.79019769198197309379, 313648.30601466225925832987, 0.99878199603375317217 },
+        { 313775.70869052573107182980, 134.82511249083370330482, 134.79019877059539567199, 313648.30008120083948597312, 0.99878199607986462016 },
+        { 313772.17715740547282621264, 134.82543299230408706535, 134.79051927083230566495, 313646.54128309502266347408, 0.99878200978130537990 }
+    },{
+        { -1, 0 },{ 2, 0 },
+        { 333958.47237982071237638593, 90.00000000000000000000, 90.00000000000000000000, 333804.87081722758011892438, 0.99862030063297502203 },
+        { 333958.47237967920955270529, 90.00000000000000000000, 90.00000000000000000000, 333804.87081722758011892438, 0.99862030063297502203 },
+        { 333958.47237982566002756357, 90.00000000000000000000, 90.00000000000000000000, 333804.87081722758011892438, 0.99862030063297502203 },
+        { 333958.47237982566002756357, 90.00000000000000000000, 90.00000000000000000000, 333804.87081722758011892438, 0.99862030063297502203 }
+    },{
+        { -1, 0 },{ 2, 2 },
+        { 400487.62166116386651992798, 56.46443555420994186989, 56.51681282834063324572, 400222.73679570353124290705, 0.99801604059094650712 },
+        { 400487.62155657896073535085, 56.46443555419775606197, 56.51681282832842612152, 400222.73680666065774857998, 0.99801604059094761734 },
+        { 400487.62116943788714706898, 56.46443456250134573793, 56.51681183466060787168, 400222.72635490150423720479, 0.99801604069467231284 },
+        { 400484.31980028707766905427, 56.46436757836366382435, 56.51674485214859089410, 400222.02724612434394657612, 0.99801604768666130063 }
+    },{
+        { -1, 0 },{ 0, 2 },
+        { 247576.47264764847932383418, 26.70874628270370010341, 26.72620180544743817563, 247513.88750887679634615779, 0.99924166327570262958 },
+        { 247576.47247877591871656477, 26.70874628267113948255, 26.72620180541484202763, 247513.88751566910650581121, 0.99924166327570307367 },
+        { 247576.47204536388744600117, 26.70874541733553186873, 26.72620093942252594843, 247513.88563483985490165651, 0.99924166328723251773 },
+        { 247573.34468778557493351400, 26.70817834948789482041, 26.72563387223648589952, 247512.65741612159763462842, 0.99924167083675718981 }
+    },{
+        { -1, 0 },{ -2, 2 },
+        { 247576.47264764847932383418, -26.70874628270370010341, -26.72620180544743817563, 247513.88750887679634615779, 0.99924166327570262958 },
+        { 247576.47247877591871656477, -26.70874628267113948255, -26.72620180541484202763, 247513.88751566910650581121, 0.99924166327570307367 },
+        { 247576.47204536388744600117, -26.70874541733554607958, -26.72620093942252594843, 247513.88563483985490165651, 0.99924166328723251773 },
+        { 247573.34468778557493351400, -26.70817834948789482041, -26.72563387223651432123, 247512.65741612159763462842, 0.99924167083675718981 }
+    },{
+        { -1, 0 },{ -2, 0 },
+        { 111319.49079327356594149023, -90.00000000000000000000, -90.00000000000000000000, 111313.80114861292531713843, 0.99984666872462601983 },
+        { 111319.49079322640318423510, -90.00000000000000000000, -90.00000000000000000000, 111313.80114861292531713843, 0.99984666872462601983 },
+        { 111319.49079326246283017099, -90.00000000000000000000, -90.00000000000000000000, 111313.80114861292531713843, 0.99984666872462601983 },
+        { 111319.49079326246283017099, -90.00000000000000000000, -90.00000000000000000000, 111313.80114861292531713843, 0.99984666872462601983 }
+    },{
+        { -1, 0 },{ -2, -2 },
+        { 247576.47264764847932383418, -153.29125371729628568573, -153.27379819455256892979, 247513.88750887679634615779, 0.99924166327570262958 },
+        { 247576.47247877591871656477, -153.29125371732885696474, -153.27379819458516863051, 247513.88751566907740198076, 0.99924166327570307367 },
+        { 247576.47204536388744600117, -153.29125458266443615685, -153.27379906057745984072, 247513.88563483985490165651, 0.99924166328723251773 },
+        { 247573.34468778557493351400, -153.29182165051210517959, -153.27436612776349988962, 247512.65741612162673845887, 0.99924167083675718981 }
+    },{
+        { -1, 0 },{ 0, -2 },
+        { 247576.47264764847932383418, 153.29125371729628568573, 153.27379819455256892979, 247513.88750887679634615779, 0.99924166327570262958 },
+        { 247576.47247877591871656477, 153.29125371732885696474, 153.27379819458516863051, 247513.88751566907740198076, 0.99924166327570307367 },
+        { 247576.47204536388744600117, 153.29125458266446457856, 153.27379906057745984072, 247513.88563483985490165651, 0.99924166328723251773 },
+        { 247573.34468778557493351400, 153.29182165051210517959, 153.27436612776349988962, 247512.65741612162673845887, 0.99924167083675718981 }
+    },{
+        { -1, 0 },{ 2, -2 },
+        { 400487.62166116386651992798, 123.53556444579005813011, 123.48318717165936675428, 400222.73679570353124290705, 0.99801604059094650712 },
+        { 400487.62155657896073535085, 123.53556444580226525431, 123.48318717167158808934, 400222.73680666059954091907, 0.99801604059094761734 },
+        { 400487.62116943788714706898, 123.53556543749867557835, 123.48318816533939923374, 400222.72635490150423720479, 0.99801604069467231284 },
+        { 400484.31980028707766905427, 123.53563242163632196480, 123.48325514785142331675, 400222.02724612434394657612, 0.99801604768666130063 }
+    },{
+        { 1, 0 },{ 2, 0 },
+        { 111319.49079327356594149023, 90.00000000000000000000, 90.00000000000000000000, 111313.80114861292531713843, 0.99984666872462601983 },
+        { 111319.49079322640318423510, 90.00000000000000000000, 90.00000000000000000000, 111313.80114861292531713843, 0.99984666872462601983 },
+        { 111319.49079326246283017099, 90.00000000000000000000, 90.00000000000000000000, 111313.80114861292531713843, 0.99984666872462601983 },
+        { 111319.49079326246283017099, 90.00000000000000000000, 90.00000000000000000000, 111313.80114861292531713843, 0.99984666872462601983 }
+    },{
+        { 1, 0 },{ 2, 2 },
+        { 247576.47264764847932383418, 26.70874628270370010341, 26.72620180544743817563, 247513.88750887679634615779, 0.99924166327570262958 },
+        { 247576.47247877591871656477, 26.70874628267113948255, 26.72620180541484202763, 247513.88751566910650581121, 0.99924166327570307367 },
+        { 247576.47204536388744600117, 26.70874541733553186873, 26.72620093942252594843, 247513.88563483985490165651, 0.99924166328723251773 },
+        { 247573.34468778557493351400, 26.70817834948789482041, 26.72563387223648589952, 247512.65741612159763462842, 0.99924167083675718981 }
+    },{
+        { 1, 0 },{ 0, 2 },
+        { 247576.47264764847932383418, -26.70874628270370010341, -26.72620180544743817563, 247513.88750887679634615779, 0.99924166327570262958 },
+        { 247576.47247877591871656477, -26.70874628267113948255, -26.72620180541484202763, 247513.88751566910650581121, 0.99924166327570307367 },
+        { 247576.47204536388744600117, -26.70874541733554607958, -26.72620093942252594843, 247513.88563483985490165651, 0.99924166328723251773 },
+        { 247573.34468778557493351400, -26.70817834948789482041, -26.72563387223651432123, 247512.65741612159763462842, 0.99924167083675718981 }
+    },{
+        { 1, 0 },{ -2, 2 },
+        { 400487.62166116386651992798, -56.46443555420994186989, -56.51681282834063324572, 400222.73679570353124290705, 0.99801604059094650712 },
+        { 400487.62155657896073535085, -56.46443555419775606197, -56.51681282832842612152, 400222.73680666065774857998, 0.99801604059094761734 },
+        { 400487.62116943788714706898, -56.46443456250133863250, -56.51681183466062918797, 400222.72635490167886018753, 0.99801604069467231284 },
+        { 400484.31980028707766905427, -56.46436757836366382435, -56.51674485214855536697, 400222.02724612399470061064, 0.99801604768666130063 }
+    },{
+        { 1, 0 },{ -2, 0 },
+        { 333958.47237982071237638593, -90.00000000000000000000, -90.00000000000000000000, 333804.87081722758011892438, 0.99862030063297502203 },
+        { 333958.47237967920955270529, -90.00000000000000000000, -90.00000000000000000000, 333804.87081722758011892438, 0.99862030063297502203 },
+        { 333958.47237982566002756357, -90.00000000000000000000, -90.00000000000000000000, 333804.87081722758011892438, 0.99862030063297502203 },
+        { 333958.47237982566002756357, -90.00000000000000000000, -90.00000000000000000000, 333804.87081722758011892438, 0.99862030063297502203 }
+    },{
+        { 1, 0 },{ -2, -2 },
+        { 400487.62166116386651992798, -123.53556444579005813011, -123.48318717165936675428, 400222.73679570353124290705, 0.99801604059094650712 },
+        { 400487.62155657896073535085, -123.53556444580226525431, -123.48318717167158808934, 400222.73680666059954091907, 0.99801604059094761734 },
+        { 400487.62116943788714706898, -123.53556543749867557835, -123.48318816533941344460, 400222.72635490121319890022, 0.99801604069467231284 },
+        { 400484.31980028707766905427, -123.53563242163632196480, -123.48325514785139489504, 400222.02724612463498488069, 0.99801604768666130063 }
+    },{
+        { 1, 0 },{ 0, -2 },
+        { 247576.47264764847932383418, -153.29125371729628568573, -153.27379819455256892979, 247513.88750887679634615779, 0.99924166327570262958 },
+        { 247576.47247877591871656477, -153.29125371732885696474, -153.27379819458516863051, 247513.88751566907740198076, 0.99924166327570307367 },
+        { 247576.47204536388744600117, -153.29125458266443615685, -153.27379906057745984072, 247513.88563483985490165651, 0.99924166328723251773 },
+        { 247573.34468778557493351400, -153.29182165051210517959, -153.27436612776349988962, 247512.65741612162673845887, 0.99924167083675718981 }
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+        { 156899.56829134028521366417, -134.80323267835512979218, -134.81195977064112412336, 156883.63778222308610565960, 0.99969540695122460772 },
+        { 156899.56827460310887545347, -134.80323267837425760263, -134.81195977066025193380, 156883.63778401838499121368, 0.99969540695121617002 },
+        { 156899.56792193598812445998, -134.80323375780119476985, -134.81196084975837834463, 156883.63480984274065122008, 0.99969540696276681935 },
+        { 156897.79947260793414898217, -134.80355405966952275776, -134.81228115195048644637, 156882.75281389255542308092, 0.99969541038810816325 }
+    }, {
+        { -1, 1 },{ -2, -2 },
+        { 349898.53698544885264709592, -161.45195469949391053888, -161.44322493617917757547, 349721.87727315956726670265, 0.99848548653821667109 },
+        { 349898.53691851865733042359, -161.45195469952594180540, -161.44322493621118042029, 349721.87728093314217403531, 0.99848548652432367323 },
+        { 349898.53644461580552160740, -161.45195534985634822078, -161.44322558687019864010, 349721.87595040485030040145, 0.99848548653585678103 },
+        { 349894.27001174906035885215, -161.45245862756033261576, -161.44372886423073509832, 349720.85020991315832361579, 0.99848549544738174344 }
+    }, {
+        { -1, 1 },{ 0, -2 },
