// This function assumes that GetDirectedHausdorffDistance() always returns // a distance from some point in a to b. So the function mainly tests whether // the returned distance is large enough, and only does a weak test on whether // it is small enough. private static void VerifyGetDirectedHausdorffDistance(S2LatLngRect a, S2LatLngRect b) { S1Angle hausdorff_distance = a.GetDirectedHausdorffDistance(b); const double kResolution = 0.1; S1Angle max_distance = S1Angle.Zero; int sample_size_on_lat = (int)(a.Lat.GetLength() / kResolution) + 1; int sample_size_on_lng = (int)(a.Lng.GetLength() / kResolution) + 1; double delta_on_lat = a.Lat.GetLength() / sample_size_on_lat; double delta_on_lng = a.Lng.GetLength() / sample_size_on_lng; double lng = a.Lng.Lo; for (int i = 0; i <= sample_size_on_lng; ++i, lng += delta_on_lng) { double lat = a.Lat.Lo; for (int j = 0; j <= sample_size_on_lat; ++j, lat += delta_on_lat) { S2LatLng latlng = S2LatLng.FromRadians(lat, lng).Normalized(); S1Angle distance_to_b = b.GetDistance(latlng); if (distance_to_b >= max_distance) { max_distance = distance_to_b; } } } Assert.True(max_distance.Radians <= hausdorff_distance.Radians + 1e-10); Assert.True(max_distance.Radians >= hausdorff_distance.Radians - kResolution); }
private static void TestIntervalOps(S2LatLngRect x, S2LatLngRect y, string expected_relation, S2LatLngRect expected_union, S2LatLngRect expected_intersection) { // Test all of the interval operations on the given pair of intervals. // "expected_relation" is a sequence of "T" and "F" characters corresponding // to the expected results of Contains(), InteriorContains(), Intersects(), // and InteriorIntersects() respectively. Assert.Equal(x.Contains(y), expected_relation[0] == 'T'); Assert.Equal(x.InteriorContains(y), expected_relation[1] == 'T'); Assert.Equal(x.Intersects(y), expected_relation[2] == 'T'); Assert.Equal(x.InteriorIntersects(y), expected_relation[3] == 'T'); Assert.Equal(x.Contains(y), x.Union(y) == x); Assert.Equal(x.Intersects(y), !x.Intersection(y).IsEmpty()); Assert.Equal(x.Union(y), expected_union); Assert.Equal(x.Intersection(y), expected_intersection); if (y.Size() == S2LatLng.FromRadians(0, 0)) { S2LatLngRect r = x; r.AddPoint(y.Lo()); Assert.Equal(r, expected_union); } }
public void Test_S2LatLngRect_GetVertex() { S2LatLngRect r1 = new(new R1Interval(0, S2.M_PI_2), new S1Interval(-Math.PI, 0)); Assert.Equal(r1.Vertex(0), S2LatLng.FromRadians(0, Math.PI)); Assert.Equal(r1.Vertex(1), S2LatLng.FromRadians(0, 0)); Assert.Equal(r1.Vertex(2), S2LatLng.FromRadians(S2.M_PI_2, 0)); Assert.Equal(r1.Vertex(3), S2LatLng.FromRadians(S2.M_PI_2, Math.PI)); // Make sure that GetVertex() returns vertices in CCW order. for (int i = 0; i < 4; ++i) { double lat = S2.M_PI_4 * (i - 2); double lng = S2.M_PI_2 * (i - 2) + 0.2; S2LatLngRect r = new( new R1Interval(lat, lat + S2.M_PI_4), new S1Interval( Math.IEEERemainder(lng, S2.M_2_PI), Math.IEEERemainder(lng + S2.M_PI_2, S2.M_2_PI))); for (int k = 0; k < 4; ++k) { Assert.True(S2Pred.Sign( r.Vertex(k - 1).ToPoint(), r.Vertex(k).ToPoint(), r.Vertex(k + 1).ToPoint()) > 0); } } }
public void testIntervalOps(S2LatLngRect x, S2LatLngRect y, String expectedRelation, S2LatLngRect expectedUnion, S2LatLngRect expectedIntersection) { // Test all of the interval operations on the given pair of intervals. // "expected_relation" is a sequence of "T" and "F" characters corresponding // to the expected results of Contains(), InteriorContains(), Intersects(), // and InteriorIntersects() respectively. assertEquals(x.Contains(y), expectedRelation[0] == 'T'); assertEquals(x.InteriorContains(y), expectedRelation[1] == 'T'); assertEquals(x.Intersects(y), expectedRelation[2] == 'T'); assertEquals(x.InteriorIntersects(y), expectedRelation[3] == 'T'); assertEquals(x.Contains(y), x.Union(y).Equals(x)); assertEquals(x.Intersects(y), !x.Intersection(y).IsEmpty); assertTrue(x.Union(y).Equals(expectedUnion)); assertTrue(x.Intersection(y).Equals(expectedIntersection)); if (y.Size == S2LatLng.FromRadians(0, 0)) { var r = x.AddPoint(y.Lo); assertTrue(r == expectedUnion); } }
