/**
* 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.
*/
private void testIntervalOps(R1Interval x, R1Interval y, String expectedRelation)
{
JavaAssert.Equal(x.Contains(y), expectedRelation[0] == 'T');
JavaAssert.Equal(x.InteriorContains(y), expectedRelation[1] == 'T');
JavaAssert.Equal(x.Intersects(y), expectedRelation[2] == 'T');
JavaAssert.Equal(x.InteriorIntersects(y), expectedRelation[3] == 'T');
JavaAssert.Equal(x.Contains(y), x.Union(y).Equals(x));
JavaAssert.Equal(x.Intersects(y), !x.Intersection(y).IsEmpty);
}
/**
* 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.
*/
private void testIntervalOps(R1Interval x, R1Interval y, String expectedRelation)
{
JavaAssert.Equal(x.Contains(y), expectedRelation[0] == 'T');
JavaAssert.Equal(x.InteriorContains(y), expectedRelation[1] == 'T');
JavaAssert.Equal(x.Intersects(y), expectedRelation[2] == 'T');
JavaAssert.Equal(x.InteriorIntersects(y), expectedRelation[3] == 'T');
JavaAssert.Equal(x.Contains(y), x.Union(y).Equals(x));
JavaAssert.Equal(x.Intersects(y), !x.Intersection(y).IsEmpty);
}
private static void TestIntervalOps(R1Interval x, R1Interval y, string expected)
{
// Test all of the interval operations on the given pair of intervals.
// "expected" is a sequence of "T" and "F" characters corresponding to
// the expected results of Contains(), InteriorContains(), Intersects(),
// and InteriorIntersects() respectively.
Assert.Equal(expected[0] == 'T', x.Contains(y));
Assert.Equal(expected[1] == 'T', x.InteriorContains(y));
Assert.Equal(expected[2] == 'T', x.Intersects(y));
Assert.Equal(expected[3] == 'T', x.InteriorIntersects(y));
Assert.Equal(x.Contains(y), x.Union(y) == x);
Assert.Equal(x.Intersects(y), !x.Intersection(y).IsEmpty());
R1Interval z = x;
z = R1Interval.AddInterval(z, y);
Assert.Equal(x.Union(y), z);
}
public void Test_S2LatLngRect_GetCentroid()
{
// Empty and full rectangles.
Assert.Equal(new S2Point(), S2LatLngRect.Empty.Centroid());
Assert.True(S2LatLngRect.Full.Centroid().Norm() <= 1e-15);
// Rectangles that cover the full longitude range.
for (int i = 0; i < 100; ++i)
{
double lat1 = S2Testing.Random.UniformDouble(-S2.M_PI_2, S2.M_PI_2);
double lat2 = S2Testing.Random.UniformDouble(-S2.M_PI_2, S2.M_PI_2);
S2LatLngRect r = new(R1Interval.FromPointPair(lat1, lat2), S1Interval.Full);
S2Point centroid = r.Centroid();
Assert2.Near(0.5 * (Math.Sin(lat1) + Math.Sin(lat2)) * r.Area(), centroid.Z, S2.DoubleError);
Assert.True(new R2Point(centroid.X, centroid.Y).GetNorm() <= 1e-15);
}
// Rectangles that cover the full latitude range.
for (int i = 0; i < 100; ++i)
{
double lng1 = S2Testing.Random.UniformDouble(-Math.PI, Math.PI);
double lng2 = S2Testing.Random.UniformDouble(-Math.PI, Math.PI);
S2LatLngRect r = new(S2LatLngRect.FullLat,
S1Interval.FromPointPair(lng1, lng2));
S2Point centroid = r.Centroid();
Assert.True(Math.Abs(centroid.Z) <= 1e-15);
Assert2.Near(r.Lng.GetCenter(), new S2LatLng(centroid).LngRadians, S2.DoubleError);
double alpha = 0.5 * r.Lng.GetLength();
// TODO(Alas): the next Assert fails sometimes
Assert2.Near(0.25 * Math.PI * Math.Sin(alpha) / alpha * r.Area(),
new R2Point(centroid.X, centroid.Y).GetNorm(), S2.DoubleError);
}
// Finally, verify that when a rectangle is recursively split into pieces,
// the centroids of the pieces add to give the centroid of their parent.