+        { 349898.53698544885264709592, 161.45195469949391053888, 161.44322493617917757547, 349721.87727315956726670265, 0.99848548653821667109 },
+        { 349898.53691851865733042359, 161.45195469952594180540, 161.44322493621118042029, 349721.87728093314217403531, 0.99848548652432367323 },
+        { 349898.53644461580552160740, 161.45195534985634822078, 161.44322558687019864010, 349721.87595040485030040145, 0.99848548653585678103 },
+        { 349894.27001174906035885215, 161.45245862756033261576, 161.44372886423073509832, 349720.85020991315832361579, 0.99848549544738174344 }
+    }, {
+        { -1, 1 },{ 2, -2 },
+        { 470675.27496387914288789034, 134.81200467523646580048, 134.78581002988772752360, 470245.32092262554215267301, 0.99726004843767979136 },
+        { 470675.27491378918057307601, 134.81200467525528097212, 134.78581002990657111695, 470245.32093124854145571589, 0.99726004841255500022 },
+        { 470675.27385581465205177665, 134.81200575348228198891, 134.78581110911991913781, 470245.31203990190988406539, 0.99726004851631100401 },
+        { 470669.97387331607751548290, 134.81232618817941215639, 134.78613154296644438546, 470242.67608062474755570292, 0.99726007931332960599 }
+    }, {
+        { 1, 1 },{ 2, 0 },
+        { 156899.56829134028521366417, 134.80323267835512979218, 134.81195977064112412336, 156883.63778222308610565960, 0.99969540695122460772 },
+        { 156899.56827460310887545347, 134.80323267837425760263, 134.81195977066025193380, 156883.63778401838499121368, 0.99969540695121617002 },
+        { 156899.56792193598812445998, 134.80323375780122319156, 134.81196084975840676634, 156883.63480984268244355917, 0.99969540696276681935 },
+        { 156897.79947260793414898217, 134.80355405966952275776, 134.81228115195048644637, 156882.75281389255542308092, 0.99969541038810816325 }
+    }, {
+        { 1, 1 },{ 2, 2 },
+        { 156876.14940188667969778180, 45.17047008417186759743, 45.19664870225999919739, 156860.22615479407249949872, 0.99969549952259184611 },
+        { 156876.14925185780157335103, 45.17047008415342190801, 45.19664870224154640255, 156860.22615983063587918878, 0.99969549954768632816 },
+        { 156876.14903264911845326424, 45.17046900832006883775, 45.19664762542461744488, 156860.22319510774104855955, 0.99969549955919845274 },
+        { 156874.38594904550700448453, 45.17014830792169988172, 45.19632692599481060824, 156859.34477911831345409155, 0.99969550299904519353 }
+    }, {
+        { 1, 1 },{ 0, 2 },
+        { 156876.14940188667969778180, -45.17047008417186759743, -45.19664870225999919739, 156860.22615479407249949872, 0.99969549952259184611 },
+        { 156876.14925185780157335103, -45.17047008415342190801, -45.19664870224154640255, 156860.22615983063587918878, 0.99969549954768632816 },
+        { 156876.14903264911845326424, -45.17046900832009015403, -45.19664762542463876116, 156860.22319510785746388137, 0.99969549955919856377 },
+        { 156874.38594904550700448453, -45.17014830792169988172, -45.19632692599484613538, 156859.34477911848807707429, 0.99969550299904519353 }
+    }, {
+        { 1, 1 },{ -2, 2 },
+        { 351676.50043935602298006415, -71.63912604463993716308, -71.71767794874519097448, 351497.13544799503870308399, 0.99847006808788085763 },
+        { 351676.50037266127765178680, -71.63912604463332911564, -71.71767794873858292704, 351497.13543194829253479838, 0.99847006821303718738 },
+        { 351676.50038040406070649624, -71.63912540344847457163, -71.71767730460118173141, 351497.12354394828435033560, 0.99847006831660412018 },
+        { 351675.39839278126601129770, -71.63924174977181280610, -71.71779365329184940947, 351499.26940268895123153925, 0.99847004976658171316 }
+    }, {
+        { 1, 1 },{ -2, 0 },
+        { 351772.23553088010521605611, -108.30322660728758421556, -108.32941323792046262042, 351592.72256732499226927757, 0.99846922689201267342 },
+        { 351772.23552330053644254804, -108.30322660729477490804, -108.32941323792763910205, 351592.72256738756550475955, 0.99846922689200567902 },
+        { 351772.23547190619865432382, -108.30322725204673872668, -108.32941388169263063901, 351592.71064292272785678506, 0.99846922699591933448 },
+        { 351771.13000385620398446918, -108.30311039505922110493, -108.32929702549677131174, 351594.87308142217807471752, 0.99846920815167106156 }
+    }, {
+        { 1, 1 },{ -2, -2 },
+        { 470675.27496387914288789034, -134.81200467523646580048, -134.78581002988772752360, 470245.32092262554215267301, 0.99726004843767979136 },
+        { 470675.27491378918057307601, -134.81200467525528097212, -134.78581002990657111695, 470245.32093124854145571589, 0.99726004841255500022 },
+        { 470675.27385581465205177665, -134.81200575348222514549, -134.78581110911991913781, 470245.31203990196809172630, 0.99726004851631100401 },
+        { 470669.97387331607751548290, -134.81232618817941215639, -134.78613154296644438546, 470242.67608062474755570292, 0.99726007931332960599 }
+    }, {
+        { 1, 1 },{ 0, -2 },
+        { 349898.53698544885264709592, -161.45195469949391053888, -161.44322493617917757547, 349721.87727315956726670265, 0.99848548653821667109 },
+        { 349898.53691851865733042359, -161.45195469952594180540, -161.44322493621118042029, 349721.87728093314217403531, 0.99848548652432367323 },
+        { 349898.53644461580552160740, -161.45195534985634822078, -161.44322558687019864010, 349721.87595040485030040145, 0.99848548653585678103 },
+        { 349894.27001174906035885215, -161.45245862756033261576, -161.44372886423073509832, 349720.85020991315832361579, 0.99848549544738174344 }
+    }, {
+        { 1, 1 },{ 2, -2 },
+        { 349898.53698544885264709592, 161.45195469949391053888, 161.44322493617917757547, 349721.87727315956726670265, 0.99848548653821667109 },
+        { 349898.53691851865733042359, 161.45195469952594180540, 161.44322493621118042029, 349721.87728093314217403531, 0.99848548652432367323 },
+        { 349898.53644461580552160740, 161.45195534985634822078, 161.44322558687019864010, 349721.87595040485030040145, 0.99848548653585678103 },
+        { 349894.27001174906035885215, 161.45245862756033261576, 161.44372886423073509832, 349720.85020991315832361579, 0.99848549544738174344 }
+    }, {
+        { 1, -1 },{ 2, 0 },
+        { 156899.56829134028521366417, 45.19676732164486310239, 45.18804022935886877121, 156883.63778222308610565960, 0.99969540695122460772 },
+        { 156899.56827460310887545347, 45.19676732162573529195, 45.18804022933975517162, 156883.63778401838499121368, 0.99969540695121617002 },
+        { 156899.56792193598812445998, 45.19676624219878391386, 45.18803915024158612823, 156883.63480984268244355917, 0.99969540696276681935 },
+        { 156897.79947260793414898217, 45.19644594033049145310, 45.18771884804951355363, 156882.75281389255542308092, 0.99969541038810816325 }
+    }, {
+        { 1, -1 },{ 2, 2 },
+        { 349898.53698544885264709592, 18.54804530050607525027, 18.55677506382081531910, 349721.87727315956726670265, 0.99848548653821667109 },
+        { 349898.53691851865733042359, 18.54804530047409016902, 18.55677506378881247429, 349721.87728093314217403531, 0.99848548652432367323 },
+        { 349898.53644461580552160740, 18.54804465014367664821, 18.55677441312981201804, 349721.87595040485030040145, 0.99848548653585678103 },
+        { 349894.27001174906035885215, 18.54754137243966738424, 18.55627113576925424354, 349720.85020991315832361579, 0.99848549544738174344 }
+    }, {
+        { 1, -1 },{ 0, 2 },
+        { 349898.53698544885264709592, -18.54804530050607525027, -18.55677506382081531910, 349721.87727315956726670265, 0.99848548653821667109 },
+        { 349898.53691851865733042359, -18.54804530047409016902, -18.55677506378881247429, 349721.87728093314217403531, 0.99848548652432367323 },
+        { 349898.53644461580552160740, -18.54804465014367664821, -18.55677441312981201804, 349721.87595040485030040145, 0.99848548653585678103 },
+        { 349894.27001174906035885215, -18.54754137243966738424, -18.55627113576928266525, 349720.85020991315832361579, 0.99848549544738174344 }
+    }, {
+        { 1, -1 },{ -2, 2 },
+        { 470675.27496387914288789034, -45.18799532476352709409, -45.21418997011227247640, 470245.32092262554215267301, 0.99726004843767979136 },
+        { 470675.27491378918057307601, -45.18799532474472613330, -45.21418997009345730476, 470245.32093124865787103772, 0.99726004841255500022 },
+        { 470675.27385581465205177665, -45.18799424651773932737, -45.21418889088011638933, 470245.31203990196809172630, 0.99726004851631100401 },
+        { 470669.97387331607751548290, -45.18767381182056652733, -45.21386845703356271997, 470242.67608062445651739836, 0.99726007931332938394 }
+    }, {
+        { 1, -1 },{ -2, 0 },
+        { 351772.23553088010521605611, -71.69677339271241578444, -71.67058676207953737958, 351592.72256732499226927757, 0.99846922689201267342 },
+        { 351772.23552330053644254804, -71.69677339270522509196, -71.67058676207236089795, 351592.72256738768192008138, 0.99846922689200567902 },
+        { 351772.23547190619865432382, -71.69677274795323285161, -71.67058611830736936099, 351592.71064292214578017592, 0.99846922699591911243 },
+        { 351771.13000385620398446918, -71.69688960494077889507, -71.67070297450322868826, 351594.87308142206165939569, 0.99846920815167106156 }
+    }, {
+        { 1, -1 },{ -2, -2 },
+        { 351676.50043935602298006415, -108.36087395536006283692, -108.28232205125480902552, 351497.13544799503870308399, 0.99847006808788085763 },
+        { 351676.50037266127765178680, -108.36087395536665667350, -108.28232205126143128382, 351497.13543194788508117199, 0.99847006821303718738 },
+        { 351676.50038040406070649624, -108.36087459655152542837, -108.28232269539881826859, 351497.12354394845897331834, 0.99847006831660412018 },
+        { 351675.39839278126601129770, -108.36075825022818719390, -108.28220634670809374711, 351499.26940269087208434939, 0.99847004976658149111 }
+    }, {
+        { 1, -1 },{ 0, -2 },
+        { 156876.14940188667969778180, -134.82952991582811819171, -134.80335129773999369718, 156860.22615479407249949872, 0.99969549952259184611 },
+        { 156876.14925185780157335103, -134.82952991584656388113, -134.80335129775846780831, 156860.22615983051946386695, 0.99969549954768643918 },
+        { 156876.14903264911845326424, -134.82953099167991695140, -134.80335237457535413341, 156860.22319510785746388137, 0.99969549955919845274 },
+        { 156874.38594904550700448453, -134.82985169207830722371, -134.80367307400518939176, 156859.34477911840076558292, 0.99969550299904519353 }
+    }, {
+        { 1, -1 },{ 2, -2 },
+        { 156876.14940188667969778180, 134.82952991582811819171, 134.80335129773999369718, 156860.22615479407249949872, 0.99969549952259184611 },
+        { 156876.14925185780157335103, 134.82952991584656388113, 134.80335129775846780831, 156860.22615983051946386695, 0.99969549954768643918 },
+        { 156876.14903264911845326424, 134.82953099167994537311, 134.80335237457538255512, 156860.22319510785746388137, 0.99969549955919856377 },