public void testConversion() { // Test special cases: poles, "date line" assertDoubleNear( new S2LatLng(S2LatLng.FromDegrees(90.0, 65.0).ToPoint()).Lat.Degrees, 90.0); assertEquals( new S2LatLng(S2LatLng.FromRadians(-S2.PiOver2, 1).ToPoint()).Lat.Radians, -S2.PiOver2); assertDoubleNear( Math.Abs(new S2LatLng(S2LatLng.FromDegrees(12.2, 180.0).ToPoint()).Lng.Degrees), 180.0); assertEquals( Math.Abs(new S2LatLng(S2LatLng.FromRadians(0.1, -S2.Pi).ToPoint()).Lng.Radians), S2.Pi); // Test a bunch of random points. for (var i = 0; i < 100000; ++i) { var p = randomPoint(); assertTrue(S2.ApproxEquals(p, new S2LatLng(p).ToPoint())); } // Test generation from E5 var test = S2LatLng.FromE5(123456, 98765); assertDoubleNear(test.Lat.Degrees, 1.23456); assertDoubleNear(test.Lng.Degrees, 0.98765); }
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))); } }
// Returns the bounding rectangle of the edge chain that connects the // vertices defined so far. This bound satisfies the guarantee made // above, i.e. if the edge chain defines a loop, then the bound contains // the S2LatLng coordinates of all S2Points contained by the loop. public S2LatLngRect GetBound() { // To save time, we ignore numerical errors in the computed S2LatLngs while // accumulating the bounds and then account for them here. // // S2LatLng(S2Point) has a maximum error of 0.955 * S2Constants.DoubleEpsilon in latitude. // In the worst case, we might have rounded "inwards" when computing the // bound and "outwards" when computing the latitude of a contained point P, // therefore we expand the latitude bounds by 2 * S2Constants.DoubleEpsilon in each // direction. (A more complex analysis shows that 1.5 * S2Constants.DoubleEpsilon is // enough, but the expansion amount should be a multiple of S2Constants.DoubleEpsilon in // order to avoid rounding errors during the expansion itself.) // // S2LatLng(S2Point) has a maximum error of S2Constants.DoubleEpsilon in longitude, which // is simply the maximum rounding error for results in the range [-Pi, Pi]. // This is true because the Gnu implementation of atan2() comes from the IBM // Accurate Mathematical Library, which implements correct rounding for this // instrinsic (i.e., it returns the infinite precision result rounded to the // nearest representable value, with ties rounded to even values). This // implies that we don't need to expand the longitude bounds at all, since // we only guarantee that the bound contains the *rounded* latitudes of // contained points. The *true* latitudes of contained points may lie up to // S2Constants.DoubleEpsilon outside of the returned bound. S2LatLng kExpansion = S2LatLng.FromRadians(2 * S2.DoubleEpsilon, 0); return(bound_.Expanded(kExpansion).PolarClosure()); }
public void Test_S2LatLngRect_GetCenterSize() { S2LatLngRect r1 = new(new R1Interval(0, S2.M_PI_2), new S1Interval(-Math.PI, 0)); Assert.Equal(r1.Center(), S2LatLng.FromRadians(S2.M_PI_4, -S2.M_PI_2)); Assert.Equal(r1.Size(), S2LatLng.FromRadians(S2.M_PI_2, Math.PI)); Assert.True(S2LatLngRect.Empty.Size().LatRadians < 0); Assert.True(S2LatLngRect.Empty.Size().LngRadians < 0); }
public override S2LatLng ToLatLng(R2Point p) { // This formula is more accurate near zero than the atan(exp()) version. double x = to_radians_ * Math.IEEERemainder(p.X, x_wrap_); double k = Math.Exp(2 * to_radians_ * p.Y); double y = double.IsInfinity(k) ? S2.M_PI_2 : Math.Asin((k - 1) / (k + 1)); return(S2LatLng.FromRadians(y, x)); }
