id,summary,reporter,owner,description,type,status,milestone,component,version,severity,resolution,keywords,cc
11931,"spherical area wrong with pole encircling polygons, etc.",Charles Karney <charles@…>,Barend Gehrels,"The following code illustrates a problem with handling of meridian
crossing and pole encircling when computing spherical areas.  It
computes the area of a 4 identical diamond shaped areas at different
positions on the sphere.  Only the area of the last of these is
computed correctly.  The others are badly wrong.

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

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

int main() {
  std::string polys[] = {
    "" 0  89,  90  89,  180  89,  270  89"",
    "" 0 -89, -90 -89, -180 -89, -270 -89"",
    ""-1   0,   0  -1,    1   0,    0   1"",
    ""89   0,  90  -1,   91   0,   90   1""
  };
  // Correct area for all these polygons is
  double area0 = 0.00060926577032113655;
  bg::model::polygon<pt, false, false> poly;
  std::cout << ""  area      relerr"" << ""\n"";
  for (int i = 0; i < 4; ++i) {
    bg::read_wkt(""POLYGON(("" + polys[i] + ""))"", poly);
    double area = bg::area(poly);
    std::cout << area << "" "" << (area - area0) / area0 << ""\n"";
  }
}
}}}

Running this code gives
{{{
  area      relerr
0.000304633 -0.5
-6.28288 -10313.2
0 -1
0.000609266 2.31338e-15
}}}

Discussion:
* I can't make out the handling of this issue in the current code at
  all.  It seems to be irredeemably broken.
* You need to do two things to handle these issues:
  1. Make sure that the sign of lambda_2 - lambda_1 correctly
    corresponds to east and west going edges (with some consistent
    convention of edges which pass over the pole).
  2. Count the number of times the polygon encircles a pole, e.g., by
    keeping track of when an edge crosses the prime meridian (again with
    some consistent treatment of cases when one or both ends of an edge
    lie on the prime meridian).
* If this count is odd, then the total area needs to be adjusted by
  2*pi, half the area of the sphere.
* I believe I've handled these issues correctly in the PolygonAreaT
  class in [http://geographiclib.sourceforge.net/html/ GeographicLib].
",Bugs,closed,Boost 1.64.0,geometry,Boost 1.58.0,Showstopper,fixed,area,charles.karney@…