// To make the code simpler we avoid rectangles that cross the 180 degree
// line of longitude.
TestCentroidSplitting(
new S2LatLngRect(S2LatLngRect.FullLat, new S1Interval(-3.14, 3.14)),
10 /*splits_left*/);
}
public void Test_R1Interval_ApproxEquals()
{
// Choose two values kLo and kHi such that it's okay to shift an endpoint by
// kLo (i.e., the resulting interval is equivalent) but not by kHi.
const double kLo = 4 * S2.DoubleEpsilon; // < max_error default
const double kHi = 6 * S2.DoubleEpsilon; // > max_error default
// Empty intervals.
R1Interval empty = R1Interval.Empty;
Assert.True(empty.ApproxEquals(empty));
Assert.True(new R1Interval(0, 0).ApproxEquals(empty));
Assert.True(empty.ApproxEquals(new R1Interval(0, 0)));
Assert.True(new R1Interval(1, 1).ApproxEquals(empty));
Assert.True(empty.ApproxEquals(new R1Interval(1, 1)));
Assert.False(empty.ApproxEquals(new R1Interval(0, 1)));
Assert.True(empty.ApproxEquals(new R1Interval(1, 1 + 2 * kLo)));
Assert.False(empty.ApproxEquals(new R1Interval(1, 1 + 2 * kHi)));
// Singleton intervals.
Assert.True(new R1Interval(1, 1).ApproxEquals(new R1Interval(1, 1)));
Assert.True(new R1Interval(1, 1).ApproxEquals(new R1Interval(1 - kLo, 1 - kLo)));
Assert.True(new R1Interval(1, 1).ApproxEquals(new R1Interval(1 + kLo, 1 + kLo)));
Assert.False(new R1Interval(1, 1).ApproxEquals(new R1Interval(1 - kHi, 1)));
Assert.False(new R1Interval(1, 1).ApproxEquals(new R1Interval(1, 1 + kHi)));
Assert.True(new R1Interval(1, 1).ApproxEquals(new R1Interval(1 - kLo, 1 + kLo)));
Assert.False(new R1Interval(0, 0).ApproxEquals(new R1Interval(1, 1)));
// Other intervals.
Assert.True(new R1Interval(1 - kLo, 2 + kLo).ApproxEquals(new R1Interval(1, 2)));
Assert.True(new R1Interval(1 + kLo, 2 - kLo).ApproxEquals(new R1Interval(1, 2)));
Assert.False(new R1Interval(1 - kHi, 2 + kLo).ApproxEquals(new R1Interval(1, 2)));
Assert.False(new R1Interval(1 + kHi, 2 - kLo).ApproxEquals(new R1Interval(1, 2)));
Assert.False(new R1Interval(1 - kLo, 2 + kHi).ApproxEquals(new R1Interval(1, 2)));
Assert.False(new R1Interval(1 + kLo, 2 - kHi).ApproxEquals(new R1Interval(1, 2)));
}
public void R1IntervalBasicTest()
{
// Constructors and accessors.
