// For shapes of dimension 2, returns the area of the shape on the unit
// sphere. The result is between 0 and 4*Pi steradians. Otherwise returns
// zero. This method has good relative accuracy for both very large and very
// small regions.
//
// All edges are modeled as spherical geodesics. The result can be converted
// to an area on the Earth's surface (with a worst-case error of 0.900% near
// the poles) using the functions in s2earth.h.
public static double GetArea(S2Shape shape)
{
if (shape.Dimension() != 2)
{
return(0.0);
}
// Since S2Shape uses the convention that the interior of the shape is to
// the left of all edges, in theory we could compute the area of the polygon
// by simply adding up all the loop areas modulo 4*Pi. The problem with
// this approach is that polygons holes typically have areas near 4*Pi,
// which can create large cancellation errors when computing the area of
// small polygons with holes. For example, a shell with an area of 4 square
// meters (1e-13 steradians) surrounding a hole with an area of 3 square
// meters (7.5e-14 sterians) would lose almost all of its accuracy if the
// area of the hole was computed as 12.566370614359098.
//
// So instead we use S2.GetSignedArea() to ensure that all loops have areas
// in the range [-2*Pi, 2*Pi].
//
// TODO(ericv): Rarely, this function returns the area of the complementary
// region (4*Pi - area). This can only happen when the true area is very
// close to zero or 4*Pi and the polygon has multiple loops. To make this
// function completely robust requires checking whether the signed area sum is
// ambiguous, and if so, determining the loop nesting structure. This allows
// the sum to be evaluated in a way that is guaranteed to have the correct
// sign.
double area = 0;
double max_error = 0;
int num_chains = shape.NumChains();
for (int chain_id = 0; chain_id < num_chains; ++chain_id)
{
GetChainVertices(shape, chain_id, out var vertices);
area += S2.GetSignedArea(vertices.ToList());
#if DEBUG
max_error += S2.GetCurvatureMaxError(new(vertices));
#endif
}
// Note that S2.GetSignedArea() guarantees that the full loop (containing
// all points on the sphere) has a very small negative area.
System.Diagnostics.Debug.Assert(Math.Abs(area) <= S2.M_4_PI + max_error);
if (area < 0.0)
{
area += S2.M_4_PI;
}
return(area);
}
public void Test_GetSignedArea_Underflow()
{
var loop = ParsePointsOrDie("0:0, 0:1e-88, 1e-88:1e-88, 1e-88:0");
Assert.True(S2.GetSignedArea(loop) > 0);
}