+        { 156874.38594904550700448453, 134.82985169207830722371, 134.80367307400518939176, 156859.34477911840076558292, 0.99969550299904519353 }
+    }, {
+        // near pole
+        {90, -45}, {90, -80},
+        {3900195.49395913071930408478, -180.00000000000000000000, -180.00000000000000000000, 3662502.04478993499651551247, 0.81922976250863166481},
+        {3900195.49395686946809291840, 180.00000000000000000000, 180.00000000000000000000, 3662502.06126507185399532318, 0.81922976508736944368},
+        {3900195.49053949490189552307, 180.00000000000000000000, 180.00000000000000000000, 3662502.06126507185399532318, 0.81922976508736944368},
+        {3900203.61653685756027698517, 180.00000000000000000000, 180.00000000000000000000, 3662507.91531699104234576225, 0.81923068138101373670}
+    }, {
+        // pole, same lon
+        {90, -45}, {90, -90},
+        {5017021.35133497882634401321, -180.00000000000000000000, -180.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017021.35133314039558172226, 180.00000000000000000000, 180.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017021.34791153203696012497, 180.00000000000000000000, 180.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017017.85355509258806705475, 180.00000000000000000000, 180.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172}
+    }, {
+        // pole
+        {90, -45}, {0, -90},
+        {5017021.35133497882634401321, -180.00000000000000000000, -90.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017021.35133314039558172226, 180.00000000000000000000, -90.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017021.34791153203696012497, 180.00000000000000000000, -90.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017017.85355509258806705475, 180.00000000000000000000, -90.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172}
+    }, {
+        // near pole
+        {90, 45}, {90, 80},
+        {3900195.49395913071930408478, 0.00000000000000000000, 0.00000000000000000000, 3662502.04478993499651551247, 0.81922976250863166481},
+        {3900195.49395686946809291840, 0.00000000000000000000, 0.00000000000000000000, 3662502.06126507185399532318, 0.81922976508736944368},
+        {3900195.49053949490189552307, 0.00000000000000000000, 0.00000000000000000000, 3662502.06126507185399532318, 0.81922976508736944368},
+        {3900203.61653685756027698517, 0.00000000000000000076, 0.00000000000000000071, 3662507.91531699104234576225, 0.81923068138101373670}
+    }, {
+        // pole, same lon
+        {90, 45}, {90, 90},
+        {5017021.35133497882634401321, 0.00000000000000000000, 0.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017021.35133314039558172226, 0.00000000000000000000, 0.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017021.34791153203696012497, 0.00000000000000000000, 0.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017017.85355509258806705475, 0.00000000000000000000, 0.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172}
+    }, {
+        // pole
+        {90, 45}, {0, 90},
+        {5017021.35133497882634401321, -0.00000000000000000000, -90.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017021.35133314039558172226, -0.00000000000000496992, -90.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017021.34791153203696012497, 0.00000000000000000000, -90.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172},
+        {5017017.85355509258806705475, 0.00000000000000000000, -90.00000000000000000000, 4517590.87884893082082271576, 0.70710678118654746172}
+    }, {
+        // pole
+        {0, -90}, {90, -45},
+        {5017021.35133497882634401321, 90.00000000000000000000, 0.00000000000000000000, 4517590.87884893082082271576, 0.70778496454077310940},
+        {5017021.35133314039558172226, 90.00000000000000000000, 0.00000000000000496992, 4517590.87884893082082271576, 0.70778494508412304054},
+        {5017021.34791153203696012497, 90.00000000000000000000, 0.00000000000000000000, 4517590.87884893082082271576, 0.70778494508412304054},
+        {5017017.85355509258806705475, 90.00000000000000000000, 0.00000000000000000000, 4517590.87884893082082271576, 0.70777827352623101653}
+    }, {
+        // pole
+        {0, 90}, {90, 45},
+        {5017021.35133497882634401321, 90.00000000000000000000, -180.00000000000000000000, 4517590.87884893082082271576, 0.70778496454077310940},
+        {5017021.35133314039558172226, 90.00000000000000000000, 180.00000000000000000000, 4517590.87884893082082271576, 0.70778494508412304054},
+        {5017021.34791153203696012497, 90.00000000000001421085, 180.00000000000000000000, 4517590.87884893082082271576, 0.70778494508412304054},
+        {5017017.85355509258806705475, 90.00000000000000000000, 180.00000000000000000000, 4517590.87884893082082271576, 0.70777827352623101653}
+    }, {
+        {90, -45}, {90, 45},
+        {9969888.75595548748970031738, 0.00000000000000000000, 0.00000000000000000000, 6361290.27546359039843082428, 0.00431189731318673553},
+        {9969888.75595730356872081757, 0.00000000000000000000, 0.00000000000000000000, 6361290.36011654324829578400, 0.00431188406305459994},
+        {9969888.73915978334844112396, 0.00000000000000000000, 0.00000000000000000000, 6361290.36011654324829578400, 0.00431188406305459994},
+        {9969881.64984572120010852814, 0.00000000000000001848, 0.00000000000000001176, 6361314.54906367603689432144, 0.00430809793699618842}
+    }, {
+        {90, 45}, {90, -45},
+        {9969888.75595548748970031738, -180.00000000000000000000, -180.00000000000000000000, 6361290.27546359039843082428, 0.00431189731318673553},
+        {9969888.75595730356872081757, 180.00000000000000000000, 180.00000000000000000000, 6361290.36011654324829578400, 0.00431188406305459994},
+        {9969888.73915978334844112396, 180.00000000000000000000, 180.00000000000000000000, 6361290.36011654324829578400, 0.00431188406305459994},
+        {9969881.64984572120010852814, 180.00000000000000000000, 180.00000000000000000000, 6361314.54906367603689432144, 0.00430809793699618842}
+    }, {
+        // pole, same lon
+        {90, 0}, {90, 90},
+        {10001965.72931272350251674652, 0.00000000000000000000, 0.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.72931179217994213104, 0.00000000000000000000, 0.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.71749142371118068695, 0.00000000000000000000, 0.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001958.67847795411944389343, 0.00000000000000000000, 0.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000}
+    }, {
+        // pole
+        {90, 0}, {0, 90},
+        {10001965.72931272350251674652, -0.00000000000000000000, -90.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.72931179217994213104, -0.00000000000000352016, -90.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.71749142371118068695, 0.00000000000000000000, -90.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001958.67847795411944389343, 0.00000000000000000000, -90.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000}
+    }, {
+        // pole, different lon
+        {0, 0}, {1, 90},
+        {10001965.72931272350251674652, 0.00000000000000000000, 1.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.72931179217994213104, 0.00000000000000006144, 1.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.71749142371118068695, 0.00000000000000000000, 0.99999999999998867573, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001958.67847795411944389343, 0.00000000000000000000, 0.99999999999998867573, 6378136.99999999906867742538, 0.00000000000000000000}
+    }, {
+        // pole, different lon
+        {0, 0}, {-1, 90},
+        {10001965.72931272350251674652, -0.00000000000000000000, -1.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.72931179217994213104, -0.00000000000000006144, -1.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.71749142371118068695, 0.00000000000000000000, -1.00000000000001421085, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001958.67847795411944389343, 0.00000000000000000000, -0.99999999999998867573, 6378136.99999999906867742538, 0.00000000000000000000}
+    }, {
+        // pole, different lon
+        {0, 0}, {1, -90},
+        {10001965.72931272350251674652, -180.00000000000000000000, 179.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.72931179217994213104, 180.00000000000000000000, 179.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.71749142371118068695, 180.00000000000000000000, 179.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001958.67847795411944389343, 180.00000000000000000000, 179.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000}
+    }, {
+        // pole, different lon
+        {0, 0}, {-1, -90},
+        {10001965.72931272350251674652, -180.00000000000000000000, -179.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.72931179217994213104, -180.00000000000000000000, -179.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001965.71749142371118068695, 180.00000000000000000000, -179.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000},
+        {10001958.67847795411944389343, 180.00000000000000000000, -179.00000000000000000000, 6378136.99999999906867742538, 0.00000000000000000000}
+    }/*, {
+        // antipodal
+        {-90, -45}, {90, 45},
+        {20003931.45862544700503349304, -180.00000000000000000000, 0.00000000000000000000, 33675.52452803074265830219, -0.99472900658949181540},
+        {19987083.06974205747246742249, 90.00000000000000000000, 90.00000000000000000000, 00000.00000000078240703623, -1.00000000000000000000},
+        {00000000.00000000000000000000, 0.00000000000000000000, 0.00000000000000000000, 000000.00000000000000000000, 1.00000000000000000000},
+        {20020712.84987257421016693115, 0.00000000000000000000, 0.00000000000000000000, 6361314.54906367603689432144, 0.00430809793699618842}
+    },{
+        // antipodal
+        {-90, 45}, {90, -45},
+        {20003931.45862544700503349304, 0.00000000000000000000, -180.00000000000000000000, 33675.52452803074265830219, -0.99472900658949181540},
+        {19987083.06974205747246742249, 90.00000000000000000000, 90.00000000000000000000, -000000.00000000078240703623, -1.00000000000000000000},
+        {00000000.00000000000000000000, 0.00000000000000000000, 0.00000000000000000000, 000000.00000000000000000000, 1.00000000000000000000},
+        {20020712.84987257421016693115, 180.00000000000000000000, 0.00000000000000000000, 33590.79639541531651047990, -1.00525773151080621837}
+    }*/
+};
+
+size_t const expected_size = sizeof(expected) / sizeof(expected_results);
+
+#endif // BOOST_GEOMETRY_TEST_INVERSE_CASES_HPP
diff --git a/src/boost/libs/geometry/test/formulas/inverse_cases_antipodal.hpp b/src/boost/libs/geometry/test/formulas/inverse_cases_antipodal.hpp
new file mode 100644
index 00000000..e66046dd
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/inverse_cases_antipodal.hpp
@@ -0,0 +1,342 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2018 Adeel Ahmad, Islamabad, Pakistan.
+
+// Contributed and/or modified by Adeel Ahmad, as part of Google Summer of Code 2018 program.
