public void Test_LoopTestBase_GetAreaConsistentWithOrientation()
{
// Test that GetArea() returns an area near 0 for degenerate loops that
// contain almost no points, and an area near 4*Pi for degenerate loops that
// contain almost all points.
const int kMaxVertices = 6;
for (int i = 0; i < 50; ++i)
{
int num_vertices = 3 + S2Testing.Random.Uniform(kMaxVertices - 3 + 1);
// Repeatedly choose N vertices that are exactly on the equator until we
// find some that form a valid loop.
S2PointLoopSpan loop = new();
do
{
for (int i2 = 0; i2 < num_vertices; ++i2)
{
// We limit longitude to the range [0, 90] to ensure that the loop is
// degenerate (as opposed to following the entire equator).
loop.Add(
S2LatLng.FromRadians(0, S2Testing.Random.RandDouble() * S2.M_PI_2).ToPoint());
}
} while (!new S2Loop(loop, S2Debug.DISABLE).IsValid());
bool ccw = S2.IsNormalized(loop);
// The error bound is sufficient for current tests but not guaranteed.
_ = i + ": " + loop.ToDebugString();
Assert2.Near(ccw ? 0 : S2.M_4_PI, S2.GetArea(loop), 1e-14);
Assert.Equal(!ccw, new S2Loop(loop).Contains(new S2Point(0, 0, 1)));
}
}
public void Test_LoopTestBase_GetCurvature()
{
Assert.Equal(-S2.M_2_PI, S2.GetCurvature(full_));
Assert.Equal(S2.M_2_PI, S2.GetCurvature(v_loop_));
CheckCurvatureInvariants(v_loop_);
// This curvature should be computed exactly.
Assert.Equal(0, S2.GetCurvature(north_hemi3_));
CheckCurvatureInvariants(north_hemi3_);
Assert2.Near(0, S2.GetCurvature(west_hemi_), 1e-15);
CheckCurvatureInvariants(west_hemi_);
// We don't have an easy way to estimate the curvature of these loops, but
// we can still check that the expected invariants hold.
CheckCurvatureInvariants(candy_cane_);
CheckCurvatureInvariants(three_leaf_clover_);
Assert2.DoubleEqual(S2.M_2_PI, S2.GetCurvature(line_triangle_));
CheckCurvatureInvariants(line_triangle_);
Assert2.DoubleEqual(S2.M_2_PI, S2.GetCurvature(skinny_chevron_));
CheckCurvatureInvariants(skinny_chevron_);
// Build a narrow spiral loop starting at the north pole. This is designed
// to test that the error in GetCurvature is linear in the number of
// vertices even when the partial sum of the curvatures gets very large.
// The spiral consists of two "arms" defining opposite sides of the loop.
// This is a pathological loop that contains many long parallel edges.
int kArmPoints = 10000; // Number of vertices in each "arm"
double kArmRadius = 0.01; // Radius of spiral.
S2PointLoopSpan spiral = new(2 * kArmPoints);
spiral[kArmPoints] = new S2Point(0, 0, 1);
for (int i = 0; i < kArmPoints; ++i)
{
double angle = (S2.M_2_PI / 3) * i;
double x = Math.Cos(angle);
double y = Math.Sin(angle);
double r1 = i * kArmRadius / kArmPoints;
double r2 = (i + 1.5) * kArmRadius / kArmPoints;
spiral[kArmPoints - i - 1] = new S2Point(r1 * x, r1 * y, 1).Normalize();
spiral[kArmPoints + i] = new S2Point(r2 * x, r2 * y, 1).Normalize();
}
// Check that GetCurvature() is consistent with GetArea() to within the
// error bound of the former. We actually use a tiny fraction of the
// worst-case error bound, since the worst case only happens when all the
// roundoff errors happen in the same direction and this test is not
// designed to achieve that. The error in GetArea() can be ignored for the
// purposes of this test since it is generally much smaller.
Assert2.Near(
S2.M_2_PI - S2.GetArea(spiral),
S2.GetCurvature(spiral),
0.01 * S2.GetCurvatureMaxError(spiral));
}
public void Test_GetArea_TinyShellAndHole()
{
// A square about 20 um on each side with a hole 10 um on each side.
double side = S1Angle.FromDegrees(1e-10).Radians;
Assert.Equal(3 * side * side, S2.GetArea(MakeLaxPolygonOrDie(
"0:0, 0:2e-10, 2e-10:2e-10, 2e-10:0; " +
"0.5e-10:0.5e-10, 1.5e-10:0.5e-10, 1.5e-10:1.5e-10, 0.5e-10:1.5e-10")));
}
public void Test_GetArea_TwoTinyShells()
{
// Two small squares with sides about 10 um (micrometers) long.
