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));
}
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);
}
// Check that the curvature is *identical* when the vertex order is
// rotated, and that the sign is inverted when the vertices are reversed.
private static void CheckCurvatureInvariants(S2PointLoopSpan loop_in)
{
S2.LoopOrder order_in = S2.GetCanonicalLoopOrder(loop_in);
var loop = loop_in.ToList();
double expected = S2.GetCurvature(loop);
for (int i = 0; i < loop.Count; ++i)
{
loop.Reverse();
Assert.Equal((expected == S2.M_2_PI) ? expected : -expected,
S2.GetCurvature(loop));
ExpectSameOrder(loop_in, order_in, loop, S2.GetCanonicalLoopOrder(loop));
loop.Reverse();
loop.RotateInPlace(1);
Assert.Equal(expected, S2.GetCurvature(loop));
ExpectSameOrder(loop_in, order_in, loop, S2.GetCanonicalLoopOrder(loop));
}
}