+
+// 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_TEST_INVERSE_CASES_ANTIPODAL_HPP
+#define BOOST_GEOMETRY_TEST_INVERSE_CASES_ANTIPODAL_HPP
+
+#include "inverse_cases.hpp"
+
+struct expected_results_antipodal
+{
+    coordinates p1;
+    coordinates p2;
+    expected_result karney;
+};
+
+/*
+ These values are collected from GeodTest which is associated with GeographicLib:
+     https://zenodo.org/record/32156
+
+ The conversion to C++ array format is done using this Python script:
+     https://github.com/adl1995/boost-geometry-extra/blob/master/geographiclib-dataset-parse-inverse.py
+
+ Geodesic scale (M12) is absent from the GeodTest dataset, so it is manually generated
+ using GeographicLib using this C++ script:
+     https://github.com/adl1995/boost-geometry-extra/blob/master/geographicLib-direct-antipodal.cpp
+*/
+expected_results_antipodal expected_antipodal[] =
+{
+    {
+        { 0, 31.394417440639 }, { 179.615601631202912322, -31.275540610835465807 },
+        { 19980218.4055399, 34.266322930672, 145.782701113414306756, 49490.8807994496209, -0.996116451012525883079717914370121434 }
+    },{
+        { 0, 29.788792273749 }, { 178.569451327813675741, -29.558013672069422725 },
+        { 19887224.5407334, 74.302205994192, 106.156240654579267308, 97043.7545600593058, -0.998624031147844926081802441331092268 }
+    },{
+        { 0, 46.471843094141 }, { 179.083144618009561276, -46.284166405924629853 },
+        { 19944337.8863917, 63.693680310665, 116.699978859005570535, 53139.140576552365, -0.997597309645591900917338534782174975 }
+    },{
+        { 0, 63.016506345929 }, { 179.862869954071637855, -63.02943882703369735 },
+        { 20000925.7533636, 153.393656073038, 26.619056019474552953, 12713.9284725111772, -1.00381317792143387457315384381217882 }
+    },{
+        { 0, 19.796231412719 }, { 179.546498474461283862, -19.470586923091672503 },
+        { 19956338.1330537, 28.272934411318, 151.789094611690988249, 87191.1749625132931, -0.997015409027664833985227232915349305 }
+    },{
+        { 0, 6.373459459035 }, { 179.240009269347556917, -6.204887833274217382 },
+        { 19946581.6983394, 56.859050230583, 123.169200847008284851, 53958.8698005263939, -0.999349049081101004077254401636309922 }
+    },{
+        { 0, 66.380766469414 }, { 179.632633596894388233, -66.27177494016956425 },
+        { 19986277.7696849, 38.646950203356, 141.550919825824399405, 22198.215635049214, -0.996949176054954366854587988200364634 }
+    },{
+        { 0, 16.483421185231 }, { 179.731567273052604726, -16.818424446748042212 },
+        { 19962737.9842573, 163.431254767325, 16.598399455529231288, 95318.4104529881431, -1.00272210232979741562076014815829694 }
+    },{
+        { 0, 4.215702155486 }, { 179.093771177769992874, -4.051917290690976764 },
+        { 19932517.393764, 65.543168480886, 114.482669479963380006, 55205.4553703842317, -0.999655858425056553784315838129259646 }
+    },{
+        { 0, 40.71372085907 }, { 179.404612926861498984, -41.047052242159400671 },
+        { 19951133.3595356, 143.672151631634, 36.54002600969304553, 70931.1530155553621, -1.00414169574077272173440178448799998 }
+    },{
+        { 0, 15.465481491654 }, { 179.020726605204181801, -14.622355549425900341 },
+        { 19877383.8879911, 36.289185640976, 143.875673907461159912, 156419.0806764376957, -0.997639074397169589580869342171354219 }
+    },{
+        { 0, 17.586197343531 }, { 179.722490735835379144, -17.731394230364437075 },
+        { 19982280.4639115, 157.929615091529, 22.089021105298661023, 69727.5357849255557, -1.00280451698301242835498214844847098 }
+    },{
+        { 0, 5.7442768247 }, { 178.85894724576868462, -6.039853564481335581 },
+        { 19902873.7431814, 116.146983678305, 63.91482549951374061, 87149.6188944111673, -1.00039332893096744037109147029696032 }
+    },{
+        { 0, 32.002904282111 }, { 179.744925422107715439, -32.297934520693132807 },
+        { 19967670.3104795, 163.052160078191, 17.004175883388454943, 78311.3164829640582, -1.00449903445302446414189034840092063 }
+    },{
+        { 0, 55.902716926362 }, { 179.300685189522463007, -55.934320218634018206 },
+        { 19970525.337607, 98.927641063414, 81.374264168520557301, 23554.0093185709067, -1.00072788779083454713259015989024192 }
+    },{
+        { 0, 22.69939784398 }, { 179.294173474584020749, -22.654875407651067149 },
+        { 19959286.1903172, 74.253870776761, 105.811588890213155275, 22369.7179951557679, -0.998972181419003457669703038845909759 }
+    },{
+        { 0, 41.312328471121 }, { 179.817186837717804928, -40.954523601529804886 },
+        { 19962690.5721867, 11.277616109847, 168.784288786443902199, 77252.6121237260201, -0.994825151471527391322524636052548885 }
+    },{
+        { 0, 27.927415327453 }, { 179.636508875679110143, -27.607314264234172721 },
+        { 19961296.8828333, 23.166421459647, 156.905194492817275222, 83096.5801709291101, -0.995959692767656723511038308060960844 }
+    },{
+        { 0, 41.567228741451 }, { 179.931812964300204608, -42.103039532074194347 },
+        { 19944253.4454809, 176.66609526064, 3.361859685835349219, 96859.08180779197, -1.00513607140487626345759508694754913 }
+    },{
+        { 0, 37.384208978567 }, { 179.225180174670992261, -36.916085670712060029 },
+        { 19928705.5911445, 39.072534864532, 141.212743814390850106, 92667.7834060578402, -0.995955516859159284415170532156480476 }
+    },{
+        { 0, 59.011868682852 }, { 179.424923485514312807, -58.82705468054708336 },
+        { 19970442.3788306, 44.970301291063, 135.333817989802309531, 38071.1136293083857, -0.996658942892707400140750451100757346 }
+    },{
+        { 0, 35.515406087737 }, { 179.50369572149476218, -35.119747127350258822 },
+        { 19948918.9139751, 28.528972431952, 151.622257906284404073, 84564.0387217601751, -0.995562861799169418475230486365035176 }
+    },{
+        { 0, 58.170252463184 }, { 179.254737571455023977, -58.372261836268550805 },
+        { 19961407.0813807, 128.021116291844, 52.399129705193347143, 43715.3070711393309, -1.00285273713280753682397516968194395 }
+    },{
+        { 0, 34.012183807959 }, { 179.83713352180447672, -34.29640782899529639 },
+        { 19970955.843065, 168.944519134772, 11.093048811826875835, 76493.5814538538151, -1.0047652354558671561335359001532197 }
+    },{
+        { 0, 45.510762948553 }, { 178.981682578823726535, -45.582753595227824235 },
+        { 19940248.3450143, 99.886784003837, 80.542330522982505877, 48555.1946627894972, -1.00083807750906350619857221317943186 }
+    },{
+        { 0, 4.19841765451 }, { 179.398024428225540172, -4.198416896099783242 },
+        { 19970496.5132933, 89.561550657928, 90.438456568689151881, 14.8790480103109, -0.999994104810285944218151144013972953 }
+    },{
+        { 0, 40.890119148103 }, { 179.6557148951668192, -41.553556264538302258 },
+        { 19926563.5817492, 165.437641169967, 14.713597527941311478, 111805.7305227545923, -1.00492294933406567380984597548376769 }
+    },{
+        { 0, 28.096672787686 }, { 178.606868012231657724, -28.472055035513955205 },
+        { 19883901.8482359, 115.174366374632, 65.257367020445564176, 107880.4353518862363, -1.00170803073331593502359737613005564 }
+    },{
+        { 0, 6.50572154271 }, { 178.926013840891647541, -6.411745140559297675 },
+        { 19917276.4101551, 79.069492719523, 100.985091481519557845, 57073.3242952680707, -0.999736666933808471036115861352300271 }
+    },{
+        { 0, .468835109567 }, { 178.325942223692180692, -.281751687044281805 },
+        { 19849380.7342734, 80.234636214474, 99.77243368342786593, 123845.4568822078908, -0.999801437209140719808431185811059549 }
+    },{
+        { 0, 1.682746325049 }, { 179.717131561406935483, -.677647430701204515 },
+        { 19890026.0274781, 10.076182752451, 169.927471515299313238, 177917.2104306563981, -0.999538055691262194990542866435134783 }
+    },{
+        { 0, 10.711305126218 }, { 179.874050163405229937, -10.349315378531556046 },
+        { 19962987.2134077, 7.528253696796, 172.480576051850009046, 104175.1095378254456, -0.998071853755238880268052525934763253 }
+    },{
+        { 0, 53.374321544652 }, { 179.729445806011012057, -53.196257519024042184 },
+        { 19980478.1457438, 23.324715976877, 156.777734080146664812, 41907.8869272231053, -0.995333596277707566279957518418086693 }
+    },{
+        { 0, 39.680221664519 }, { 179.87506206720154785, -39.256187213040660911 },
+        { 19956191.7841809, 7.075406493429, 172.967624741991546131, 86943.8110669895148, -0.994801087909667924868983845954062417 }
+    },{
+        { 0, 1.377666714083 }, { 178.994542525209058878, -1.415358715570225495 },
+        { 19925401.4931301, 95.29199069739, 84.7178724483824156, 45800.9140624827059, -0.99999803170512457928253979844157584 }
+    },{
+        { 0, 48.751426624188 }, { 179.661697715070846977, -48.688146707479475147 },
+        { 19988599.1160495, 40.252328570137, 139.808452951157199824, 26322.3790862461568, -0.995999245724129789181233718409202993 }
+    },{
+        { 0, 59.443039048494 }, { 179.247605418616998285, -59.454371825393424121 },
+        { 19969935.9534732, 93.052184108221, 87.331416513795326158, 25342.4691896499534, -1.00020727848897084122370415570912883 }
+    },{
+        { 0, 4.122408476235 }, { 179.749430572914989772, -4.689124208743755363 },
+        { 19938291.6332293, 167.73479753304, 12.274635577599782826, 127855.6475863583497, -1.00068600902837667732114823593292385 }
+    },{
+        { 0, 46.422470082432 }, { 178.857408435141563774, -46.390934261324541952 },
+        { 19931980.7029341, 86.67365350297, 93.852683224054943377, 56114.680046867064, -0.999607096116300386512421027873642743 }
+    },{
+        { 0, 32.614423729024 }, { 179.460593512880455451, -32.01874745886238612 },
+        { 19926887.3785175, 24.943814520557, 155.229917137448282531, 112355.3319340873104, -0.995562150676871926435751447570510209 }
+    },{
+        { 0, 3.242895277973 }, { 179.556428318080663113, -3.001106476068264917 },
+        { 19964490.4789049, 30.247458779683, 149.760178923092147784, 80929.0418317066044, -0.999474184270344845337774586369050667 }
+    },{
+        { 0, 6.29069210113 }, { 178.556859259685624933, -6.354208910915346725 },
+        { 19877160.8505733, 94.34299459284, 85.750059038253282986, 94127.1566760840083, -0.999976397350904933070125935046235099 }
+    },{
+        { 0, 18.232086569498 }, { 179.658073278238477245, -18.87394850776853555 },
+        { 19927978.7462175, 164.41905055334, 15.640779355822506503, 129771.1882449660559, -1.00293460439063886191490837518358603 }
+    },{
+        { 0, 12.049849333181 }, { 179.761046682699610657, -11.201990279782499264 },
+        { 19908004.4552909, 9.418096768309, 170.610608272305604585, 157761.5040571466343, -0.997761474497510958414636661473196 }
+    },{
+        { 0, 40.289465276136 }, { 179.644208494155329095, -40.370034926441385999 },
+        { 19985674.936106, 143.092606818963, 36.958610382613096419, 36200.8933724688593, -1.00414965876091266672176516294712201 }
+    },{
+        { 0, 2.197784650379 }, { 179.961199531084784854, -1.353440827124394777 },
+        { 19910509.7517973, 1.542117609437, 178.458582198505846426, 160403.6285079348996, -0.999488724639301051588802238256903365 }
+    },{
+        { 0, 1.966575272177 }, { 179.699817324905962184, -3.101125282483752618 },
+        { 19875595.6267266, 170.112968791865, 9.89572776349855838, 192355.7206665719908, -1.00015463589804554089823795948177576 }
+    },{
+        { 0, 25.078832492684 }, { 178.600804840925824646, -24.897833702325682511 },
+        { 19887997.7953866, 77.264585323781, 103.101167809583406892, 92442.9124509225839, -0.998981189838600847075156252685701475 }
+    },{
+        { 0, 31.740361941314 }, { 179.553485210731879874, -31.909206787477701871 },
+        { 19972325.3556069, 143.930820896999, 36.145242998351638503, 54883.4113710054145, -1.00379461628115951299378139083273709 }
+    },{
+        { 0, .05479250563 }, { 178.822647462220726609, .836079031223269324 },