var unit = new R1Interval(0, 1);
var negunit = new R1Interval(-1, 0);
JavaAssert.Equal(unit.Lo, 0.0);
JavaAssert.Equal(unit.Hi, 1.0);
JavaAssert.Equal(negunit.Lo, -1.0);
JavaAssert.Equal(negunit.Hi, 0.0);
// is_empty()
var half = new R1Interval(0.5, 0.5);
Assert.True(!unit.IsEmpty);
Assert.True(!half.IsEmpty);
var empty = R1Interval.Empty;
Assert.True(empty.IsEmpty);
// GetCenter(), GetLength()
JavaAssert.Equal(unit.Center, 0.5);
JavaAssert.Equal(half.Center, 0.5);
JavaAssert.Equal(negunit.Length, 1.0);
JavaAssert.Equal(half.Length, 0.0);
Assert.True(empty.Length < 0);
// contains(double), interiorContains(double)
Assert.True(unit.Contains(0.5));
Assert.True(unit.InteriorContains(0.5));
Assert.True(unit.Contains(0));
Assert.True(!unit.InteriorContains(0));
Assert.True(unit.Contains(1));
Assert.True(!unit.InteriorContains(1));
// contains(R1Interval), interiorContains(R1Interval)
// Intersects(R1Interval), InteriorIntersects(R1Interval)
testIntervalOps(empty, empty, "TTFF");
testIntervalOps(empty, unit, "FFFF");
testIntervalOps(unit, half, "TTTT");
testIntervalOps(unit, unit, "TFTT");
testIntervalOps(unit, empty, "TTFF");
testIntervalOps(unit, negunit, "FFTF");
testIntervalOps(unit, new R1Interval(0, 0.5), "TFTT");
testIntervalOps(half, new R1Interval(0, 0.5), "FFTF");
// addPoint()
R1Interval r;
r = empty.AddPoint(5);
Assert.True(r.Lo == 5.0 && r.Hi == 5.0);
r = r.AddPoint(-1);
Assert.True(r.Lo == -1.0 && r.Hi == 5.0);
r = r.AddPoint(0);
Assert.True(r.Lo == -1.0 && r.Hi == 5.0);
// fromPointPair()
JavaAssert.Equal(R1Interval.FromPointPair(4, 4), new R1Interval(4, 4));
JavaAssert.Equal(R1Interval.FromPointPair(-1, -2), new R1Interval(-2, -1));
JavaAssert.Equal(R1Interval.FromPointPair(-5, 3), new R1Interval(-5, 3));
// expanded()
JavaAssert.Equal(empty.Expanded(0.45), empty);
JavaAssert.Equal(unit.Expanded(0.5), new R1Interval(-0.5, 1.5));
// union(), intersection()
Assert.True(new R1Interval(99, 100).Union(empty).Equals(new R1Interval(99, 100)));
Assert.True(empty.Union(new R1Interval(99, 100)).Equals(new R1Interval(99, 100)));
Assert.True(new R1Interval(5, 3).Union(new R1Interval(0, -2)).IsEmpty);
Assert.True(new R1Interval(0, -2).Union(new R1Interval(5, 3)).IsEmpty);
Assert.True(unit.Union(unit).Equals(unit));
Assert.True(unit.Union(negunit).Equals(new R1Interval(-1, 1)));
Assert.True(negunit.Union(unit).Equals(new R1Interval(-1, 1)));
Assert.True(half.Union(unit).Equals(unit));
Assert.True(unit.Intersection(half).Equals(half));
Assert.True(unit.Intersection(negunit).Equals(new R1Interval(0, 0)));
Assert.True(negunit.Intersection(half).IsEmpty);
Assert.True(unit.Intersection(empty).IsEmpty);
Assert.True(empty.Intersection(unit).IsEmpty);
}
public void R1IntervalBasicTest()
{
// Constructors and accessors.
var unit = new R1Interval(0, 1);
var negunit = new R1Interval(-1, 0);
JavaAssert.Equal(unit.Lo, 0.0);
JavaAssert.Equal(unit.Hi, 1.0);
JavaAssert.Equal(negunit.Lo, -1.0);
JavaAssert.Equal(negunit.Hi, 0.0);
// is_empty()
var half = new R1Interval(0.5, 0.5);
Assert.True(!unit.IsEmpty);
Assert.True(!half.IsEmpty);
var empty = R1Interval.Empty;
Assert.True(empty.IsEmpty);
// GetCenter(), GetLength()
JavaAssert.Equal(unit.Center, 0.5);
JavaAssert.Equal(half.Center, 0.5);
JavaAssert.Equal(negunit.Length, 1.0);
JavaAssert.Equal(half.Length, 0.0);
Assert.True(empty.Length < 0);
// contains(double), interiorContains(double)
Assert.True(unit.Contains(0.5));
Assert.True(unit.InteriorContains(0.5));
Assert.True(unit.Contains(0));