double side = S1Angle.FromDegrees(1e-10).Radians;
Assert.Equal(2 * side * side, S2.GetArea(MakeLaxPolygonOrDie(
"0:0, 0:1e-10, 1e-10:1e-10, 1e-10:0; " +
"0:0, 0:-1e-10, -1e-10:-1e-10, -1e-10:0")));
}
private static void TestAreaConsistentWithCurvature(S2PointLoopSpan loop)
{
// Check that the area computed using GetArea() is consistent with the loop
// curvature. According to the Gauss-Bonnet theorem, the area of the loop
// equals 2*Pi minus its curvature.
double area = S2.GetArea(loop);
double gauss_area = S2.M_2_PI - S2.GetCurvature(loop);
// The error bound below is sufficient for current tests but not guaranteed.
_ = loop.ToDebugString();
Assert.True(Math.Abs(area - gauss_area) <= 1e-14);
}
// Returns the total area of all polygons in the index. Returns zero if no
// polygons are present. This method has good relative accuracy for both very
// large and very small regions. Note that the result may exceed 4*Pi if the
// index contains overlapping polygons.
//
// 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(S2ShapeIndex index)
{
double area = 0;
for (int i = 0; i < index.NumShapeIds(); ++i)
{
var shape = index.Shape(i);
if (shape != null)
{
area += S2.GetArea(shape);
}
}
return(area);
}
public void Test_LoopTestBase_GetAreaAndCentroid()
{
Assert.Equal(S2.M_4_PI, S2.GetArea(full_));
Assert.Equal(S2Point.Empty, S2.GetCentroid(full_));
Assert2.DoubleEqual(S2.GetArea(north_hemi_), S2.M_2_PI);
Assert2.Near(S2.M_2_PI, S2.GetArea(east_hemi_), 1e-12);
// Construct spherical caps of random height, and approximate their boundary
// with closely spaces vertices. Then check that the area and centroid are
// correct.
for (int iter = 0; iter < 50; ++iter)
{
// Choose a coordinate frame for the spherical cap.
S2Testing.GetRandomFrame(out var x, out var y, out var z);
// Given two points at latitude phi and whose longitudes differ by dtheta,
// the geodesic between the two points has a maximum latitude of
// atan(tan(phi) / cos(dtheta/2)). This can be derived by positioning
// the two points at (-dtheta/2, phi) and (dtheta/2, phi).
//
// We want to position the vertices close enough together so that their
// maximum distance from the boundary of the spherical cap is kMaxDist.
// Thus we want Math.Abs(atan(tan(phi) / cos(dtheta/2)) - phi) <= kMaxDist.
const double kMaxDist = 1e-6;
double height = 2 * S2Testing.Random.RandDouble();
double phi = Math.Asin(1 - height);
double max_dtheta = 2 * Math.Acos(Math.Tan(Math.Abs(phi)) / Math.Tan(Math.Abs(phi) + kMaxDist));
max_dtheta = Math.Min(Math.PI, max_dtheta); // At least 3 vertices.
S2PointLoopSpan loop = new();
for (double theta = 0; theta < S2.M_2_PI;
theta += S2Testing.Random.RandDouble() * max_dtheta)
{
loop.Add(Math.Cos(theta) * Math.Cos(phi) * x +
Math.Sin(theta) * Math.Cos(phi) * y +
Math.Sin(phi) * z);
}
double area = S2.GetArea(loop);
S2Point centroid = S2.GetCentroid(loop);
double expected_area = S2.M_2_PI * height;
Assert.True(Math.Abs(area - expected_area) <= S2.M_2_PI * kMaxDist);
S2Point expected_centroid = expected_area * (1 - 0.5 * height) * z;
Assert.True((centroid - expected_centroid).Norm() <= 2 * kMaxDist);
}
}
public void Test_GetArea_FullPolygon()
{
Assert.Equal(4 * Math.PI, S2.GetArea(MakeLaxPolygonOrDie("full")));
Assert.Equal(4 * Math.PI,
S2.GetArea(new S2Polygon.OwningShape(MakePolygonOrDie("full"))));
}
public void Test_GetArea_EmptyPolygon()
{
Assert.Equal(0.0, S2.GetArea(MakeLaxPolygonOrDie("empty")));
}
public void Test_GetArea_WrongDimension()
{
Assert.Equal(0.0, S2.GetArea(MakeIndexOrDie("0:0 # #").Shape(0)));
Assert.Equal(0.0, S2.GetArea(MakeLaxPolylineOrDie("0:0, 0:1, 1:0")));
}