+        { 19858049.4780499, 41.349430623518, 138.645259065012502544, 169078.442370111714, -0.9997793696948588104689292777038645 }
+    },{
+        { 0, 36.685139871608 }, { 179.366667224014334712, -36.6833040833258687 },
+        { 19968965.6773632, 89.167975517493, 90.921025521408327068, 13327.2156799476918, -0.999916537946348604748436628142371774 }
+    },{
+        { 0, 3.451199399671 }, { 179.107509334399258305, -3.459003521120242021 },
+        { 19938203.3838544, 91.541212417048, 88.476282464773035164, 32316.1747698810781, -1.00000397484395819880376166111091152 }
+    },{
+        { 0, 27.692898794247 }, { 178.512356615673144314, -27.666009301228316555 },
+        { 19883493.6699045, 88.406440883665, 92.036345087713397961, 94128.7880896190836, -0.999736458322951659916100197733612731 }
+    },{
+        { 0, 17.363238291869 }, { 179.567921315455829491, -17.288872648596950413 },
+        { 19980749.7638027, 39.697196316589, 140.321938237586060826, 46975.9359427664379, -0.997687691981715030209443284547887743 }
+    },{
+        { 0, 37.006775102539 }, { 179.191103068859169842, -37.156365616364686838 },
+        { 19949309.9180043, 116.455543532607, 63.771817992036617793, 45856.1961421018701, -1.00221962858918423044940482213860378 }
+    },{
+        { 0, 45.572883540957 }, { 179.224707765088686272, -45.94675931323086696 },
+        { 19940027.8586414, 137.627256708444, 42.723991162977357301, 74208.4359612889496, -1.00380887786447159371050474874209613 }
+    },{
+        { 0, 43.63393981955 }, { 178.878236417027994157, -43.642335115130514773 },
+        { 19931045.2914508, 91.203625101465, 89.268780774643462256, 55253.5406349861764, -1.00002974153150514524668324156664312 }
+    },{
+        { 0, 38.4995307019 }, { 179.143856004445269342, -39.042223438550921467 },
+        { 19918391.2222193, 141.232864609445, 39.117947060740562295, 102217.2563106863077, -1.00388164115732947401227193040540442 }
+    },{
+        { 0, 27.55015339382 }, { 179.596220103573824099, -27.587412128122249651 },
+        { 19986004.7358853, 137.025135713548, 42.992898351962011956, 33938.7346646670654, -1.00316044390281167153489150223322213 }
+    },{
+        { 0, 1.54507498314 }, { 179.567115633151308577, -1.448861185025252004 },
+        { 19978593.3191777, 36.816106412092, 143.185763012309022403, 56320.5800276739168, -0.999770499462467210349814195069484413 }
+    },{
+        { 0, 45.217063644222 }, { 179.807382581661125, -45.086424050571516283 },
+        { 19987042.0782465, 18.114645812265, 161.928120141429818658, 45544.2915061261936, -0.994974179414854997816064496873877943 }
+    },{
+        { 0, 13.473522450751 }, { 179.726941062277208626, -13.570372758027936877 },
+        { 19987364.078382, 156.839609002403, 23.170293747820406391, 65329.9068132034472, -1.00219093189506569530067281448282301 }
+    },{
+        { 0, 6.287741997374 }, { 179.071252372259552052, -6.743450924917895817 },
+        { 19912159.8245954, 132.954797451112, 47.100789519677419746, 104772.4027498097375, -1.00071252411103017720961361192166805 }
+    },{
+        { 0, 7.639709001531 }, { 179.616156296978583335, -7.48702643786017917 },
+        { 19976374.3699535, 29.731916588299, 150.279582966919438164, 69224.6591757209539, -0.998789792086741234911073661351110786 }
+    },{
+        { 0, 5.893688050348 }, { 179.586212000450856399, -4.888408917114795625 },
+        { 19886907.2520668, 14.653438882877, 165.371181401863458848, 177183.5330818593022, -0.998794647031120752522781458537792787 }
+    },{
+        { 0, 61.997076235476 }, { 179.605779116829636081, -62.19593758437129915 },
+        { 19976288.2901729, 149.562797049254, 30.65850204223272625, 36696.2853801462176, -1.00373071432437144245852778112748638 }
+    },{
+        { 0, 50.507637741656 }, { 179.893569206021038536, -50.721890799900161112 },
+        { 19979542.5263293, 171.564028344478, 8.4746613464253591, 50644.5234828162697, -1.00508881632281776852266830246662721 }
+    },{
+        { 0, 7.484475238477 }, { 178.638400003000590878, -6.926155588124333461 },
+        { 19867425.2906303, 57.020570370985, 123.087267812322270238, 132929.2775641349633, -0.999097042677338120775232255255104974 }
+    },{
+        { 0, 56.851165323215 }, { 179.587046628550073045, -56.875248360744638525 },
+        { 19988235.9960515, 112.345749045605, 67.744017057185404441, 9971.0934553515518, -1.00182859249871403228837607457535341 }
+    },{
+        { 0, 10.692273150738 }, { 178.709520715733071393, -10.851727623036704339 },
+        { 19893210.3050033, 102.824601316946, 77.308514969817191459, 83032.7122948051111, -1.00034345584508432835946223349310458 }
+    },{
+        { 0, 46.694739303788 }, { 179.926838145841924189, -46.948618153686522669 },
+        { 19975447.9283188, 174.663684259477, 5.361568174833475454, 59614.5876209460645, -1.00520484875201732144489596976200119 }
+    },{
+        { 0, 15.804386137005 }, { 178.367587635209819128, -15.522042847777054984 },
+        { 19855850.8800526, 74.932089158884, 105.357235560913450667, 123350.4326645237628, -0.999091578546475345135036150168161839 }
+    },{
+        { 0, 4.371450175299 }, { 179.780887420199549421, -4.566109732313098407 },
+        { 19979071.1035552, 164.163592252794, 15.840695025950408814, 84137.2115482558728, -1.00076323969894742660358133434783667 }
+    },{
+        { 0, 30.894388279688 }, { 179.375426183521944524, -30.871308884744172663 },
+        { 19968681.8321577, 77.35154610481, 102.709506078439532936, 14048.0277985734058, -0.998975176336422854284080585784977302 }
+    },{
+        { 0, 9.541166838639 }, { 178.432934555386452839, -10.09982228112793472 },
+        { 19848553.7844137, 118.441353539081, 61.736686215549403663, 144831.1911566651614, -1.00060548620110489892454097571317106 }
+    },{
+        { 0, 8.489292700054 }, { 179.906698338023119097, -8.559237750032113623 },
+        { 19995477.1669578, 171.963952699866, 8.037517851139094467, 72192.60793572974, -1.00152068486306466965629624610301107 }
+    },{
+        { 0, 19.562401114224 }, { 178.838724116996037606, -20.05038360490599475 },
+        { 19893208.1788508, 126.362762598128, 53.875560227496658204, 112181.7524188837615, -1.00185202668802775249901060306001455 }
+    },{
+        { 0, 42.260350252749 }, { 179.807860448877064601, -42.79985897702184353 },
+        { 19942715.0054774, 170.703419847646, 9.377654670896439828, 96336.3477142010769, -1.00508642406443549077721399953588843 }
+    },{
+        { 0, 24.511403144656 }, { 178.957598444862223515, -24.616808725039883945 },
+        { 19924809.5184876, 102.913211410163, 77.297538210434837096, 55403.453072179318, -1.0008408309188838725134473861544393 }
+    },{
+        { 0, 20.844284170708 }, { 179.069258863637226633, -20.321320573298341477 },
+        { 19909084.6340808, 44.172784008084, 136.01627115731728436, 111009.0987238994608, -0.997389183621778974142557672166731209 }
+    },{
+        { 0, 2.426010809098 }, { 178.236397468862000784, -2.513715200833756776 },
+        { 19840940.6924189, 94.315194952561, 85.734896842737189557, 130002.6104886615638, -0.999825249844991659209370027383556589 }
+    },{
+        { 0, 6.600682554664 }, { 179.646475458013797028, -7.699164822656561787 },
+        { 19878412.28273, 168.167678684515, 11.861035812918738552, 187426.3958525886692, -1.00098284856064978498579876031726599 }
+    },{
+        { 0, 23.372339802326 }, { 179.499422665106094027, -24.239465200482591299 },
+        { 19899498.4582543, 161.197647943542, 18.932355367478826536, 151863.2545535951091, -1.00347666868431395492677893344080076 }
+    },{
+        { 0, 16.194668264095 }, { 179.115193814080201851, -17.129419031459576897 },
+        { 19874825.6683239, 148.942349959054, 31.225656401221968078, 166033.3161394594622, -1.00222032222233647935638600756647065 }
+    },{
+        { 0, 1.528726471528 }, { 178.791047180477802091, -1.282203000582034597 },
+        { 19897803.9939987, 69.212891442493, 110.802928803578167132, 85252.8333849204133, -0.999827144228156883265512533398577943 }
+    },{
+        { 0, 6.297249676078 }, { 178.623258703845895437, -5.709470001196540278 },
+        { 19864042.0495193, 56.274639904925, 123.817184177744186806, 137475.1283083659258, -0.999190450178399580671850799262756482 }
+    },{
+        { 0, 17.393540327984 }, { 179.330156510680163326, -17.431100690958209424 },
+        { 19962624.6302607, 107.855062015266, 72.181322855288535245, 19320.5501845044839, -1.00091841779689127989172447996679693 }
+    },{
+        { 0, 46.284685151236 }, { 179.852534804091121255, -46.176234945675219984 },
+        { 19990422.3478916, 14.758013867151, 165.271681964991897184, 42614.1796365710104, -0.994894592261839960656288894824683666 }
+    },{
+        { 0, 14.924320176299 }, { 179.195663739713760883, -14.125476432252858442 },
+        { 19891861.8615337, 31.446544793174, 148.678916887199611191, 149419.6596309045804, -0.997620142585332936313591289945179597 }
+    },{
+        { 0, 23.668824656069 }, { 179.409875478773990359, -24.107855233601412399 },
+        { 19938736.4442268, 148.091483667618, 32.02919257641173958, 97771.7687385830819, -1.00323262872000595891108787327539176 }
+    },{
+        { 0, 46.986276695896 }, { 179.92040916864362177, -47.301644191214905832 },
+        { 19968596.0414782, 174.796708941456, 5.234240076649939638, 66113.7417494369769, -1.00519095452608087093437916337279603 }
+    },{
+        { 0, 65.946144289524 }, { 179.808282612725835525, -65.871840130833632868 },
+        { 19993734.5109736, 25.375428509648, 154.703163938350061652, 18355.2254271672769, -0.996436935914610577569305860379245132 }
+    },{
+        { 0, 10.950298933293 }, { 179.624609619829763098, -10.787771536605316781 },
+        { 19975919.5586889, 28.779018914489, 151.238005588662201946, 70291.1998404303581, -0.998272071834115148902810688014142215 }
+    },{
+        { 0, 13.609869340778 }, { 179.035623147420893383, -14.023624108675206222 },
+        { 19913213.8514358, 129.616021271129, 50.506400999466711623, 97596.7664002074776, -1.00146664642314031645753402699483559 }
+    },{
+        { 0, 48.701427557433 }, { 179.385565054218238481, -48.735316652259656533 },
+        { 19972955.2699173, 102.875149183407, 77.294384444682547869, 18461.7742226227697, -1.00114676855429074464609584538266063 }
+    },{
+        { 0, 31.519172055785 }, { 179.555251675378549409, -31.140142027808697534 },
+        { 19952318.3772514, 26.247105619999, 153.865822276646938125, 86354.7117605101002, -0.995739948399825047786748655198607594 }
+    },{
+        { 0, 31.863784754278 }, { 179.722489476483407524, -31.826935359797657785 },
+        { 19993324.8682601, 29.572313410211, 150.440607907359037187, 41427.6181613499234, -0.995888009001147267440501309465616941 }
+    },{
+        { 0, 76.434608546092 }, { 179.918287057674124459, -76.48787937532808951 },
+        { 19997750.023578, 167.428385412814, 12.621032110142724567, 9619.5267710862108, -1.00233963893091582164629471662919968 }
+    },{
+        { 0, 73.114273316483 }, { 179.576736605988553624, -73.098788070892914568 },
+        { 19992866.6147806, 78.154765899661, 102.085693546950923465, 8580.6475692800946, -0.999384143308475469957841141876997426 }
+    },{
+        { 0, 1.125639056292 }, { 178.426819580880619395, -.694775021853292564 },
+        { 19852573.5442848, 67.184842289382, 112.831314850896246589, 132932.8743502563937, -0.999732957962833457266071945923613384 }
+    }
+};
+
+size_t const expected_size_antipodal = sizeof(expected_antipodal) / sizeof(expected_results_antipodal);
+
+#endif // BOOST_GEOMETRY_TEST_INVERSE_CASES_ANTIPODAL_HPP
diff --git a/src/boost/libs/geometry/test/formulas/inverse_cases_small_angles.hpp b/src/boost/libs/geometry/test/formulas/inverse_cases_small_angles.hpp
new file mode 100644
index 00000000..6fe9557c
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/inverse_cases_small_angles.hpp
@@ -0,0 +1,106 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2018 Adeel Ahmad, Islamabad, Pakistan.
