public void Test_S2_AreaMethods()
{
S2Point pz = new(0, 0, 1);
S2Point p000 = new(1, 0, 0);
S2Point p045 = new S2Point(1, 1, 0).Normalize();
S2Point p090 = new(0, 1, 0);
S2Point p180 = new(-1, 0, 0);
Assert2.Near(S2.Area(p000, p090, pz), S2.M_PI_2);
Assert2.Near(S2.Area(p045, pz, p180), 3 * S2.M_PI_4);
// Make sure that Area() has good *relative* accuracy even for
// very small areas.
const double eps = 1e-10;
S2Point pepsx = new S2Point(eps, 0, 1).Normalize();
S2Point pepsy = new S2Point(0, eps, 1).Normalize();
double expected1 = 0.5 * eps * eps;
Assert2.Near(S2.Area(pepsx, pepsy, pz), expected1, 1e-14 * expected1);
// Make sure that it can handle degenerate triangles.
S2Point pr = new S2Point(0.257, -0.5723, 0.112).Normalize();
S2Point pq = new S2Point(-0.747, 0.401, 0.2235).Normalize();
Assert.Equal(0, S2.Area(pr, pr, pr));
// The following test is not exact due to rounding error.
Assert2.Near(S2.Area(pr, pq, pr), 0, S2.DoubleError);
Assert.Equal(0, S2.Area(p000, p045, p090));
double max_girard = 0;
for (int i = 0; i < 10000; ++i)
{
S2Point p0 = S2Testing.RandomPoint();
S2Point d1 = S2Testing.RandomPoint();
S2Point d2 = S2Testing.RandomPoint();
S2Point p1 = (p0 + S2.DoubleError * d1).Normalize();
S2Point p2 = (p0 + S2.DoubleError * d2).Normalize();
// The actual displacement can be as much as 1.2e-15 due to roundoff.
// This yields a maximum triangle area of about 0.7e-30.
Assert.True(S2.Area(p0, p1, p2) <= 0.7e-30);
max_girard = Math.Max(max_girard, S2.GirardArea(p0, p1, p2));
}
// This check only passes if GirardArea() uses RobustCrossProd().
Assert.True(max_girard <= 1e-14);
// Try a very long and skinny triangle.
S2Point p045eps = new S2Point(1, 1, eps).Normalize();
double expected2 = 5.8578643762690495119753e-11; // Mathematica.
Assert2.Near(S2.Area(p000, p045eps, p090), expected2, 1e-9 * expected2);
// Triangles with near-180 degree edges that sum to a quarter-sphere.
const double eps2 = 1e-14;
S2Point p000eps2 = new S2Point(1, 0.1 * eps2, eps2).Normalize();
double quarter_area1 = S2.Area(p000eps2, p000, p045) +
S2.Area(p000eps2, p045, p180) +
S2.Area(p000eps2, p180, pz) +
S2.Area(p000eps2, pz, p000);
Assert2.Near(quarter_area1, Math.PI);
// Four other triangles that sum to a quarter-sphere.
S2Point p045eps2 = new S2Point(1, 1, eps2).Normalize();
double quarter_area2 = S2.Area(p045eps2, p000, p045) +
S2.Area(p045eps2, p045, p180) +
S2.Area(p045eps2, p180, pz) +
S2.Area(p045eps2, pz, p000);
Assert2.Near(quarter_area2, Math.PI);
// Compute the area of a hemisphere using four triangles with one near-180
// degree edge and one near-degenerate edge.
for (int i = 0; i < 100; ++i)
{
double lng = S2.M_2_PI * S2Testing.Random.RandDouble();
S2Point p0 = S2LatLng.FromRadians(1e-20, lng).Normalized().ToPoint();
S2Point p1 = S2LatLng.FromRadians(0, lng).Normalized().ToPoint();
double p2_lng = lng + S2Testing.Random.RandDouble();
S2Point p2 = S2LatLng.FromRadians(0, p2_lng).Normalized().ToPoint();
S2Point p3 = S2LatLng.FromRadians(0, lng + Math.PI).Normalized().ToPoint();
S2Point p4 = S2LatLng.FromRadians(0, lng + 5.0).Normalized().ToPoint();
double area = (S2.Area(p0, p1, p2) + S2.Area(p0, p2, p3) +
S2.Area(p0, p3, p4) + S2.Area(p0, p4, p1));
Assert2.Near(area, S2.M_2_PI, 2e-15);
}
// This tests a case where the triangle has zero area, but S2.Area()
// computes (dmin > 0) due to rounding errors.
