Why is boost::geometry geographic Vincenty distance inaccurate around the Equator?
<p>I require a function to calculate the distance between a pair of <a href="http://www.icao.int/safety/pbn/documentation/eurocontrol/eurocontrol%20wgs%2084%20implementation%20manual.pdf" rel="noreferrer">WGS 84</a> positions to a high degree of accuracy and I was planning to use the <code>geographic</code> functions from <a href="http://www.boost.org/doc/libs/1_60_0/libs/geometry/doc/html/index.html" rel="noreferrer">boost geometry</a>.</p>

<p>The <a href="http://www.boost.org/doc/libs/1_60_0/libs/geometry/doc/html/geometry/design.html" rel="noreferrer">boost geometry Design Rational</a> states:</p>

<blockquote>
  <p>There is the Andoyer method, fast and precise, and there is the Vincenty method, slower and more precise..</p>
</blockquote>

<p>However, when testing the <code>boost::geometry::distance</code> function with both the <code>Andoyer</code> and <code>Vincenty</code> strategies, I got the following results:</p>

<pre><code>WGS 84 values (metres)
    Semimajor axis:         6378137.000000
    Flattening:             0.003353
    Semiminor axis:         6356752.314245

    Semimajor distance:     20037508.342789
    Semiminor distance:     19970326.371123

Boost geometry near poles
Andoyer function:
    Semimajor distance:     20037508.151445
    Semiminor distance:     20003917.164970
Vincenty function:
    Semimajor distance:     **19970326.180419**
    Semiminor distance:     20003931.266635

Boost geometry at poles
Andoyer function:
    Semimajor distance:     0.000000
    Semiminor distance:     0.000000
Vincenty function:
    Semimajor distance:     **19970326.371122**
    Semiminor distance:     20003931.458623
</code></pre>

<p>The <code>Vincenty</code> distances along the Semimajor axis (i.e. around the Equator) are less than the distance around the Semiminor axis between the North and South Poles. That can't be correct.</p>

<p>The Semiminor  and <code>Andoyer</code> distances look reasonable. Except when the points are on opposing side of the Earth, when the <code>boost</code> <code>Andoyer</code> function returns zero!</p>

<p>Is the problem in: the <code>Vincenty</code> algorithm, the <code>boost geometry</code> implementation of it, or my test code?  </p>

<p>Test code:</p>

<pre><code>/// boost geometry WGS84 distance issue

// Note: M_PI is not part of the C or C++ standards, _USE_MATH_DEFINES enables it
#define _USE_MATH_DEFINES
#include &lt;boost/geometry.hpp&gt;
#include &lt;cmath&gt;
#include &lt;iostream&gt;
#include &lt;ios&gt;

// WGS 84 parameters from: Eurocontrol WGS 84 Implementation Manual
// Version 2.4 Chapter 3, page 14

/// The Semimajor axis measured in metres.
/// This is the radius at the equator.
constexpr double a = 6378137.0;

/// Flattening, a ratio.
/// This is the flattening of the ellipse at the poles
constexpr double f = 1.0/298.257223563;

/// The Semiminor axis measured in metres.
/// This is the radius at the poles.
/// Note: this is derived from the Semimajor axis and the flattening.
/// See WGS 84 Implementation Manual equation B-2, page 69.
constexpr double b = a * (1.0 - f);

int main(int /*argc*/, char ** /*argv*/)
{
  std::cout.setf(std::ios::fixed);

  std::cout &lt;&lt; "WGS 84 values (metres)\n";
  std::cout &lt;&lt; "\tSemimajor axis:\t\t"   &lt;&lt; a &lt;&lt; "\n";
  std::cout &lt;&lt; "\tFlattening:\t\t"       &lt;&lt; f &lt;&lt; "\n";
  std::cout &lt;&lt; "\tSemiminor axis:\t\t"   &lt;&lt; b &lt;&lt; "\n\n";

  std::cout &lt;&lt; "\tSemimajor distance:\t" &lt;&lt; M_PI * a &lt;&lt; "\n";
  std::cout &lt;&lt; "\tSemiminor distance:\t" &lt;&lt; M_PI * b &lt;&lt; "\n";
  std::cout &lt;&lt; std::endl;

