private static void TestInterpolate(S2Point a, S2Point b, double t, S2Point expected)
{
a = a.Normalize();
b = b.Normalize();
expected = expected.Normalize();
// We allow a bit more than the usual 1e-15 error tolerance because
// interpolation uses trig functions.
S1Angle kError = S1Angle.FromRadians(3e-15);
Assert.True(new S1Angle(S2.Interpolate(a, b, t), expected) <= kError);
// Now test the other interpolation functions.
S1Angle r = t * new S1Angle(a, b);
Assert.True(new S1Angle(S2.GetPointOnLine(a, b, r), expected) <= kError);
if (a.DotProd(b) == 0)
{ // Common in the test cases below.
Assert.True(new S1Angle(S2.GetPointOnRay(a, b, r), expected) <= kError);
}
if (r.Radians >= 0 && r.Radians < 0.99 * S2.M_PI)
{
S1ChordAngle r_ca = new(r);
Assert.True(new S1Angle(S2.GetPointOnLine(a, b, r_ca), expected) <= kError);
if (a.DotProd(b) == 0)
{
Assert.True(new S1Angle(S2.GetPointOnRay(a, b, r_ca), expected) <= kError);
}
}
}
public void Test_S2_InterpolateCanExtrapolate()
{
S2Point i = new(1, 0, 0);
S2Point j = new(0, 1, 0);
// Initial vectors at 90 degrees.
TestInterpolate(i, j, 0, new S2Point(1, 0, 0));
TestInterpolate(i, j, 1, new S2Point(0, 1, 0));
TestInterpolate(i, j, 1.5, new S2Point(-1, 1, 0));
TestInterpolate(i, j, 2, new S2Point(-1, 0, 0));
TestInterpolate(i, j, 3, new S2Point(0, -1, 0));
TestInterpolate(i, j, 4, new S2Point(1, 0, 0));
// Negative values of t.
TestInterpolate(i, j, -1, new S2Point(0, -1, 0));
TestInterpolate(i, j, -2, new S2Point(-1, 0, 0));
TestInterpolate(i, j, -3, new S2Point(0, 1, 0));
TestInterpolate(i, j, -4, new S2Point(1, 0, 0));
// Initial vectors at 45 degrees.
TestInterpolate(i, new S2Point(1, 1, 0), 2, new S2Point(0, 1, 0));
TestInterpolate(i, new S2Point(1, 1, 0), 3, new S2Point(-1, 1, 0));
TestInterpolate(i, new S2Point(1, 1, 0), 4, new S2Point(-1, 0, 0));
// Initial vectors at 135 degrees.
TestInterpolate(i, new S2Point(-1, 1, 0), 2, new S2Point(0, -1, 0));
// Take a small fraction along the curve.
S2Point p = S2.Interpolate(i, j, 0.001);
// We should get back where we started.
TestInterpolate(i, p, 1000, j);
}
public void Test_S2Cell_GetMaxDistanceToEdge()
{
// Test an edge for which its antipode crosses the cell. Validates both the
// standard and brute force implementations for this case.
S2Cell cell = S2Cell.FromFacePosLevel(0, 0, 20);
S2Point a = -S2.Interpolate(2.0, cell.Center(), cell.Vertex(0));
S2Point b = -S2.Interpolate(2.0, cell.Center(), cell.Vertex(2));
S1ChordAngle actual = cell.MaxDistance(a, b);
S1ChordAngle expected = GetMaxDistanceToEdgeBruteForce(cell, a, b);
Assert2.Near(expected.Radians(), S1ChordAngle.Straight.Radians(), S2.DoubleError);
Assert2.Near(actual.Radians(), S1ChordAngle.Straight.Radians(), S2.DoubleError);
}
public void Test_S2_RepeatedInterpolation()
{
// Check that points do not drift away from unit length when repeated
// interpolations are done.
for (int i = 0; i < 100; ++i)
{
S2Point a = S2Testing.RandomPoint();
S2Point b = S2Testing.RandomPoint();
for (int j = 0; j < 1000; ++j)
{
a = S2.Interpolate(a, b, 0.01);
}
Assert.True(a.IsUnitLength());
}
}
public void Test_S2Cell_ConsistentWithS2CellIdFromPoint()
{
// Construct many points that are nearly on an S2Cell edge, and verify that
// S2Cell(S2CellId(p)).Contains(p) is always true.