public void Test_S2LatLngRect_AddPoint() { S2LatLngRect p = S2LatLngRect.Empty; p = p.AddPoint(S2LatLng.FromDegrees(0, 0)); Assert.True(p.IsPoint()); p = p.AddPoint(S2LatLng.FromRadians(0, -S2.M_PI_2)); Assert.False(p.IsPoint()); p = p.AddPoint(S2LatLng.FromRadians(S2.M_PI_4, -Math.PI)); p = p.AddPoint(new S2Point(0, 0, 1)); Assert.Equal(p, RectFromDegrees(0, -180, 90, 0)); }
// Returns the maximum error in GetBound() provided that the result does // not include either pole. It is only to be used for testing purposes // (e.g., by passing it to S2LatLngRect.ApproxEquals). public static S2LatLng MaxErrorForTests() { // The maximum error in the latitude calculation is // 3.84 * DBL_EPSILON for the cross product calculation (see above) // 0.96 * S2Constants.DoubleEpsilon for the Latitude() calculation // 5 * S2Constants.DoubleEpsilon added by AddPoint/GetBound to compensate for error // ------------------ // 9.80 * S2Constants.DoubleEpsilon maximum error in result // // The maximum error in the longitude calculation is S2Constants.DoubleEpsilon. GetBound // does not do any expansion because this isn't necessary in order to // bound the *rounded* longitudes of contained points. return(S2LatLng.FromRadians(10 * S2.DoubleEpsilon, 1 * S2.DoubleEpsilon)); }
public void Test_S2LatLng_TestBasic() { S2LatLng ll_rad = S2LatLng.FromRadians(S2.M_PI_4, S2.M_PI_2); Assert.Equal(S2.M_PI_4, ll_rad.LatRadians); Assert.Equal(S2.M_PI_2, ll_rad.LngRadians); Assert.True(ll_rad.IsValid()); S2LatLng ll_deg = S2LatLng.FromDegrees(45, 90); Assert.Equal(ll_rad, ll_deg); Assert.True(ll_deg.IsValid()); Assert.False(S2LatLng.FromDegrees(-91, 0).IsValid()); Assert.False(S2LatLng.FromDegrees(0, 181).IsValid()); S2LatLng bad = S2LatLng.FromDegrees(120, 200); Assert.False(bad.IsValid()); S2LatLng better = bad.Normalized(); Assert.True(better.IsValid()); Assert.Equal(S1Angle.FromDegrees(90), better.Lat()); Assert2.DoubleEqual(S1Angle.FromDegrees(-160).Radians, better.LngRadians); bad = S2LatLng.FromDegrees(-100, -360); Assert.False(bad.IsValid()); better = bad.Normalized(); Assert.True(better.IsValid()); Assert.Equal(S1Angle.FromDegrees(-90), better.Lat()); Assert2.DoubleEqual(0.0, better.LngRadians); Assert.True((S2LatLng.FromDegrees(10, 20) + S2LatLng.FromDegrees(20, 30)). ApproxEquals(S2LatLng.FromDegrees(30, 50))); Assert.True((S2LatLng.FromDegrees(10, 20) - S2LatLng.FromDegrees(20, 30)). ApproxEquals(S2LatLng.FromDegrees(-10, -10))); Assert.True((0.5 * S2LatLng.FromDegrees(10, 20)). ApproxEquals(S2LatLng.FromDegrees(5, 10))); // Check that Invalid() returns an invalid point. S2LatLng invalid = S2LatLng.Invalid; Assert.False(invalid.IsValid()); // Check that the default constructor sets latitude and longitude to 0. S2LatLng default_ll = S2LatLng.Center; Assert.True(default_ll.IsValid()); Assert.Equal(0, default_ll.LatRadians); Assert.Equal(0, default_ll.LngRadians); }
public void Test_MercatorProjection_ProjectUnproject() { MercatorProjection proj = new(180); double inf = double.PositiveInfinity; TestProjectUnproject(proj, new R2Point(0, 0), new S2Point(1, 0, 0)); TestProjectUnproject(proj, new R2Point(180, 0), new S2Point(-1, 0, 0)); TestProjectUnproject(proj, new R2Point(90, 0), new S2Point(0, 1, 0)); TestProjectUnproject(proj, new R2Point(-90, 0), new S2Point(0, -1, 0)); TestProjectUnproject(proj, new R2Point(0, inf), new S2Point(0, 0, 1)); TestProjectUnproject(proj, new R2Point(0, -inf), new S2Point(0, 0, -1)); // Test one arbitrary point as a sanity check. TestProjectUnproject(proj, new R2Point(0, 70.255578967830246), S2LatLng.FromRadians(1, 0).ToPoint()); }