+
+// Contributed and/or modified by Adeel Ahmad, as part of Google Summer of Code 2018 program.
+
+// 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_TEST_INVERSE_CASES_SMALL_ANGLES_HPP
+#define BOOST_GEOMETRY_TEST_INVERSE_CASES_SMALL_ANGLES_HPP
+
+#include "inverse_cases.hpp"
+
+struct expected_results_small_angles
+{
+    coordinates p1;
+    coordinates p2;
+    expected_result karney;
+};
+
+/*
+ These values are collected from GeodTest which is associated with GeographicLib:
+     https://zenodo.org/record/32156
+
+ The conversion to C++ array format is done using this Python script:
+     https://github.com/adl1995/boost-geometry-extra/blob/master/geographiclib-dataset-parse-inverse.py
+
+ Geodesic scale (M12) is absent from the GeodTest dataset, so it is manually generated
+ using GeographicLib using this C++ script:
+     https://github.com/adl1995/boost-geometry-extra/blob/master/geographicLib-direct-small_angles.cpp
+*/
+expected_results_small_angles expected_small_angles[] =
+{
+    {
+        { 180, 0 },{ 0, 0 },
+        { 20003931.45862544700503349304, -0.00000000000000000000, 180.00000000000000000000, 67125.61229850351810455322266, -1.00000000000000000000 },
+    },{
+        { 180, 0 },{ 1e-300, 0 },
+        { 20003931.45862544700503349304, -0.00000000000000000000, 180.00000000000000000000, 67125.61229850351810455322266, -1.00000000000000000000 },
+    },{
+        { 180, 0 },{ 1e-200, 0 },
+        { 20003931.45862544700503349304, -0.00000000000000000000, 180.00000000000000000000, 67125.61229850351810455322266, -1.00000000000000000000 },
+    },{
+        { 180, 0 },{ 1e-100, 0 },
+        { 20003931.45862544700503349304, -0.00000000000000000000, 180.00000000000000000000, 67125.61229850351810455322266, -1.00000000000000000000 },
+    },{
+        { 180, 0 },{ 1e-50, 0 },
+        { 20003931.45862544700503349304, -0.00000000000000000000, 180.00000000000000000000, 67125.61229850351810455322266, -1.00000000000000000000 },
+    },{
+        { 180, 0 },{ 1e-20, 0 },
+        { 20003931.45862544700503349304, -0.00000000000000000000, 180.00000000000000000000, 67125.61229850351810455322266, -1.00000000000000000000 },
+    },{
+        { 180, 0 },{ 1e-10, 0 },
+        { 20003931.45862544700503349304, -9.501793528220011062168943853e-09, -179.9999999904981962117744843, 67125.61229850351810455322266, -1 },
+    },{
+        { 0, 1e-100 },{ 170, 1e-200},
+        { 18924313.43485650792717933655, 90, 90, 1041298.808552250848151743412, -0.9864919282563420210863114335 },
+    },{
+        { 0, 1e-300 },{ 170, 1e-50},
+        { 18924313.43485650792717933655, 90, 90, 1041298.808552250848151743412, -0.9864919282563420210863114335 },
+    },{
+        { 0, 1e-300 },{ 170, 1e-10},
+        { 18924313.43485650792717933655, 89.99999999939157646622334141, 90.00000000060019544889655663, 1041298.80855225014965981245, -0.9864919282563420210863114335 },
+    },{
+        { 0, 1e-100 },{ 170, 1e-50},
+        { 18924313.43485650792717933655, 90, 90, 1041298.808552250848151743412, -0.9864919282563420210863114335 },
+    },{
+        { 0, 1e-200 },{ 170, 1e-50},
+        { 18924313.43485650792717933655, 90, 90, 1041298.808552250848151743412, -0.9864919282563420210863114335 },
+    },{
+        { 0, 0.0 },{ 170, 1e-10},
+        { 18924313.43485650792717933655, 89.99999999939157646622334141, 90.00000000060019544889655663, 1041298.80855225014965981245, -0.9864919282563420210863114335 },
+    },{
+        { 0, 1e-20 },{ 170, 1e-100},
+        { 18924313.43485650792717933655, 90, 90, 1041298.808552250848151743412, -0.9864919282563420210863114335 },
+    },{
+        { 0, 1e-100 },{ 170, 0.0},
+        { 18924313.43485650792717933655, 90, 90, 1041298.808552250848151743412, -0.9864919282563420210863114335 },
+    },{
+        { 0, 1e-10 },{ 170, 1e-300},
+        { 18924313.43485650792717933655, 89.99999999939980455110344337, 90.00000000060842353377665859, 1041298.80855225014965981245, -0.9864919282563420210863114335 },
+    },{
+        { 0, 1e-300 },{ 170, 1e-100},
+        { 18924313.43485650792717933655, 90, 90, 1041298.808552250848151743412, -0.9864919282563420210863114335 },
+    },{
+        { 0, 1e-200 },{ 170, 1e-100},
+        { 18924313.43485650792717933655, 90, 90, 1041298.808552250848151743412, -0.9864919282563420210863114335 },
+    },{
+        { 0, 1e-10 },{ 170, 1e-50},
+        { 18924313.43485650792717933655, 89.99999999939980455110344337, 90.00000000060842353377665859, 1041298.80855225014965981245, -0.9864919282563420210863114335 },
+    },{
+        { 0, 0.0 },{ 170, 1e-200},
+        { 18924313.43485650792717933655, 90, 90, 1041298.808552250848151743412, -0.9864919282563420210863114335 },
+    },{
+        { 0, 0.0 },{ 170, 1e-10},
+        { 18924313.43485650792717933655, 89.99999999939157646622334141, 90.00000000060019544889655663, 1041298.80855225014965981245, -0.9864919282563420210863114335 },
+    }
+};
+
+
+size_t const expected_size_small_angles = sizeof(expected_small_angles) / sizeof(expected_results_small_angles);
+
+#endif // BOOST_GEOMETRY_TEST_INVERSE_CASES_SMALL_ANGLES_HPP
diff --git a/src/boost/libs/geometry/test/formulas/inverse_karney.cpp b/src/boost/libs/geometry/test/formulas/inverse_karney.cpp
new file mode 100644
index 00000000..f7004b64
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/inverse_karney.cpp
@@ -0,0 +1,104 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2016-2019 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle
+
+// Copyright (c) 2018 Adeel Ahmad, Islamabad, Pakistan.