Assert.True(!unit.InteriorContains(0));
Assert.True(unit.Contains(1));
Assert.True(!unit.InteriorContains(1));
// contains(R1Interval), interiorContains(R1Interval)
// Intersects(R1Interval), InteriorIntersects(R1Interval)
testIntervalOps(empty, empty, "TTFF");
testIntervalOps(empty, unit, "FFFF");
testIntervalOps(unit, half, "TTTT");
testIntervalOps(unit, unit, "TFTT");
testIntervalOps(unit, empty, "TTFF");
testIntervalOps(unit, negunit, "FFTF");
testIntervalOps(unit, new R1Interval(0, 0.5), "TFTT");
testIntervalOps(half, new R1Interval(0, 0.5), "FFTF");
// addPoint()
R1Interval r;
r = empty.AddPoint(5);
Assert.True(r.Lo == 5.0 && r.Hi == 5.0);
r = r.AddPoint(-1);
Assert.True(r.Lo == -1.0 && r.Hi == 5.0);
r = r.AddPoint(0);
Assert.True(r.Lo == -1.0 && r.Hi == 5.0);
// fromPointPair()
JavaAssert.Equal(R1Interval.FromPointPair(4, 4), new R1Interval(4, 4));
JavaAssert.Equal(R1Interval.FromPointPair(-1, -2), new R1Interval(-2, -1));
JavaAssert.Equal(R1Interval.FromPointPair(-5, 3), new R1Interval(-5, 3));
// expanded()
JavaAssert.Equal(empty.Expanded(0.45), empty);
JavaAssert.Equal(unit.Expanded(0.5), new R1Interval(-0.5, 1.5));
// union(), intersection()
Assert.True(new R1Interval(99, 100).Union(empty).Equals(new R1Interval(99, 100)));
Assert.True(empty.Union(new R1Interval(99, 100)).Equals(new R1Interval(99, 100)));
Assert.True(new R1Interval(5, 3).Union(new R1Interval(0, -2)).IsEmpty);
Assert.True(new R1Interval(0, -2).Union(new R1Interval(5, 3)).IsEmpty);
Assert.True(unit.Union(unit).Equals(unit));
Assert.True(unit.Union(negunit).Equals(new R1Interval(-1, 1)));
Assert.True(negunit.Union(unit).Equals(new R1Interval(-1, 1)));
Assert.True(half.Union(unit).Equals(unit));
Assert.True(unit.Intersection(half).Equals(half));
Assert.True(unit.Intersection(negunit).Equals(new R1Interval(0, 0)));
Assert.True(negunit.Intersection(half).IsEmpty);
Assert.True(unit.Intersection(empty).IsEmpty);
Assert.True(empty.Intersection(unit).IsEmpty);
}
public void AddPoint(S2Point b)
{
// assert (S2.isUnitLength(b));
var bLatLng = new S2LatLng(b);
if (bound.IsEmpty)
{
bound = bound.AddPoint(bLatLng);
}
else
{
// We can't just call bound.addPoint(bLatLng) here, since we need to
// ensure that all the longitudes between "a" and "b" are included.
bound = bound.Union(S2LatLngRect.FromPointPair(aLatLng, bLatLng));
// Check whether the Min/Max latitude occurs in the edge interior.
// We find the normal to the plane containing AB, and then a vector
// "dir" in this plane that also passes through the equator. We use
// RobustCrossProd to ensure that the edge normal is accurate even
// when the two points are very close together.
var aCrossB = S2.RobustCrossProd(a, b);
var dir = S2Point.CrossProd(aCrossB, new S2Point(0, 0, 1));
var da = dir.DotProd(a);
var db = dir.DotProd(b);
if (da*db < 0)
{
// Minimum/maximum latitude occurs in the edge interior. This affects
// the latitude bounds but not the longitude bounds.
var absLat = Math.Acos(Math.Abs(aCrossB[2]/aCrossB.Norm));
var lat = bound.Lat;
if (da < 0)
{
// It's possible that absLat < lat.lo() due to numerical errors.
lat = new R1Interval(lat.Lo, Math.Max(absLat, bound.Lat.Hi));
}
else
{
lat = new R1Interval(Math.Min(-absLat, bound.Lat.Lo), lat.Hi);
}
bound = new S2LatLngRect(lat, bound.Lng);
}
}
a = b;
aLatLng = bLatLng;
}
private static double SampleInterval(R1Interval x) => S2Testing.Random.UniformDouble(x.Lo, x.Hi);
public void Test_R1Interval_TestBasic()
{
// Constructors and accessors.
