id	summary	reporter	owner	description	type	status	milestone	component	version	severity	resolution	keywords	cc
11929	haversine method is inaccurate	Charles Karney <charles@…>	Barend Gehrels	"The following code illustrates a problem with the ""haversine"" strategy
for computing great circle distances.  It computes the distance
between two nearly antipodal points.  The relative error in the result
is 5e-9, much bigger than you should get with doubles.

{{{#!c++
#include <iostream>
#include <cmath>
#include <boost/geometry.hpp>

namespace bg = boost::geometry;
typedef bg::model::point<double, 2,
                         bg::cs::spherical_equatorial<bg::degree> > pt;

int main() {
  double pi = std::atan2(0, -1);
  bg::model::segment<pt> seg;
  bg::read_wkt(""SEGMENT(0.000001 0, 180 0)"", seg);
  double
    // arc length (converted to degrees
    dist = bg::distance(seg.first, seg.second) * (180/pi),
    // the correct distance (in degrees)
    dist0 = 180 - 0.000001,
    err = (dist - dist0) / dist0;
  std::cout << ""relative error "" << err << ""\n"";
}
}}}

Running this code gives
{{{
relative error 5.55556e-09
}}}

Discussion:
* Surely it's a bad idea to embed the algorithm name ""haversine"" into
  the API.  Computing great circle distances is really
  straightforward.  Users shouldn't need to worry about the underlying
  method.  Maybe ""great_circle"" is a better name.
* The use of haversines made sense when the log haversine function was
  tabulated and you needed trigonometric identities involving products
  to facilitate computations with log tables.  Nowadays there are
  better formulas.
* As this example illustrates, it's subject to a loss of accuracy with
  antipodal points (it takes the arcsine of a number close to 1.)
* A good formula to use is given in the Wikipedia article on
  [https://en.wikipedia.org/wiki/Great-circle_navigation#cite_note-2 great-circle navigation]
  (this is the formula used by Vincenty in the spherical limit
  -- but it surely doesn't originate with him).
"	Bugs	new	To Be Determined	geometry	Boost 1.58.0	Problem		distance	charles.karney@…