public void Test_S2LatLng_TestConversion() { // Test special cases: poles, "date line" Assert2.DoubleEqual(90.0, new S2LatLng(S2LatLng.FromDegrees(90.0, 65.0).ToPoint()).Lat().GetDegrees()); Assert.Equal(-S2.M_PI_2, new S2LatLng(S2LatLng.FromRadians(-S2.M_PI_2, 1).ToPoint()).LatRadians); Assert2.DoubleEqual(180.0, Math.Abs(new S2LatLng(S2LatLng.FromDegrees(12.2, 180.0).ToPoint()).Lng().GetDegrees())); Assert.Equal(Math.PI, Math.Abs(new S2LatLng(S2LatLng.FromRadians(0.1, -Math.PI).ToPoint()).LngRadians)); // Test a bunch of random points. for (int i = 0; i < 100000; ++i) { S2Point p = S2Testing.RandomPoint(); Assert.True(S2.ApproxEquals(p, new S2LatLng(p).ToPoint())); } }
public void Test_S2LatLngRect_Contains() { // Contains(S2LatLng), InteriorContains(S2LatLng), Contains() S2LatLng eq_m180 = S2LatLng.FromRadians(0, -Math.PI); S2LatLng north_pole = S2LatLng.FromRadians(S2.M_PI_2, 0); S2LatLngRect r1 = new(eq_m180, north_pole); Assert.True(r1.Contains(S2LatLng.FromDegrees(30, -45))); Assert.True(r1.InteriorContains(S2LatLng.FromDegrees(30, -45))); Assert.False(r1.Contains(S2LatLng.FromDegrees(30, 45))); Assert.False(r1.InteriorContains(S2LatLng.FromDegrees(30, 45))); Assert.True(r1.Contains(eq_m180)); Assert.False(r1.InteriorContains(eq_m180)); Assert.True(r1.Contains(north_pole)); Assert.False(r1.InteriorContains(north_pole)); Assert.True(r1.Contains(new S2Point(0.5, -0.3, 0.1))); Assert.False(r1.Contains(new S2Point(0.5, 0.2, 0.1))); }
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); } }
public void testBasic() { var llRad = S2LatLng.FromRadians(S2.PiOver4, S2.PiOver2); assertTrue(llRad.Lat.Radians == S2.PiOver4); assertTrue(llRad.Lng.Radians == S2.PiOver2); assertTrue(llRad.IsValid); var llDeg = S2LatLng.FromDegrees(45, 90); assertEquals(llDeg, llRad); assertTrue(llDeg.IsValid); assertTrue(!S2LatLng.FromDegrees(-91, 0).IsValid); assertTrue(!S2LatLng.FromDegrees(0, 181).IsValid); var bad = S2LatLng.FromDegrees(120, 200); assertTrue(!bad.IsValid); var better = bad.Normalized; assertTrue(better.IsValid); assertEquals(better.Lat, S1Angle.FromDegrees(90)); assertDoubleNear(better.Lng.Radians, S1Angle.FromDegrees(-160).Radians); bad = S2LatLng.FromDegrees(-100, -360); assertTrue(!bad.IsValid); better = bad.Normalized; assertTrue(better.IsValid); assertEquals(better.Lat, S1Angle.FromDegrees(-90)); assertDoubleNear(better.Lng.Radians, 0); assertTrue((S2LatLng.FromDegrees(10, 20) + S2LatLng.FromDegrees(20, 30)).ApproxEquals( S2LatLng.FromDegrees(30, 50))); assertTrue((S2LatLng.FromDegrees(10, 20) - S2LatLng.FromDegrees(20, 30)).ApproxEquals( S2LatLng.FromDegrees(-10, -10))); assertTrue((S2LatLng.FromDegrees(10, 20) * 0.5).ApproxEquals(S2LatLng.FromDegrees(5, 10))); }
// Expands a bound returned by GetBound() so that it is guaranteed to // contain the bounds of any subregion whose bounds are computed using // this class. For example, consider a loop L that defines a square. // GetBound() ensures that if a point P is contained by this square, then // S2LatLng(P) is contained by the bound. But now consider a diamond // shaped loop S contained by L. It is possible that GetBound() returns a // *larger* bound for S than it does for L, due to rounding errors. This // method expands the bound for L so that it is guaranteed to contain the // bounds of any subregion S. // // More precisely, if L is a loop that does not contain either pole, and S // is a loop such that L.Contains(S), then // // ExpandForSubregions(RectBound(L)).Contains(RectBound(S)). public static S2LatLngRect ExpandForSubregions(S2LatLngRect bound) { // Empty bounds don't need expansion. if (bound.IsEmpty()) { return(bound); } // First we need to check whether the bound B contains any nearly-antipodal // points (to within 4.309 * S2Constants.DoubleEpsilon). If so then we need to return // S2LatLngRect.Full(), since the subregion might have an edge between two // such points, and AddPoint() returns Full() for such edges. Note