  // Min value for delta. 0.000000014 causes Andoyer to fail.
  const double DELTA(0.000000015);

  // For boost::geometry:
  typedef boost::geometry::cs::geographic&lt;boost::geometry::radian&gt; Wgs84Coords;
  typedef boost::geometry::model::point&lt;double, 2, Wgs84Coords&gt; GeographicPoint;
  // Note boost points are Long &amp; Lat NOT Lat &amp; Long
  GeographicPoint near_north_pole   (0.0,  M_PI_2 - DELTA);
  GeographicPoint near_south_pole   (0.0, -M_PI_2 + DELTA);

  GeographicPoint near_equator_east ( M_PI_2 - DELTA, 0.0);
  GeographicPoint near_equator_west (-M_PI_2 + DELTA, 0.0);

  // Note: the default boost geometry spheroid is WGS84
  // #include &lt;boost/geometry/core/srs.hpp&gt;
  typedef boost::geometry::srs::spheroid&lt;double&gt; SpheroidType;
  SpheroidType spheriod;

  //#include &lt;boost/geometry/strategies/geographic/distance_andoyer.hpp&gt;
  typedef boost::geometry::strategy::distance::andoyer&lt;SpheroidType&gt;
                                                               AndoyerStrategy;
  AndoyerStrategy andoyer(spheriod);

  std::cout &lt;&lt; "Boost geometry near poles\n";
  std::cout &lt;&lt; "Andoyer function:\n";
  double andoyer_major(boost::geometry::distance(near_equator_east, near_equator_west, andoyer));
  std::cout &lt;&lt; "\tSemimajor distance:\t" &lt;&lt; andoyer_major &lt;&lt; "\n";
  double andoyer_minor(boost::geometry::distance(near_north_pole, near_south_pole, andoyer));
  std::cout &lt;&lt; "\tSemiminor distance:\t" &lt;&lt; andoyer_minor &lt;&lt; "\n";

  //#include &lt;boost/geometry/strategies/geographic/distance_vincenty.hpp&gt;
  typedef boost::geometry::strategy::distance::vincenty&lt;SpheroidType&gt;
                                                               VincentyStrategy;
  VincentyStrategy vincenty(spheriod);

  std::cout &lt;&lt; "Vincenty function:\n";
  double vincenty_major(boost::geometry::distance(near_equator_east, near_equator_west, vincenty));
  std::cout &lt;&lt; "\tSemimajor distance:\t" &lt;&lt; vincenty_major &lt;&lt; "\n";
  double vincenty_minor(boost::geometry::distance(near_north_pole, near_south_pole, vincenty));
  std::cout &lt;&lt; "\tSemiminor distance:\t" &lt;&lt; vincenty_minor &lt;&lt; "\n\n";

  // Note boost points are Long &amp; Lat NOT Lat &amp; Long
  GeographicPoint north_pole   (0.0,  M_PI_2);
  GeographicPoint south_pole   (0.0, -M_PI_2);

  GeographicPoint equator_east ( M_PI_2, 0.0);
  GeographicPoint equator_west (-M_PI_2, 0.0);

  std::cout &lt;&lt; "Boost geometry at poles\n";
  std::cout &lt;&lt; "Andoyer function:\n";
  andoyer_major = boost::geometry::distance(equator_east, equator_west, andoyer);
  std::cout &lt;&lt; "\tSemimajor distance:\t" &lt;&lt; andoyer_major &lt;&lt; "\n";
  andoyer_minor = boost::geometry::distance(north_pole, south_pole, andoyer);
  std::cout &lt;&lt; "\tSemiminor distance:\t" &lt;&lt; andoyer_minor &lt;&lt; "\n";

  std::cout &lt;&lt; "Vincenty function:\n";
  vincenty_major = boost::geometry::distance(equator_east, equator_west, vincenty);
  std::cout &lt;&lt; "\tSemimajor distance:\t" &lt;&lt; vincenty_major &lt;&lt; "\n";
  vincenty_minor = boost::geometry::distance(north_pole, south_pole, vincenty);
  std::cout &lt;&lt; "\tSemiminor distance:\t" &lt;&lt; vincenty_minor &lt;&lt; "\n";

  return 0;
}
</code></pre>