private static S1Angle bruteForceRectPointDistance(S2LatLngRect a, S2LatLng b)
{
if (a.Contains(b))
{
return(S1Angle.FromRadians(0));
}
var bToLoLat = getDistance(b, a.LatLo, a.Lng);
var bToHiLat = getDistance(b, a.LatHi, a.Lng);
var bToLoLng =
S2EdgeUtil.GetDistance(b.ToPoint(), new S2LatLng(a.LatLo, a.LngLo).ToPoint(),
new S2LatLng(a.LatHi, a.LngLo).ToPoint());
var bToHiLng =
S2EdgeUtil.GetDistance(b.ToPoint(), new S2LatLng(a.LatLo, a.LngHi).ToPoint(),
new S2LatLng(a.LatHi, a.LngHi).ToPoint());
return(S1Angle.Min(bToLoLat, S1Angle.Min(bToHiLat, S1Angle.Min(bToLoLng, bToHiLng))));
}
private static S1Angle bruteForceRectPointDistance(S2LatLngRect a, S2LatLng b)
{
if (a.Contains(b))
{
return S1Angle.FromRadians(0);
}
var bToLoLat = getDistance(b, a.LatLo, a.Lng);
var bToHiLat = getDistance(b, a.LatHi, a.Lng);
var bToLoLng =
S2EdgeUtil.GetDistance(b.ToPoint(), new S2LatLng(a.LatLo, a.LngLo).ToPoint(),
new S2LatLng(a.LatHi, a.LngLo).ToPoint());
var bToHiLng =
S2EdgeUtil.GetDistance(b.ToPoint(), new S2LatLng(a.LatLo, a.LngHi).ToPoint(),
new S2LatLng(a.LatHi, a.LngHi).ToPoint());
return S1Angle.Min(bToLoLat, S1Angle.Min(bToHiLat, S1Angle.Min(bToLoLng, bToHiLng)));
}
// This method is called to add a vertex to the chain when the vertex is
// represented as an S2LatLng. Repeated vertices are ignored.
public void AddLatLng(S2LatLng b_latlng)
{
AddInternal(b_latlng.ToPoint(), b_latlng);
}
// 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;
}