that // this can happen even if B is not Full(); for example, consider a loop // that defines a 10km strip straddling the equator extending from // longitudes -100 to +100 degrees. // // It is easy to check whether B contains any antipodal points, but checking // for nearly-antipodal points is trickier. Essentially we consider the // original bound B and its reflection through the origin B', and then test // whether the minimum distance between B and B' is less than 4.309 * // S2Constants.DoubleEpsilon. // "lng_gap" is a lower bound on the longitudinal distance between B and its // reflection B'. (2.5 * S2Constants.DoubleEpsilon is the maximum combined error of the // endpoint longitude calculations and the GetLength() call.) double lng_gap = Math.Max(0.0, Math.PI - bound.Lng.GetLength() - 2.5 * S2.DoubleEpsilon); // "min_abs_lat" is the minimum distance from B to the equator (if zero or // negative, then B straddles the equator). double min_abs_lat = Math.Max(bound.Lat.Lo, -bound.Lat.Hi); // "lat_gap1" and "lat_gap2" measure the minimum distance from B to the // south and north poles respectively. double lat_gap1 = S2.M_PI_2 + bound.Lat.Lo; double lat_gap2 = S2.M_PI_2 - bound.Lat.Hi; if (min_abs_lat >= 0) { // The bound B does not straddle the equator. In this case the minimum // distance is between one endpoint of the latitude edge in B closest to // the equator and the other endpoint of that edge in B'. The latitude // distance between these two points is 2*min_abs_lat, and the longitude // distance is lng_gap. We could compute the distance exactly using the // Haversine formula, but then we would need to bound the errors in that // calculation. Since we only need accuracy when the distance is very // small (close to 4.309 * S2Constants.DoubleEpsilon), we substitute the Euclidean // distance instead. This gives us a right triangle XYZ with two edges of // length x = 2*min_abs_lat and y ~= lng_gap. The desired distance is the // length of the third edge "z", and we have // // z ~= Math.Sqrt(x^2 + y^2) >= (x + y) / Math.Sqrt(2) // // Therefore the region may contain nearly antipodal points only if // // 2*min_abs_lat + lng_gap < Math.Sqrt(2) * 4.309 * S2Constants.DoubleEpsilon // ~= 1.354e-15 // // Note that because the given bound B is conservative, "min_abs_lat" and // "lng_gap" are both lower bounds on their true values so we do not need // to make any adjustments for their errors. if (2 * min_abs_lat + lng_gap < 1.354e-15) { return(S2LatLngRect.Full); } } else if (lng_gap >= S2.M_PI_2) { // B spans at most Pi/2 in longitude. The minimum distance is always // between one corner of B and the diagonally opposite corner of B'. We // use the same distance approximation that we used above; in this case // we have an obtuse triangle XYZ with two edges of length x = lat_gap1 // and y = lat_gap2, and angle Z >= Pi/2 between them. We then have // // z >= Math.Sqrt(x^2 + y^2) >= (x + y) / Math.Sqrt(2) // // Unlike the case above, "lat_gap1" and "lat_gap2" are not lower bounds // (because of the extra addition operation, and because S2Constants.M_PI_2 is not // exactly equal to Pi/2); they can exceed their true values by up to // 0.75 * S2Constants.DoubleEpsilon. Putting this all together, the region may // contain nearly antipodal points only if // // lat_gap1 + lat_gap2 < (Math.Sqrt(2) * 4.309 + 1.5) * S2Constants.DoubleEpsilon // ~= 1.687e-15 if (lat_gap1 + lat_gap2 < 1.687e-15) { return(S2LatLngRect.Full); } } else { // Otherwise we know that (1) the bound straddles the equator and (2) its // width in longitude is at least Pi/2. In this case the minimum // distance can occur either between a corner of B and the diagonally // opposite corner of B' (as in the case above), or between a corner of B // and the opposite