for (int iter = 0; iter < 1000; ++iter)
{
S2Cell cell = new(S2Testing.GetRandomCellId());
int i = S2Testing.Random.Uniform(4);
S2Point v1 = cell.Vertex(i);
S2Point v2 = S2Testing.SamplePoint(
new S2Cap(cell.Vertex(i + 1), S1Angle.FromRadians(1e-15)));
S2Point p = S2.Interpolate(S2Testing.Random.RandDouble(), v1, v2);
Assert.True(new S2Cell(new S2CellId(p)).Contains(p));
}
}
public void Test_S2_CollinearEdgesThatDontTouch()
{
int kIters = 500;
for (int iter = 0; iter < kIters; ++iter)
{
S2Point a = S2Testing.RandomPoint();
S2Point d = S2Testing.RandomPoint();
S2Point b = S2.Interpolate(0.05, a, d);
S2Point c = S2.Interpolate(0.95, a, d);
Assert.True(0 > S2.CrossingSign(a, b, c, d));
Assert.True(0 > S2.CrossingSign(a, b, c, d));
S2EdgeCrosser crosser = new(a, b, c);
Assert.True(0 > crosser.CrossingSign(d));
Assert.True(0 > crosser.CrossingSign(c));
}
}
public void Test_S2_Interpolate()
{
// Choose test points designed to expose floating-point errors.
S2Point p1 = new S2Point(0.1, 1e-30, 0.3).Normalize();
S2Point p2 = new S2Point(-0.7, -0.55, -1e30).Normalize();
// A zero-length edge.
TestInterpolate(p1, p1, 0, p1);
TestInterpolate(p1, p1, 1, p1);
// Start, end, and middle of a medium-length edge.
TestInterpolate(p1, p2, 0, p1);
TestInterpolate(p1, p2, 1, p2);
TestInterpolate(p1, p2, 0.5, 0.5 * (p1 + p2));
// Test that interpolation is done using distances on the sphere rather than
// linear distances.
TestInterpolate(new S2Point(1, 0, 0), new S2Point(0, 1, 0), 1.0 / 3,
new S2Point(Math.Sqrt(3), 1, 0));
TestInterpolate(new S2Point(1, 0, 0), new S2Point(0, 1, 0), 2.0 / 3,
new S2Point(1, Math.Sqrt(3), 0));
// Test that interpolation is accurate on a long edge (but not so long that
// the definition of the edge itself becomes too unstable).
{
double kLng = Math.PI - 1e-2;
S2Point a = S2LatLng.FromRadians(0, 0).ToPoint();
S2Point b = S2LatLng.FromRadians(0, kLng).ToPoint();
for (double f = 0.4; f > 1e-15; f *= 0.1)
{
TestInterpolate(a, b, f, S2LatLng.FromRadians(0, f * kLng).ToPoint());
TestInterpolate(a, b, 1 - f, S2LatLng.FromRadians(0, (1 - f) * kLng).ToPoint());
}
}
// Test that interpolation on a 180 degree edge (antipodal endpoints) yields
// a result with the correct distance from each endpoint.
for (double t = 0; t <= 1; t += 0.125)
{
S2Point actual = S2.Interpolate(p1, -p1, t);
Assert2.Near(new S1Angle(actual, p1).Radians, t * Math.PI, 3e-15);
}
}
private S1ChordAngle EstimateMaxError(R2Point pa, S2Point a, R2Point pb, S2Point b)
{
// See the algorithm description at the top of this file.
// We always tessellate edges longer than 90 degrees on the sphere, since the
// approximation below is not robust enough to handle such edges.
if (a.DotProd(b) < -1e-14)
{
return(S1ChordAngle.Infinity);
}
const double t1 = kInterpolationFraction;
const double t2 = 1 - kInterpolationFraction;
S2Point mid1 = S2.Interpolate(a, b, t1);
S2Point mid2 = S2.Interpolate(a, b, t2);
S2Point pmid1 = proj_.Unproject(Projection.Interpolate(t1, pa, pb));
S2Point pmid2 = proj_.Unproject(Projection.Interpolate(t2, pa, pb));
return(S1ChordAngle.Max(new S1ChordAngle(mid1, pmid1), new S1ChordAngle(mid2, pmid2)));
}
public void Test_S2Cell_RectBoundIsLargeEnough()
{
// Construct many points that are nearly on an S2Cell edge, and verify that
// whenever the cell contains a point P then its bound contains S2LatLng(P).
for (int iter = 0; iter < 1000; /* advanced in loop below */)
{
S2Cell cell = new(S2Testing.GetRandomCellId());
int i = S2Testing.Random.Uniform(4);
S2Point v1 = cell.Vertex(i);
S2Point v2 = S2Testing.SamplePoint(
new S2Cap(cell.Vertex(i + 1), S1Angle.FromRadians(1e-15)));
S2Point p = S2.Interpolate(S2Testing.Random.RandDouble(), v1, v2);
if (new S2Loop(cell).Contains(p))
{
Assert.True(cell.GetRectBound().Contains(new S2LatLng(p)));
++iter;
}
}
}