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// Copyright 2017 Google Inc. All rights reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
package s2
import (
"math"
"github.com/golang/geo/r1"
"github.com/golang/geo/r3"
"github.com/golang/geo/s1"
)
// RectBounder is used to compute a bounding rectangle that contains all edges
// defined by a vertex chain (v0, v1, v2, ...). All vertices must be unit length.
// Note that the bounding rectangle of an edge can be larger than the bounding
// rectangle of its endpoints, e.g. consider an edge that passes through the North Pole.
//
// The bounds are calculated conservatively to account for numerical errors
// when points are converted to LatLngs. More precisely, this function
// guarantees the following:
// Let L be a closed edge chain (Loop) such that the interior of the loop does
// not contain either pole. Now if P is any point such that L.ContainsPoint(P),
// then RectBound(L).ContainsPoint(LatLngFromPoint(P)).
type RectBounder struct {
// The previous vertex in the chain.
a Point
// The previous vertex latitude longitude.
aLL LatLng
bound Rect
}
// NewRectBounder returns a new instance of a RectBounder.
func NewRectBounder() *RectBounder {
return &RectBounder{
bound: EmptyRect(),
}
}
// maxErrorForTests returns the maximum error in RectBound provided that the
// result does not include either pole. It is only used for testing purposes
func (r *RectBounder) maxErrorForTests() LatLng {
// The maximum error in the latitude calculation is
// 3.84 * dblEpsilon for the PointCross calculation
// 0.96 * dblEpsilon for the Latitude calculation
// 5 * dblEpsilon added by AddPoint/RectBound to compensate for error
// -----------------
// 9.80 * dblEpsilon maximum error in result
//
// The maximum error in the longitude calculation is dblEpsilon. RectBound
// does not do any expansion because this isn't necessary in order to
// bound the *rounded* longitudes of contained points.
return LatLng{10 * dblEpsilon * s1.Radian, 1 * dblEpsilon * s1.Radian}
}
// AddPoint adds the given point to the chain. The Point must be unit length.
func (r *RectBounder) AddPoint(b Point) {
bLL := LatLngFromPoint(b)
if r.bound.IsEmpty() {
r.a = b
r.aLL = bLL
r.bound = r.bound.AddPoint(bLL)
return
}
// 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 RobustSign
// 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.
n := r.a.Sub(b.Vector).Cross(r.a.Add(b.Vector)) // 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 * dblEpsilon, and it is desirable to
// have the total error be a multiple of dblEpsilon, we have chosen to
// limit the maximum error in the normal to be 3.84 * dblEpsilon.
// It is possible to show that the error is less than this when
//
// n.Norm() >= 8 * sqrt(3) / (3.84 - 0.5 - sqrt(3)) * dblEpsilon
// = 1.91346e-15 (about 8.618 * dblEpsilon)
nNorm := n.Norm()
if nNorm < 1.91346e-15 {
// A and B are either nearly identical or nearly antipodal (to within
// 4.309 * dblEpsilon, or about 6 nanometers on the earth's surface).
if r.a.Dot(b.Vector) < 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.
r.bound = FullRect()
} else {
// The two points are nearly identical (to within 4.309 * dblEpsilon).
// In this case we can just use the bounding rectangle of the points,
// since after the expansion done by GetBound this Rect is
// guaranteed to include the (lat,lng) values of all points along AB.
r.bound = r.bound.Union(RectFromLatLng(r.aLL).AddPoint(bLL))
}
r.a = b
r.aLL = bLL
return
}
// Compute the longitude range spanned by AB.
lngAB := s1.EmptyInterval().AddPoint(r.aLL.Lng.Radians()).AddPoint(bLL.Lng.Radians())
if lngAB.Length() >= math.Pi-2*dblEpsilon {
// The points lie on nearly opposite lines of longitude to within the
// maximum error of the calculation. The easiest solution is to assume
// that AB could go on either side of the pole.
lngAB = s1.FullInterval()
}
// Next we compute the latitude range spanned by the edge AB. We start
// with the range spanning the two endpoints of the edge:
latAB := r1.IntervalFromPoint(r.aLL.Lat.Radians()).AddPoint(bLL.Lat.Radians())
// 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.
m := n.Cross(r3.Vector{0, 0, 1})
mA := m.Dot(r.a.Vector)
mB := m.Dot(b.Vector)
// We want to test the signs of "mA" and "mB", so we need to bound
// the error in these calculations. It is possible to show that the
// total error is bounded by
//
// (1 + sqrt(3)) * dblEpsilon * nNorm + 8 * sqrt(3) * (dblEpsilon**2)
// = 6.06638e-16 * nNorm + 6.83174e-31
mError := 6.06638e-16*nNorm + 6.83174e-31
if mA*mB < 0 || math.Abs(mA) <= mError || math.Abs(mB) <= mError {
// 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 * dblEpsilon)
// - 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 * dblEpsilon
// individually, but it is possible to show by a more complex analysis
// that together they can add up to at most 1.16 * dblEpsilon, for a
// total error of 5 * dblEpsilon.