longitudinal edge reflected in B'. It is sufficient // to only consider the corner-edge case, since this distance is also a // lower bound on the corner-corner distance when that case applies. // Consider the spherical triangle XYZ where X is a corner of B with // minimum absolute latitude, Y is the closest pole to X, and Z is the // point closest to X on the opposite longitudinal edge of B'. This is a // right triangle (Z = Pi/2), and from the spherical law of sines we have // // sin(z) / sin(Z) = sin(y) / sin(Y) // sin(max_lat_gap) / 1 = sin(d_min) / sin(lng_gap) // sin(d_min) = sin(max_lat_gap) * sin(lng_gap) // // where "max_lat_gap" = Math.Max(lat_gap1, lat_gap2) and "d_min" is the // desired minimum distance. Now using the facts that sin(t) >= (2/Pi)*t // for 0 <= t <= Pi/2, that we only need an accurate approximation when // at least one of "max_lat_gap" or "lng_gap" is extremely small (in // which case sin(t) ~= t), and recalling that "max_lat_gap" has an error // of up to 0.75 * S2Constants.DoubleEpsilon, we want to test whether // // max_lat_gap * lng_gap < (4.309 + 0.75) * (Pi/2) * S2Constants.DoubleEpsilon // ~= 1.765e-15 if (Math.Max(lat_gap1, lat_gap2) * lng_gap < 1.765e-15) { return(S2LatLngRect.Full); } } // Next we need to check whether the subregion might contain any edges that // span (Math.PI - 2 * S2Constants.DoubleEpsilon) radians or more in longitude, since AddPoint // sets the longitude bound to Full() in that case. This corresponds to // testing whether (lng_gap <= 0) in "lng_expansion" below. // Otherwise, the maximum latitude error in AddPoint is 4.8 * S2Constants.DoubleEpsilon. // In the worst case, the errors when computing the latitude bound for a // subregion could go in the opposite direction as the errors when computing // the bound for the original region, so we need to double this value. // (More analysis shows that it's okay to round down to a multiple of // S2Constants.DoubleEpsilon.) // // For longitude, we rely on the fact that atan2 is correctly rounded and // therefore no additional bounds expansion is necessary. double lat_expansion = 9 * S2.DoubleEpsilon; double lng_expansion = (lng_gap <= 0) ? Math.PI : 0; return(bound.Expanded(S2LatLng.FromRadians(lat_expansion, lng_expansion)).PolarClosure()); }
public override S2LatLng ToLatLng(R2Point p) { var rem = Math.IEEERemainder(p.X, x_wrap_); return(S2LatLng.FromRadians(to_radians_ * p.Y, to_radians_ * rem)); }
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 testBasic() { // Most of the S2LatLngRect methods have trivial implementations that // use the R1Interval and S1Interval classes, so most of the testing // is done in those unit tests. // Test basic properties of empty and full caps. var empty = S2LatLngRect.Empty; var full = S2LatLngRect.Full; assertTrue(empty.IsValid); assertTrue(empty.IsEmpty); assertTrue(full.IsValid); assertTrue(full.IsFull); // assertTrue various constructors and accessor methods. var d1 = rectFromDegrees(-90, 0, -45, 180); assertDoubleNear(d1.LatLo.Degrees, -90); assertDoubleNear(d1.LatHi.Degrees, -45); assertDoubleNear(d1.LngLo.Degrees, 0); assertDoubleNear(d1.LngHi.Degrees, 180); assertTrue(d1.Lat.Equals(new R1Interval(-S2.PiOver2, -S2.PiOver4))); assertTrue(d1.Lng.Equals(new S1Interval(0, S2.Pi))); // FromCenterSize() assertTrue( S2LatLngRect.FromCenterSize(S2LatLng.FromDegrees(80, 170), S2LatLng.FromDegrees(40, 60)) .ApproxEquals(rectFromDegrees(60, 140, 90, -160))); assertTrue(S2LatLngRect .FromCenterSize(S2LatLng.FromDegrees(10, 40), S2LatLng.FromDegrees(210, 400)).IsFull); assertTrue( S2LatLngRect.FromCenterSize(S2LatLng.FromDegrees(-90, 180), S2LatLng.FromDegrees(20, 50)) .ApproxEquals(rectFromDegrees(-90, 155, -80, -155))); // FromPoint(), FromPointPair() assertEquals(S2LatLngRect.FromPoint(d1.Lo), new