R1Interval unit = new(0, 1);
R1Interval negunit = new(-1, 0);
Assert.Equal(0, unit.Lo);
Assert.Equal(1, unit.Hi);
Assert.Equal(-1, negunit[0]);
Assert.Equal(0, negunit[1]);
R1Interval ten = new(0, 10);
Assert.Equal(new R1Interval(0, 10), ten);
ten = new R1Interval(-10, ten.Hi);
Assert.Equal(new R1Interval(-10, 10), ten);
ten = new R1Interval(ten.Lo, 0);
Assert.True(new R1Interval(-10, 0) == ten);
ten = new R1Interval(0, 10);
Assert.Equal(new R1Interval(0, 10), ten);
// IsEmpty
R1Interval half = new(0.5, 0.5);
Assert.False(unit.IsEmpty());
Assert.False(half.IsEmpty());
R1Interval empty = R1Interval.Empty;
Assert.True(empty.IsEmpty());
// == and !=
Assert.True(empty == R1Interval.Empty);
Assert.True(unit == new R1Interval(0, 1));
Assert.True(unit != empty);
Assert.True(new R1Interval(1, 2) != new R1Interval(1, 3));
// Check that the default R1Interval is identical to Empty().
R1Interval default_empty = R1Interval.Empty;
Assert.True(default_empty.IsEmpty());
Assert.Equal(empty.Lo, default_empty.Lo);
Assert.Equal(empty.Hi, default_empty.Hi);
// GetCenter(), GetLength()
Assert.Equal(0.5, unit.GetCenter());
Assert.Equal(0.5, half.GetCenter());
Assert.Equal(1.0, negunit.GetLength());
Assert.Equal(0, half.GetLength());
Assert.True(empty.GetLength() < 0);
// Contains(double), InteriorContains(double)
Assert.True(unit.Contains(0.5));
Assert.True(unit.InteriorContains(0.5));
Assert.True(unit.Contains(0));
Assert.False(unit.InteriorContains(0));
Assert.True(unit.Contains(1));
Assert.False(unit.InteriorContains(1));
// Contains(R1Interval), InteriorContains(R1Interval)
// Intersects(R1Interval), InteriorIntersects(R1Interval)
TestIntervalOps(empty, empty, "TTFF");
TestIntervalOps(empty, unit, "FFFF");
TestIntervalOps(unit, half, "TTTT");
TestIntervalOps(unit, unit, "TFTT");
TestIntervalOps(unit, empty, "TTFF");
TestIntervalOps(unit, negunit, "FFTF");
TestIntervalOps(unit, new R1Interval(0, 0.5), "TFTT");
TestIntervalOps(half, new R1Interval(0, 0.5), "FFTF");
// AddPoint()
R1Interval r = empty;
r = R1Interval.AddPoint(r, 5);
Assert.Equal(5, r.Lo);
Assert.Equal(5, r.Hi);
r = R1Interval.AddPoint(r, -1);
Assert.Equal(-1, r.Lo);
Assert.Equal(5, r.Hi);
r = R1Interval.AddPoint(r, 0);
Assert.Equal(-1, r.Lo);
Assert.Equal(5, r.Hi);
// Project()
Assert.Equal(0.3, new R1Interval(0.1, 0.4).Project(0.3));
Assert.Equal(0.1, new R1Interval(0.1, 0.4).Project(-7.0));
Assert.Equal(0.4, new R1Interval(0.1, 0.4).Project(0.6));
// FromPointPair()
Assert.Equal(new R1Interval(4, 4), R1Interval.FromPointPair(4, 4));
Assert.Equal(new R1Interval(-2, -1), R1Interval.FromPointPair(-1, -2));
Assert.Equal(new R1Interval(-5, 3), R1Interval.FromPointPair(-5, 3));
// Expanded()
Assert.Equal(empty, empty.Expanded(0.45));
Assert.Equal(new R1Interval(-0.5, 1.5), unit.Expanded(0.5));
Assert.Equal(new R1Interval(0.5, 0.5), unit.Expanded(-0.5));
Assert.True(unit.Expanded(-0.51).IsEmpty());
Assert.True(unit.Expanded(-0.51).Expanded(0.51).IsEmpty());
// Union(), Intersection()
Assert.Equal(new R1Interval(99, 100), new R1Interval(99, 100).Union(empty));
Assert.Equal(new R1Interval(99, 100), empty.Union(new R1Interval(99, 100)));
Assert.True(new R1Interval(5, 3).Union(new R1Interval(0, -2)).IsEmpty());
Assert.True(new R1Interval(0, -2).Union(new R1Interval(5, 3)).IsEmpty());
Assert.Equal(unit, unit.Union(unit));
Assert.Equal(new R1Interval(-1, 1), unit.Union(negunit));
Assert.Equal(new R1Interval(-1, 1), negunit.Union(unit));
Assert.Equal(unit, half.Union(unit));
Assert.Equal(half, unit.Intersection(half));
Assert.Equal(new R1Interval(0, 0), unit.Intersection(negunit));
Assert.True(negunit.Intersection(half).IsEmpty());
Assert.True(unit.Intersection(empty).IsEmpty());
Assert.True(empty.Intersection(unit).IsEmpty());
}
// Common back end for AddPoint() and AddLatLng(). b and b_latlng
// must refer to the same vertex.