Assert.Equal(0.0, S2.Area(S2LatLng.FromDegrees(-45, -170).ToPoint(),
S2LatLng.FromDegrees(45, -170).ToPoint(),
S2LatLng.FromDegrees(0, -170).ToPoint()));
}
public void testAngleArea()
{
var pz = new S2Point(0, 0, 1);
var p000 = new S2Point(1, 0, 0);
var p045 = new S2Point(1, 1, 0);
var p090 = new S2Point(0, 1, 0);
var p180 = new S2Point(-1, 0, 0);
assertDoubleNear(S2.Angle(p000, pz, p045), S2.PiOver4);
assertDoubleNear(S2.Angle(p045, pz, p180), 3 * S2.PiOver4);
assertDoubleNear(S2.Angle(p000, pz, p180), S2.Pi);
assertDoubleNear(S2.Angle(pz, p000, pz), 0);
assertDoubleNear(S2.Angle(pz, p000, p045), S2.PiOver2);
assertDoubleNear(S2.Area(p000, p090, pz), S2.PiOver2);
assertDoubleNear(S2.Area(p045, pz, p180), 3 * S2.PiOver4);
// Make sure that area() has good *relative* accuracy even for
// very small areas.
var eps = 1e-10;
var pepsx = new S2Point(eps, 0, 1);
var pepsy = new S2Point(0, eps, 1);
var expected1 = 0.5 * eps * eps;
assertDoubleNear(S2.Area(pepsx, pepsy, pz), expected1, 1e-14 * expected1);
// Make sure that it can handle degenerate triangles.
var pr = new S2Point(0.257, -0.5723, 0.112);
var pq = new S2Point(-0.747, 0.401, 0.2235);
assertEquals(S2.Area(pr, pr, pr), 0.0);
// TODO: The following test is not exact in optimized mode because the
// compiler chooses to mix 64-bit and 80-bit intermediate results.
assertDoubleNear(S2.Area(pr, pq, pr), 0);
assertEquals(S2.Area(p000, p045, p090), 0.0);
double maxGirard = 0;
for (var i = 0; i < 10000; ++i)
{
var p0 = randomPoint();
var d1 = randomPoint();
var d2 = randomPoint();
var p1 = p0 + (d1 * 1e-15);
var p2 = p0 + (d2 * 1e-15);
// The actual displacement can be as much as 1.2e-15 due to roundoff.
// This yields a maximum triangle area of about 0.7e-30.
assertTrue(S2.Area(p0, p1, p2) < 0.7e-30);
maxGirard = Math.Max(maxGirard, S2.GirardArea(p0, p1, p2));
}
Console.WriteLine("Worst case Girard for triangle area 1e-30: " + maxGirard);
// Try a very long and skinny triangle.
var p045eps = new S2Point(1, 1, eps);
var expected2 = 5.8578643762690495119753e-11; // Mathematica.
assertDoubleNear(S2.Area(p000, p045eps, p090), expected2, 1e-9 * expected2);
// Triangles with near-180 degree edges that sum to a quarter-sphere.
var eps2 = 1e-10;
var p000eps2 = new S2Point(1, 0.1 * eps2, eps2);
var quarterArea1 =
S2.Area(p000eps2, p000, p090) + S2.Area(p000eps2, p090, p180) + S2.Area(p000eps2, p180, pz)
+ S2.Area(p000eps2, pz, p000);
assertDoubleNear(quarterArea1, S2.Pi);
// Four other triangles that sum to a quarter-sphere.
var p045eps2 = new S2Point(1, 1, eps2);
var quarterArea2 =
S2.Area(p045eps2, p000, p090) + S2.Area(p045eps2, p090, p180) + S2.Area(p045eps2, p180, pz)
+ S2.Area(p045eps2, pz, p000);
assertDoubleNear(quarterArea2, S2.Pi);
}