S2LatLngRect(d1.Lo, d1.Lo)); assertEquals( S2LatLngRect.FromPointPair(S2LatLng.FromDegrees(-35, -140), S2LatLng.FromDegrees(15, 155)), rectFromDegrees(-35, 155, 15, -140)); assertEquals( S2LatLngRect.FromPointPair(S2LatLng.FromDegrees(25, -70), S2LatLng.FromDegrees(-90, 80)), rectFromDegrees(-90, -70, 25, 80)); // GetCenter(), GetVertex(), Contains(S2LatLng), InteriorContains(S2LatLng). var eqM180 = S2LatLng.FromRadians(0, -S2.Pi); var northPole = S2LatLng.FromRadians(S2.PiOver2, 0); var r1 = new S2LatLngRect(eqM180, northPole); assertEquals(r1.Center, S2LatLng.FromRadians(S2.PiOver4, -S2.PiOver2)); assertEquals(r1.GetVertex(0), S2LatLng.FromRadians(0, S2.Pi)); assertEquals(r1.GetVertex(1), S2LatLng.FromRadians(0, 0)); assertEquals(r1.GetVertex(2), S2LatLng.FromRadians(S2.PiOver2, 0)); assertEquals(r1.GetVertex(3), S2LatLng.FromRadians(S2.PiOver2, S2.Pi)); assertTrue(r1.Contains(S2LatLng.FromDegrees(30, -45))); assertTrue(!r1.Contains(S2LatLng.FromDegrees(30, 45))); assertTrue(!r1.InteriorContains(eqM180) && !r1.InteriorContains(northPole)); assertTrue(r1.Contains(new S2Point(0.5, -0.3, 0.1))); assertTrue(!r1.Contains(new S2Point(0.5, 0.2, 0.1))); // Make sure that GetVertex() returns vertices in CCW order. for (var i = 0; i < 4; ++i) { var lat = S2.PiOver4 * (i - 2); var lng = S2.PiOver2 * (i - 2) + 0.2; var r = new S2LatLngRect(new R1Interval(lat, lat + S2.PiOver4), new S1Interval( Math.IEEERemainder(lng, 2 * S2.Pi), Math.IEEERemainder(lng + S2.PiOver2, 2 * S2.Pi))); for (var k = 0; k < 4; ++k) { assertTrue( S2.SimpleCcw(r.GetVertex((k - 1) & 3).ToPoint(), r.GetVertex(k).ToPoint(), r.GetVertex((k + 1) & 3).ToPoint())); } } // Contains(S2LatLngRect), InteriorContains(S2LatLngRect), // Intersects(), InteriorIntersects(), Union(), Intersection(). // // Much more testing of these methods is done in s1interval_unittest // and r1interval_unittest. var r1Mid = rectFromDegrees(45, -90, 45, -90); var reqM180 = new S2LatLngRect(eqM180, eqM180); var rNorthPole = new S2LatLngRect(northPole, northPole); testIntervalOps(r1, r1Mid, "TTTT", r1, r1Mid); testIntervalOps(r1, reqM180, "TFTF", r1, reqM180); testIntervalOps(r1, rNorthPole, "TFTF", r1, rNorthPole); assertTrue(r1.Equals(rectFromDegrees(0, -180, 90, 0))); testIntervalOps(r1, rectFromDegrees(-10, -1, 1, 20), "FFTT", rectFromDegrees(-10, -180, 90, 20), rectFromDegrees(0, -1, 1, 0)); testIntervalOps(r1, rectFromDegrees(-10, -1, 0, 20), "FFTF", rectFromDegrees(-10, -180, 90, 20), rectFromDegrees(0, -1, 0, 0)); testIntervalOps(r1, rectFromDegrees(-10, 0, 1, 20), "FFTF", rectFromDegrees(-10, -180, 90, 20), rectFromDegrees(0, 0, 1, 0)); testIntervalOps(rectFromDegrees(-15, -160, -15, -150), rectFromDegrees(20, 145, 25, 155), "FFFF", rectFromDegrees(-15, 145, 25, -150), empty); testIntervalOps(rectFromDegrees(70, -10, 90, -140), rectFromDegrees(60, 175, 80, 5), "FFTT", rectFromDegrees(60, -180, 90, 180), rectFromDegrees(70, 175, 80, 5)); // assertTrue that the intersection of two rectangles that overlap in // latitude // but not longitude is valid, and vice versa. testIntervalOps(rectFromDegrees(12, 30, 60, 60), rectFromDegrees(0, 0, 30, 18), "FFFF", rectFromDegrees(0, 0, 60, 60), empty); testIntervalOps(rectFromDegrees(0, 0, 18, 42), rectFromDegrees(30, 12, 42, 60), "FFFF", rectFromDegrees(0, 0, 42, 60), empty); // AddPoint() var p = S2LatLngRect.Empty; p = p.AddPoint(S2LatLng.FromDegrees(0, 0)); p = p.AddPoint(S2LatLng.FromRadians(0, -S2.PiOver2)); p = p.AddPoint(S2LatLng.FromRadians(S2.PiOver4, -S2.Pi)); p = p.AddPoint(new S2Point(0, 0, 1)); assertTrue(p.Equals(r1)); // Expanded() assertTrue( rectFromDegrees(70, 150, 80, 170).Expanded(S2LatLng.FromDegrees(20, 30)).ApproxEquals( rectFromDegrees(50, 120, 90, -160))); assertTrue(S2LatLngRect.Empty.Expanded(S2LatLng.FromDegrees(20, 