/**
* Helper method to get area and optionally centroid.
*/
private S2AreaCentroid GetAreaCentroid(bool doCentroid)
{
// Don't crash even if loop is not well-defined.
if (NumVertices < 3)
{
return(new S2AreaCentroid(0D));
}
// The triangle area calculation becomes numerically unstable as the length
// of any edge approaches 180 degrees. However, a loop may contain vertices
// that are 180 degrees apart and still be valid, e.g. a loop that defines
// the northern hemisphere using four points. We handle this case by using
// triangles centered around an origin that is slightly displaced from the
// first vertex. The amount of displacement is enough to get plenty of
// accuracy for antipodal points, but small enough so that we still get
// accurate areas for very tiny triangles.
//
// Of course, if the loop contains a point that is exactly antipodal from
// our slightly displaced vertex, the area will still be unstable, but we
// expect this case to be very unlikely (i.e. a polygon with two vertices on
// opposite sides of the Earth with one of them displaced by about 2mm in
// exactly the right direction). Note that the approximate point resolution
// using the E7 or S2CellId representation is only about 1cm.
var origin = Vertex(0);
var axis = (origin.LargestAbsComponent + 1) % 3;
var slightlyDisplaced = origin[axis] + S2.E * 1e-10;
origin =
new S2Point((axis == 0) ? slightlyDisplaced : origin.X,
(axis == 1) ? slightlyDisplaced : origin.Y, (axis == 2) ? slightlyDisplaced : origin.Z);
origin = S2Point.Normalize(origin);
double areaSum = 0;
var centroidSum = new S2Point(0, 0, 0);
for (var i = 1; i <= NumVertices; ++i)
{
areaSum += S2.SignedArea(origin, Vertex(i - 1), Vertex(i));
if (doCentroid)
{
// The true centroid is already premultiplied by the triangle area.
var trueCentroid = S2.TrueCentroid(origin, Vertex(i - 1), Vertex(i));
centroidSum = centroidSum + trueCentroid;
}
}
// The calculated area at this point should be between -4*Pi and 4*Pi,
// although it may be slightly larger or smaller than this due to
// numerical errors.
// assert (Math.abs(areaSum) <= 4 * S2.M_PI + 1e-12);
if (areaSum < 0)
{
// If the area is negative, we have computed the area to the right of the
// loop. The area to the left is 4*Pi - (-area). Amazingly, the centroid
// does not need to be changed, since it is the negative of the integral
// of position over the region to the right of the loop. This is the same
// as the integral of position over the region to the left of the loop,
// since the integral of position over the entire sphere is (0, 0, 0).
areaSum += 4 * S2.Pi;
}
// The loop's sign() does not affect the return result and should be taken
// into account by the caller.
S2Point?centroid = null;
if (doCentroid)
{
centroid = centroidSum;
}
return(new S2AreaCentroid(areaSum, centroid));
}