private void AddInternal(S2Point b, S2LatLng b_latlng)
{
// Simple consistency check to verify that b and b_latlng are alternate
// representations of the same vertex.
System.Diagnostics.Debug.Assert(S2.ApproxEquals(b, b_latlng.ToPoint()));
if (bound_.IsEmpty())
{
bound_ = bound_.AddPoint(b_latlng);
}
else
{
// First compute the cross product N = A x B robustly. This is the normal
// to the great circle through A and B. We don't use S2.RobustCrossProd()
// since that method returns an arbitrary vector orthogonal to A if the two
// vectors are proportional, and we want the zero vector in that case.
var n = (a_ - b).CrossProd(a_ + b); // N = 2 * (A x B)
// The relative error in N gets large as its norm gets very small (i.e.,
// when the two points are nearly identical or antipodal). We handle this
// by choosing a maximum allowable error, and if the error is greater than
// this we fall back to a different technique. Since it turns out that
// the other sources of error in converting the normal to a maximum
// latitude add up to at most 1.16 * S2Constants.DoubleEpsilon (see below), and it is
// desirable to have the total error be a multiple of S2Constants.DoubleEpsilon, we have
// chosen to limit the maximum error in the normal to 3.84 * S2Constants.DoubleEpsilon.
// It is possible to show that the error is less than this when
//
// n.Norm >= 8 * Math.Sqrt(3) / (3.84 - 0.5 - Math.Sqrt(3)) * S2Constants.DoubleEpsilon
// = 1.91346e-15 (about 8.618 * S2Constants.DoubleEpsilon)
var n_norm = n.Norm();
if (n_norm < 1.91346e-15)
{
// A and B are either nearly identical or nearly antipodal (to within
// 4.309 * S2Constants.DoubleEpsilon, or about 6 nanometers on the earth's surface).
if (a_.DotProd(b) < 0)
{
// The two points are nearly antipodal. The easiest solution is to
// assume that the edge between A and B could go in any direction
// around the sphere.
bound_ = S2LatLngRect.Full;
}
else
{
// The two points are nearly identical (to within 4.309 * S2Constants.DoubleEpsilon).
// In this case we can just use the bounding rectangle of the points,
// since after the expansion done by GetBound() this rectangle is
// guaranteed to include the (lat,lng) values of all points along AB.
bound_ = bound_.Union(S2LatLngRect.FromPointPair(a_latlng_, b_latlng));
}
}
else
{
// Compute the longitude range spanned by AB.
var lng_ab = S1Interval.FromPointPair(a_latlng_.LngRadians, b_latlng.LngRadians);
if (lng_ab.GetLength() >= Math.PI - 2 * S2.DoubleEpsilon)
{
// The points lie on nearly opposite lines of longitude to within the
// maximum error of the calculation. (Note that this test relies on
// the fact that Math.PI is slightly less than the true value of Pi, and
// that representable values near Math.PI are 2 * S2Constants.DoubleEpsilon apart.)