30)).IsEmpty); assertTrue(S2LatLngRect.Full.Expanded(S2LatLng.FromDegrees(20, 30)).IsFull); assertTrue( rectFromDegrees(-90, 170, 10, 20).Expanded(S2LatLng.FromDegrees(30, 80)).ApproxEquals( rectFromDegrees(-90, -180, 40, 180))); // ConvolveWithCap() var llr1 = new S2LatLngRect(S2LatLng.FromDegrees(0, 170), S2LatLng.FromDegrees(0, -170)) .ConvolveWithCap(S1Angle.FromDegrees(15)); var llr2 = new S2LatLngRect(S2LatLng.FromDegrees(-15, 155), S2LatLng.FromDegrees(15, -155)); assertTrue(llr1.ApproxEquals(llr2)); llr1 = new S2LatLngRect(S2LatLng.FromDegrees(60, 150), S2LatLng.FromDegrees(80, 10)) .ConvolveWithCap(S1Angle.FromDegrees(15)); llr2 = new S2LatLngRect(S2LatLng.FromDegrees(45, -180), S2LatLng.FromDegrees(90, 180)); assertTrue(llr1.ApproxEquals(llr2)); // GetCapBound(), bounding cap at center is smaller: assertTrue(new S2LatLngRect(S2LatLng.FromDegrees(-45, -45), S2LatLng.FromDegrees(45, 45)).CapBound.ApproxEquals(S2Cap.FromAxisHeight(new S2Point(1, 0, 0), 0.5))); // GetCapBound(), bounding cap at north pole is smaller: assertTrue(new S2LatLngRect(S2LatLng.FromDegrees(88, -80), S2LatLng.FromDegrees(89, 80)).CapBound.ApproxEquals(S2Cap.FromAxisAngle(new S2Point(0, 0, 1), S1Angle.FromDegrees(2)))); // GetCapBound(), longitude span > 180 degrees: assertTrue( new S2LatLngRect(S2LatLng.FromDegrees(-30, -150), S2LatLng.FromDegrees(-10, 50)).CapBound .ApproxEquals(S2Cap.FromAxisAngle(new S2Point(0, 0, -1), S1Angle.FromDegrees(80)))); // Contains(S2Cell), MayIntersect(S2Cell), Intersects(S2Cell) // Special cases. testCellOps(empty, S2Cell.FromFacePosLevel(3, (byte)0, 0), 0); testCellOps(full, S2Cell.FromFacePosLevel(2, (byte)0, 0), 4); testCellOps(full, S2Cell.FromFacePosLevel(5, (byte)0, 25), 4); // This rectangle includes the first quadrant of face 0. It's expanded // slightly because cell bounding rectangles are slightly conservative. var r4 = rectFromDegrees(-45.1, -45.1, 0.1, 0.1); testCellOps(r4, S2Cell.FromFacePosLevel(0, (byte)0, 0), 3); testCellOps(r4, S2Cell.FromFacePosLevel(0, (byte)0, 1), 4); testCellOps(r4, S2Cell.FromFacePosLevel(1, (byte)0, 1), 0); // This rectangle intersects the first quadrant of face 0. var r5 = rectFromDegrees(-10, -45, 10, 0); testCellOps(r5, S2Cell.FromFacePosLevel(0, (byte)0, 0), 3); testCellOps(r5, S2Cell.FromFacePosLevel(0, (byte)0, 1), 3); testCellOps(r5, S2Cell.FromFacePosLevel(1, (byte)0, 1), 0); // Rectangle consisting of a single point. testCellOps(rectFromDegrees(4, 4, 4, 4), S2Cell.FromFacePosLevel(0, (byte)0, 0), 3); // Rectangles that intersect the bounding rectangle of a face // but not the face itself. testCellOps(rectFromDegrees(41, -87, 42, -79), S2Cell.FromFacePosLevel(2, (byte)0, 0), 1); testCellOps(rectFromDegrees(-41, 160, -40, -160), S2Cell.FromFacePosLevel(5, (byte)0, 0), 1); { // This is the leaf cell at the top right hand corner of face 0. // It has two angles of 60 degrees and two of 120 degrees. var cell0tr = new S2Cell(new S2Point(1 + 1e-12, 1, 1)); var bound0tr = cell0tr.RectBound; var v0 = new S2LatLng(cell0tr.GetVertexRaw(0)); testCellOps( rectFromDegrees(v0.Lat.Degrees - 1e-8, v0.Lng.Degrees - 1e-8, v0.Lat.Degrees - 2e-10, v0.Lng.Degrees + 1e-10), cell0tr, 1); } // Rectangles that intersect a face but where no vertex of one region // is contained by the other region. The first one passes through // a corner of one of the face cells. testCellOps(rectFromDegrees(-37, -70, -36, -20), S2Cell.FromFacePosLevel(5, (byte)0, 0), 2); { // These two intersect like a diamond and a square. var cell202 = S2Cell.FromFacePosLevel(2, (byte)0, 2); var bound202 = cell202.RectBound; testCellOps( rectFromDegrees(bound202.Lo.Lat.Degrees + 3, bound202.Lo.Lng.Degrees + 3, bound202.Hi.Lat.Degrees - 3, bound202.Hi.Lng.Degrees - 3), cell202, 2); } }