+
+// Contributed and/or modified by Adeel Ahmad, as part of Google Summer of Code 2018 program
+
+// 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)
+
+
+#include <sstream>
+
+#include "test_formula.hpp"
+#include "inverse_cases.hpp"
+#include "inverse_cases_antipodal.hpp"
+#include "inverse_cases_small_angles.hpp"
+
+#include <boost/geometry/formulas/karney_inverse.hpp>
+
+#include <boost/geometry/srs/spheroid.hpp>
+
+template <typename Result>
+void check_inverse(std::string const& name,
+                   Result const& results,
+                   bg::formula::result_inverse<double> const& result,
+                   expected_result const& expected,
+                   expected_result const& reference,
+                   double reference_error)
+{
+    std::stringstream ss;
+    ss << "(" << results.p1.lon << " " << results.p1.lat << ")->(" << results.p2.lon << " " << results.p2.lat << ")";
+
+    check_one(name + "_d  " + ss.str(),
+              result.distance, expected.distance, reference.distance, reference_error);
+    check_one(name + "_a  " + ss.str(),
+              result.azimuth, expected.azimuth, reference.azimuth, reference_error, true);
+    check_one(name + "_ra " + ss.str(),
+              result.reverse_azimuth, expected.reverse_azimuth, reference.reverse_azimuth, reference_error, true);
+    check_one(name + "_rl " + ss.str(),
+              result.reduced_length, expected.reduced_length, reference.reduced_length, reference_error);
+    check_one(name + "_gs " + ss.str(),
+              result.geodesic_scale, expected.geodesic_scale, reference.geodesic_scale, reference_error);
+}
+
+void test_all(expected_results const& results)
+{
+    double lon1d = results.p1.lon;
+    double lat1d = results.p1.lat;
+    double lon2d = results.p2.lon;
+    double lat2d = results.p2.lat;
+
+    // WGS84
+    bg::srs::spheroid<double> spheroid(6378137.0, 6356752.3142451793);
+
+    bg::formula::result_inverse<double> result_k;
+
+    typedef bg::formula::karney_inverse<double, true, true, true, true, true, 8> ka_t;
+    result_k = ka_t::apply(lon1d, lat1d, lon2d, lat2d, spheroid);
+    check_inverse("karney", results, result_k, results.vincenty, results.reference, 0.0000001);
+}
+
+template <typename ExpectedResults>
+void test_karney(ExpectedResults const& results)
+{
+    double lon1d = results.p1.lon;
+    double lat1d = results.p1.lat;
+    double lon2d = results.p2.lon;
+    double lat2d = results.p2.lat;
+
+    // WGS84
+    bg::srs::spheroid<double> spheroid(6378137.0, 6356752.3142451793);
+
+    bg::formula::result_inverse<double> result;
+
+    typedef bg::formula::karney_inverse<double, true, true, true, true, true, 8> ka_t;
+    result = ka_t::apply(lon1d, lat1d, lon2d, lat2d, spheroid);
+    check_inverse("karney", results, result, results.karney, results.karney, 0.0000001);
+}
+
+int test_main(int, char*[])
+{
+    for (size_t i = 0; i < expected_size; ++i)
+    {
+        test_all(expected[i]);
+    }
+
+    for (size_t i = 0; i < expected_size_antipodal; ++i)
+    {
+        test_karney(expected_antipodal[i]);
+    }
+
+    for (size_t i = 0; i < expected_size_small_angles; ++i)
+    {
+        test_karney(expected_small_angles[i]);
+    }
+
+    return 0;
+}
diff --git a/src/boost/libs/geometry/test/formulas/test_formula.hpp b/src/boost/libs/geometry/test/formulas/test_formula.hpp
new file mode 100644
index 00000000..9e41bfca
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/test_formula.hpp
@@ -0,0 +1,97 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2016-2019 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_TEST_FORMULA_HPP
+#define BOOST_GEOMETRY_TEST_FORMULA_HPP
+
+#include <geometry_test_common.hpp>
+
+#include <boost/geometry/util/math.hpp>
+
+void normalize_deg(double & deg)
+{
+    while (deg > 180.0)
+        deg -= 360.0;
+    while (deg <= -180.0)
+        deg += 360.0;
+}
+
+
+#define BOOST_GEOMETRY_CHECK_CLOSE( L, R, T, M )        BOOST_TEST_TOOL_IMPL( 0, \
+    ::boost::test_tools::check_is_close_t(), M, CHECK, CHECK_MSG, (L)(R)(::boost::math::fpc::percent_tolerance(T)) )
+
+
+void check_one(std::string const& name, double result, double expected)
+{
+    std::string id = name.empty() ? "" : (name + " : ");
+
+    double eps = std::numeric_limits<double>::epsilon();
+    double abs_result = bg::math::abs(result);
+    double abs_expected = bg::math::abs(expected);
+    double res_max = (std::max)(abs_result, abs_expected);
+    double res_min = (std::min)(abs_result, abs_expected);
+    if (res_min <= eps) // including 0
+    {
+        bool is_close = abs_result <= 30 * eps && abs_expected <= 30 * eps;
+        BOOST_CHECK_MESSAGE((is_close),
+            id << std::setprecision(20) << "result {" << result << "} different than expected {" << expected << "}.");
+    }
+    else if (res_max > 100 * eps)
+    {
+        BOOST_GEOMETRY_CHECK_CLOSE(result, expected, 0.1,
+            id << std::setprecision(20) << "result {" << result << "} different than expected {" << expected << "}.");
+    }
+    else if (res_max > 10 * eps)
+    {
+        BOOST_GEOMETRY_CHECK_CLOSE(result, expected, 10,
+            id << std::setprecision(20) << "result {" << result << "} different than expected {" << expected << "}.");
+    }
+    else if (res_max > eps)
+    {
+        BOOST_GEOMETRY_CHECK_CLOSE(result, expected, 1000,
+            id << std::setprecision(20) << "result {" << result << "} different than expected {" << expected << "}.");
+    }
+}
+
+void check_one(std::string const& name,
+               double result, double expected, double reference, double reference_error,
+               bool normalize = false, bool check_reference_only = false)
+{
+    std::string id = name.empty() ? "" : (name + " : ");
+
+    if (normalize)
+    {
+        normalize_deg(result);
+        normalize_deg(expected);
+        normalize_deg(reference);
+    }
+
+    if (! check_reference_only)
+    {
+        check_one(name, result, expected);
+    }
+
+    // NOTE: in some cases it probably will be necessary to normalize
+    //       the differences between the result and expected result
+    double ref_diff = bg::math::abs(result - reference);
+    double ref_max = (std::max)(bg::math::abs(result), bg::math::abs(reference));
+    bool is_ref_close = ref_diff <= reference_error || ref_diff <= reference_error * ref_max;
+    BOOST_CHECK_MESSAGE((is_ref_close),
+        id << std::setprecision(20) << "result {" << result << "} and reference {" << reference << "} not close enough.");
+}
+
+void check_one(double result, double expected, double reference, double reference_error,
+               bool normalize = false, bool check_reference_only = false)
+{
+    check_one("", result, expected, reference, reference_error, normalize, check_reference_only);
+}
+
+#endif // BOOST_GEOMETRY_TEST_FORMULA_HPP
diff --git a/src/boost/libs/geometry/test/formulas/vertex_longitude.cpp b/src/boost/libs/geometry/test/formulas/vertex_longitude.cpp
new file mode 100644
index 00000000..1aadc4a3
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/vertex_longitude.cpp
@@ -0,0 +1,154 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2017 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Vissarion Fysikopoulos, 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)
+
+#include <geometry_test_common.hpp>
+
+#include "test_formula.hpp"
+#include "vertex_longitude_cases.hpp"
+
+#include <boost/geometry/formulas/vertex_latitude.hpp>
+#include <boost/geometry/formulas/vertex_longitude.hpp>
+#include <boost/geometry/formulas/vincenty_inverse.hpp>
+#include <boost/geometry/formulas/thomas_inverse.hpp>
+#include <boost/geometry/formulas/andoyer_inverse.hpp>
+
+#include <boost/geometry/strategies/strategies.hpp>
+
+#include <boost/geometry/util/math.hpp>
+
+#define BOOST_GEOMETRY_TEST_DEBUG
+
+namespace bg = boost::geometry;
+
+template<typename CT>
+CT test_vrt_lon_sph(CT lon1r,
+                    CT lat1r,
+                    CT lon2r,
+                    CT lat2r)
+{
+    CT a1 = bg::formula::spherical_azimuth(lon1r, lat1r, lon2r, lat2r);
+
+    typedef bg::model::point<CT, 2,
+            bg::cs::spherical_equatorial<bg::radian> > point;
+
+    bg::model::segment<point> segment(point(lon1r, lat1r),
+                                      point(lon2r, lat2r));
+    bg::model::box<point> box;
+    bg::envelope(segment, box);
+
+    CT vertex_lat;
+    CT lat_sum = lat1r + lat2r;
+    if (lat_sum > CT(0))
+    {
+        vertex_lat = bg::get_as_radian<bg::max_corner, 1>(box);
+    } else {
+        vertex_lat = bg::get_as_radian<bg::min_corner, 1>(box);
+    }
+
+    bg::strategy::azimuth::spherical<> azimuth;
+
+    return bg::formula::vertex_longitude
+            <CT, bg::spherical_equatorial_tag>::
+            apply(lon1r, lat1r,
+                  lon2r, lat2r,
+                  vertex_lat,
+                  a1,
+                  azimuth);
+}
+
+template
+<
+        template <typename, bool, bool, bool, bool, bool> class FormulaPolicy,
+        typename CT
+        >
+CT test_vrt_lon_geo(CT lon1r,
+                    CT lat1r,
+                    CT lon2r,
+                    CT lat2r)
+{
+    // WGS84
+    bg::srs::spheroid<CT> spheroid(6378137.0, 6356752.3142451793);
+
+    typedef FormulaPolicy<CT, false, true, false, false, false> formula;
+    CT a1 = formula::apply(lon1r, lat1r, lon2r, lat2r, spheroid).azimuth;
+
+    typedef bg::model::point<CT, 2, bg::cs::geographic<bg::radian> > geo_point;
+
+    bg::model::segment<geo_point> segment(geo_point(lon1r, lat1r),
+                                          geo_point(lon2r, lat2r));
+    bg::model::box<geo_point> box;
+    bg::envelope(segment, box);
+
+    CT vertex_lat;
+    CT lat_sum = lat1r + lat2r;
+    if (lat_sum > CT(0))
+    {
+        vertex_lat = bg::get_as_radian<bg::max_corner, 1>(box);
+    } else {
+        vertex_lat = bg::get_as_radian<bg::min_corner, 1>(box);
+    }
+
+    bg::strategy::azimuth::geographic<> azimuth_geographic;
+
+    return bg::formula::vertex_longitude
+            <CT, bg::geographic_tag>::apply(lon1r, lat1r,
+                                            lon2r, lat2r,
+                                            vertex_lat,
+                                            a1,
+                                            azimuth_geographic);
+
+}
+
+void test_all(expected_results const& results)
+{
+    double const d2r = bg::math::d2r<double>();
+
+    double lon1r = results.p1.lon * d2r;
+    double lat1r = results.p1.lat * d2r;
+    double lon2r = results.p2.lon * d2r;
+    double lat2r = results.p2.lat * d2r;
+
+    if(lon1r > lon2r)
+    {
+        std::swap(lon1r, lon2r);
+        std::swap(lat1r, lat2r);
+    }
+
+    double res_an = test_vrt_lon_geo<bg::formula::andoyer_inverse>
+            (lon1r, lat1r, lon2r, lat2r);
+    double res_th = test_vrt_lon_geo<bg::formula::thomas_inverse>
+            (lon1r, lat1r, lon2r, lat2r);
+    double res_vi = test_vrt_lon_geo<bg::formula::vincenty_inverse>
+            (lon1r, lat1r, lon2r, lat2r);
+    double res_sh = test_vrt_lon_sph(lon1r, lat1r, lon2r, lat2r);
+
+    bg::math::normalize_longitude<bg::radian, double>(res_an);
+    bg::math::normalize_longitude<bg::radian, double>(res_th);
+    bg::math::normalize_longitude<bg::radian, double>(res_vi);
+    bg::math::normalize_longitude<bg::radian, double>(res_sh);
+
+    check_one(res_an, results.andoyer * d2r, res_vi, 0.001);
+    check_one(res_th, results.thomas * d2r, res_vi, 0.01);//in some tests thomas gives low accuracy
+    check_one(res_vi, results.vincenty * d2r, res_vi, 0.0000001);
+    check_one(res_sh, results.spherical * d2r, res_vi, 1);
+}
+
+int test_main(int, char*[])
+{
+
+    for (size_t i = 0; i < expected_size; ++i)
+    {
+        test_all(expected[i]);
+    }
+
+    return 0;
+}
+
diff --git a/src/boost/libs/geometry/test/formulas/vertex_longitude_cases.hpp b/src/boost/libs/geometry/test/formulas/vertex_longitude_cases.hpp
new file mode 100644
index 00000000..2357c2b1
--- /dev/null
+++ b/src/boost/libs/geometry/test/formulas/vertex_longitude_cases.hpp
@@ -0,0 +1,457 @@
+// Boost.Geometry
+// Unit Test
+
+// Copyright (c) 2017 Oracle and/or its affiliates.