// The easiest solution is to assume that AB could go on either side
// of the pole.
lng_ab = S1Interval.Full;
}
// Next we compute the latitude range spanned by the edge AB. We start
// with the range spanning the two endpoints of the edge:
var lat_ab = R1Interval.FromPointPair(a_latlng_.LatRadians, b_latlng.LatRadians);
// This is the desired range unless the edge AB crosses the plane
// through N and the Z-axis (which is where the great circle through A
// and B attains its minimum and maximum latitudes). To test whether AB
// crosses this plane, we compute a vector M perpendicular to this
// plane and then project A and B onto it.
var m = n.CrossProd(new S2Point(0, 0, 1));
var m_a = m.DotProd(a_);
var m_b = m.DotProd(b);
// We want to test the signs of "m_a" and "m_b", so we need to bound
// the error in these calculations. It is possible to show that the
// total error is bounded by
//
// (1 + Math.Sqrt(3)) * S2Constants.DoubleEpsilon * n_norm + 8 * Math.Sqrt(3) * (S2Constants.DoubleEpsilon**2)
// = 6.06638e-16 * n_norm + 6.83174e-31
double m_error = 6.06638e-16 * n_norm + 6.83174e-31;
if (m_a * m_b < 0 || Math.Abs(m_a) <= m_error || Math.Abs(m_b) <= m_error)
{
// Minimum/maximum latitude *may* occur in the edge interior.
//
// The maximum latitude is 90 degrees minus the latitude of N. We
// compute this directly using atan2 in order to get maximum accuracy
// near the poles.
//
// Our goal is compute a bound that contains the computed latitudes of
// all S2Points P that pass the point-in-polygon containment test.
// There are three sources of error we need to consider:
// - the directional error in N (at most 3.84 * S2Constants.DoubleEpsilon)
// - converting N to a maximum latitude
// - computing the latitude of the test point P
// The latter two sources of error are at most 0.955 * S2Constants.DoubleEpsilon
// individually, but it is possible to show by a more complex analysis
// that together they can add up to at most 1.16 * S2Constants.DoubleEpsilon, for a
// total error of 5 * S2Constants.DoubleEpsilon.
//
// We add 3 * S2Constants.DoubleEpsilon to the bound here, and GetBound() will pad
// the bound by another 2 * S2Constants.DoubleEpsilon.
var max_lat = Math.Min(
Math.Atan2(Math.Sqrt(n[0] * n[0] + n[1] * n[1]), Math.Abs(n[2])) + 3 * S2.DoubleEpsilon,
S2.M_PI_2);
// In order to get tight bounds when the two points are close together,
// we also bound the min/max latitude relative to the latitudes of the
// endpoints A and B. First we compute the distance between A and B,
// and then we compute the maximum change in latitude between any two
// points along the great circle that are separated by this distance.
// This gives us a latitude change "budget". Some of this budget must
// be spent getting from A to B; the remainder bounds the round-trip
// distance (in latitude) from A or B to the min or max latitude
// attained along the edge AB.
//
// There is a maximum relative error of 4.5 * DBL_EPSILON in computing
// the squared distance (a_ - b), which means a maximum error of (4.5
// / 2 + 0.5) == 2.75 * DBL_EPSILON in computing Norm(). The sin()
// and multiply each have a relative error of 0.5 * DBL_EPSILON which
// we round up to a total of 4 * DBL_EPSILON.
var lat_budget_z = 0.5 * (a_ - b).Norm() * Math.Sin(max_lat);
const double folded = (1 + 4 * S2.DoubleEpsilon);
var lat_budget = 2 * Math.Asin(Math.Min(folded * lat_budget_z, 1.0));
var max_delta = 0.5 * (lat_budget - lat_ab.GetLength()) + S2.DoubleEpsilon;
// Test whether AB passes through the point of maximum latitude or
// minimum latitude. If the dot product(s) are small enough then the
// result may be ambiguous.
if (m_a <= m_error && m_b >= -m_error)
{
lat_ab = new R1Interval(lat_ab.Lo, Math.Min(max_lat, lat_ab.Hi + max_delta));
}
if (m_b <= m_error && m_a >= -m_error)
{
lat_ab = new R1Interval(Math.Max(-max_lat, lat_ab.Lo - max_delta), lat_ab.Lo);
}
}
bound_ = bound_.Union(new S2LatLngRect(lat_ab, lng_ab));
}
}
a_ = b;
a_latlng_ = b_latlng;
}