+
+// Contributed and/or modified by Vissarion Fysikopoulos, 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_TEST_VERTEX_LONGITUDE_CASES_HPP
+#define BOOST_GEOMETRY_TEST_VERTEX_LONGITUDE_CASES_HPP
+
+struct coordinates
+{
+    double lon;
+    double lat;
+};
+
+struct expected_result
+{
+    double lon;
+    double lat;
+};
+
+struct expected_results
+{
+    coordinates p1;
+    coordinates p2;
+    double andoyer;
+    double thomas;
+    double vincenty;
+    double spherical;
+};
+
+expected_results expected[] =
+{ 
+    { //ascenting segments (wrt pole)
+      { 1, 1 },{ 100, 2 },
+      66.25553538,
+      66.25594187,
+      66.25594273,
+      66.39744208
+    },{
+        { 1, 1 },{ 90, 2 },
+        64.09051435,
+        64.09082287,
+        64.09082352,
+        64.24414382
+    },{
+        { 0, 1 },{ 50, 1 },
+        24.99994265,
+        24.99999906,
+        25,
+        25
+    },{
+        { 0, 1 },{ 50, 1.1 },
+        30.79039009,
+        30.79049758,
+        30.79049828,
+        30.83209056
+    },{
+        { 0, 1 },{ 50, 1.2 },
+        35.95625867,
+        35.95640445,
+        35.95640493,
+        36.0343576
+    },{
+        { 0, 1 },{ 50, 1.3 },
+        40.52204781,
+        40.52222064,
+        40.52222094,
+        40.63120292
+    },{
+        { 0, 1 },{ 50, 1.4 },
+        44.53768967,
+        44.53788045,
+        44.53788061,
+        44.6729585
+    },{
+        { 0, 1 },{ 50, 1.5 },
+        48.06382018,
+        48.0640219,
+        48.06402195,
+        48.22088385
+    },{
+        { 0, 1 },{ 50, 1.6 },
+        50,
+        50,
+        50,
+        50
+    },{ //descending segment (wrt pole)
+        { 50, 1 },{ 0, 1.1 },
+        19.20950181,
+        19.20950054,
+        19.20950172,
+        19.16790944
+    },{
+        { 50, 1 },{ 0, 1.2 },
+        14.04365003,
+        14.04359367,
+        14.04359507,
+        13.9656424
+    },{
+        { 50, 1 },{ 0, 1.3 },
+        9.477883847,
+        9.477777483,
+        9.477779056,
+        9.368797077
+    },{
+        { 50, 1 },{ 0, 1.4 },
+        5.462267984,
+        5.462117673,
+        5.462119386,
+        5.327041497
+    },{
+        { 50, 1 },{ 0, 1.5 },
+        1.936164363,
+        1.935976226,
+        1.93597805,
+        1.779116148
+    },{
+        { 3, 5 },{ 150, 0},
+        60.29182988,
+        60.29785309,
+        60.29785255,
+        60
+    },{
+        { 3, 5 },{ 150, 0.5},
+        63.11344576,
+        63.11900045,
+        63.11899891,
+        62.87000766
+    },{
+        { 3, 5 },{ 150, 1},
+        65.51880171,
+        65.52391866,
+        65.52391623,
+        65.31813729
+    },{
+        { 3, 5 },{ 150, 5},
+        76.49727275,
+        76.50000657,
+        76.5,
+        76.5
+    },{ //segments parallel to equator
+        { 0, 1 },{ 4, 1 },
+        1.999999973,
+        1.999999925,
+        2,
+        2
+    },{
+        { 0, 1 },{ 10, 1 },
+        4.999999569,
+        4.999999812,
+        5,
+        5
+    },{
+        { 0, 1 },{ 60, 1 },
+        29.9998978,
+        29.99999887,
+        30,
+        30
+    },{
+        { 0, 1 },{ 90, 1 },
+        44.99960266,
+        45.22272756,//thomas low accuracy
+        45,
+        45
+    },{
+        { 0, 1 },{ 120, 1 },
+        59.99878311,
+        59.99999778,
+        60,
+        60
+    },{
+        { 0, 1 },{ 180, 1 },
+        90,
+        90,
+        90,
+        90
+    },{
+        { 0, 1 },{ 270, 1 },
+        -44.99960266,
+        -45.08931472,//thomas low accuracy
+        -45,
+        -45
+    },{
+        { 0, 1 },{ 290, 1 },
+        -34.9998314,
+        -34.99999868,
+        -35,
+        -35
+    },{
+        { 0, 1 },{ 150, 1 },
+        74.99598515,
+        74.99999794,
+        75,
+        75
+    },{
+        { 0, 1 },{ 180, 1 },
+        90,
+        90,
+        90,
+        90
+    },{ //in equator vertex is any point on segment
+        { 1, 0 },{ 10, 0 },
+        1,
+        1,
+        1,
+        1
+    },{// one point on equator (descending)
+       { 150, 0},{ 3, 1 },
+       60.29513726,
+       60.30158943,
+       60.3015947,
+       60
+    },{// one point on equator (ascending)
+       { 3, 0 },{ 150, 1},
+       92.69840523,
+       92.6984053,
+       92.6984053,
+       93
+    },{ //meridian
+        { 1, 1 },{ 1, 2 },
+        1,
+        1,
+        1,
+        1
+    },{ //nearly meridian
+        { 1, 1 },{ 1.001, 2 },
+        1.001,
+        1.001,
+        1.001,
+        1.001
+    },{ //vertex is a segment's endpoint
+        { 1, 1 },{ 10, 2 },
+        10,
+        10,
+        10,
+        10
+    },{
+        { 10, 1 },{ 1, 2 },
+        1,
+        1,
+        1,
+        1
+    },{ //South hemisphere, ascending
+        { 0, -1 },{ 50, -1.4 },
+        44.53768958,
+        44.53788035,
+        44.53788052,
+        44.6729585
+    },{ //South hemisphere, descending
+        { 0, -1.5 },{ 50, -1 },
+        1.936164356,
+        1.935976219,
+        1.935978042,
+        1.779116148
+    },{ //South hemisphere, same latitude
+        { 0, -1 },{ 50, -1 },
+        24.99994261,
+        24.99999901,
+        24.99999995,
+        25
+    },{//Both hemispheres, vertex on the northern
+       //A desc vertex north
+       { 3, 5 },{ 150, -3},
+       27.357069,
+       27.36422922,
+       27.36423549,
+       26.74999989
+    },{//B asc vertex north
+       { 3, -3 },{ 150, 5},
+       125.6403436,
+       125.6357677,
+       125.6357659,
+       126.2500001
+    },{//C desc vertex south
+       { 3, -5 },{ 150, 3},
+       27.3570679,
+       27.36422812,
+       27.36423439,
+       26.74999989
+    },{//D asc vertex south
+       { 3, 3 },{ 150, -5},
+       125.6403423,
+       125.6357664,
+       125.6357645,
+       126.2500001
+    },{//E asc vertex south
+       { 3, 3 },{ 184, -5},
+       -88.00743796,
+       -88.0660268,
+       -88.0558747,
+       -88.49315894
+    },{
+        { 3, 5 },{ 150, -3.5},
+        17.96722293,
+        17.97322407,
+        17.97323051,
+        17.3742464
+    },{
+        { 3, 5 },{ 150, -1},
+        52.9706038,
+        52.97759463,
+        52.9775964,
+        52.56504545
+    },{ //Both hemispheres, vertex on the southern
+        { 3, 3},{ 5, -5},
+        5,
+        5,
+        5,
+        5
+    },{
+        { 3, -5 },{ 150, 1}, //symmetric to { 3, 5 },{ 150, -1}
+        52.97060093,
+        52.97759176,
+        52.97759353,
+        52.56504545
+    },{// fix p1 lon, lat and p2 lon and vary p2 lat
+       { 3, 5 },{ 150, 1},
+       65.51880171,
+       65.52391866,
+       65.52391623,
+       65.31813729
+    },{
+        { 3, 5 },{ 150, 0},
+        60.29182988,
+        60.29785309,
+        60.29785255,
+        60
+    },{
+        { 3, 5 },{ 150, -0.1},
+        59.66911673,
+        59.67523649,
+        59.67523616,
+        59.36690727
+    },{
+        { 3, 5 },{ 150, -1},
+        52.9706038,
+        52.97759463,
+        52.9775964,
+        52.56504545
+    },{
+        { 3, 5 },{ 150, -4.15},
+        4.481947557,
+        4.485467841,
+        4.485473295,
+        3.981178967
+    },{
+        { 3, 5 },{ 150, -4.2},
+        3,
+        3,
+        3,
+        3
+    },{//symmetry of geodesics:
+           // (i) case A same as C and B same as D
+           // (ii) longitude diff between vertex and p2 in A, C equals
+           //      longitude diff between vertex and p1 in B, D by symmetry
+           // case (A)
+           { 0, 5 },{ 30, 5.5},
+           25.06431998,
+           25.0644277,
+           25.06442787,
+           25.13253724
+        },{// case (B)
+           { 0, 5.5 },{ 30, 5},
+           4.935667094,
+           4.935571216,
+           4.93557213,
+           4.867462762
+        },{// case (C)
+           { 0, -5 },{ 30, -5.5},
+           25.06431885,
+           25.06442657,
+           25.06442674,
+           25.13253724
+        },{// case (D)
+           { 0, -5.5 },{ 30, -5},
+           4.935666841,
+           4.935570963,
+           4.935571877,
+           4.867462762
+        },{//crossing meridian
+           { -10, 1 },{ 50, 1.1},
+           24.68113946,
+           24.68127641,
+           24.68127733,
+           24.71605263
+        },{
+           { 350, 1 },{ 50, 1.1},
+           24.68113946,
+           24.68127641,
+           24.68127733,
+           24.71605263
+        },{//crossing antimeridian
+           { 130, 1 },{ 190, 1.1},
+           164.6811395,
+           164.6812764,
+           164.6812773,
+           164.7160526
+        },{
+           { 130, 1 },{ -170, 1.1},
+           164.6811395,
+           164.6812764,
+           164.6812773,
+           164.7160526
+        },{//crossing meridian both hemispheres
+           { -10, -5 },{ 150, 1},
+           55.61285835,
+           55.62727853,
+           55.62725182,
+           55.19943725
+        },{
+            { 350, -5 },{ 150, 1},
+            55.6243632,
+            55.6272619,
+            55.627257,
+            55.1994373
+        },{//crossing anti-meridian both hemispheres
+           { 90, -5 },{ 210, 1},
+           109.4997596,
+           109.5011987,
+           109.5012031,
+           109.1354089
+        },{
+            { 90, -5 },{ -150, 1},
+            109.4997596,
+            109.5011987,
+            109.5012031,
+            109.1354089
+        },{
+            { -150, -5 },{ 90, 1},
+            -169.4997596,
+            -169.5011987,
+            -169.5012031,
+            -169.1354089
+        },{
+            { 90, 1 },{ 210, -5},
+            -169.5008004,
+            -169.5012037,
+            -169.501204,
+            -169.1354089
+        },{
+            { 0, 1 },{ 120, -5},
+            100.4991996,
+            100.4987963,
+            100.498796,
+            100.8645911
+        }
+};
+
+size_t const expected_size = sizeof(expected) / sizeof(expected_results);
+
+#endif // BOOST_GEOMETRY_TEST_VERTEX_LONGITUDE_CASES_HPP