//
// We add 3 * dblEpsilon to the bound here, and GetBound() will pad
// the bound by another 2 * dblEpsilon.
maxLat := math.Min(
math.Atan2(math.Sqrt(n.X*n.X+n.Y*n.Y), math.Abs(n.Z))+3*dblEpsilon,
math.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.
latBudget := 2 * math.Asin(0.5*(r.a.Sub(b.Vector)).Norm()*math.Sin(maxLat))
maxDelta := 0.5*(latBudget-latAB.Length()) + dblEpsilon
// 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 mA <= mError && mB >= -mError {
latAB.Hi = math.Min(maxLat, latAB.Hi+maxDelta)
}
if mB <= mError && mA >= -mError {
latAB.Lo = math.Max(-maxLat, latAB.Lo-maxDelta)
}
}
r.a = b
r.aLL = bLL
r.bound = r.bound.Union(Rect{latAB, lngAB})
}
// RectBound 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 LatLng coordinates of all Points contained by the loop.
func (r *RectBounder) RectBound() Rect {
return r.bound.expanded(LatLng{s1.Angle(2 * dblEpsilon), 0}).PolarClosure()
}
// ExpandForSubregions expands a bounding Rect so that it is guaranteed to
// contain the bounds of any subregion whose bounds are computed using
// ComputeRectBound. For example, consider a loop L that defines a square.
// GetBound ensures that if a point P is contained by this square, then
// LatLngFromPoint(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(L.RectBound).Contains(S.RectBound).
//
func ExpandForSubregions(bound Rect) Rect {
// 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 * dblEpsilon). If so then we need to return
// FullRect, 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 * dblEpsilon.
// lngGap is a lower bound on the longitudinal distance between B and its
// reflection B'. (2.5 * dblEpsilon is the maximum combined error of the
// endpoint longitude calculations and the Length call.)
lngGap := math.Max(0, math.Pi-bound.Lng.Length()-2.5*dblEpsilon)
// minAbsLat is the minimum distance from B to the equator (if zero or
// negative, then B straddles the equator).
minAbsLat := math.Max(bound.Lat.Lo, -bound.Lat.Hi)
// latGapSouth and latGapNorth measure the minimum distance from B to the
// south and north poles respectively.
latGapSouth := math.Pi/2 + bound.Lat.Lo
latGapNorth := math.Pi/2 - bound.Lat.Hi
if minAbsLat >= 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*minAbsLat, and the longitude
// distance is lngGap. 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 * dblEpsilon), we substitute the Euclidean
// distance instead. This gives us a right triangle XYZ with two edges of
// length x = 2*minAbsLat and y ~= lngGap. The desired distance is the
// length of the third edge z, and we have
//
// z ~= sqrt(x^2 + y^2) >= (x + y) / sqrt(2)
//
// Therefore the region may contain nearly antipodal points only if
//
// 2*minAbsLat + lngGap < sqrt(2) * 4.309 * dblEpsilon
// ~= 1.354e-15
//
// Note that because the given bound B is conservative, minAbsLat and
// lngGap are both lower bounds on their true values so we do not need
// to make any adjustments for their errors.
if 2*minAbsLat+lngGap < 1.354e-15 {
return FullRect()
}
} else if lngGap >= math.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 = latGapSouth
// and y = latGapNorth, and angle Z >= Pi/2 between them. We then have
//
// z >= sqrt(x^2 + y^2) >= (x + y) / sqrt(2)
//
// Unlike the case above, latGapSouth and latGapNorth are not lower bounds
// (because of the extra addition operation, and because math.Pi/2 is not
// exactly equal to Pi/2); they can exceed their true values by up to
// 0.75 * dblEpsilon. Putting this all together, the region may contain
// nearly antipodal points only if
//
// latGapSouth + latGapNorth < (sqrt(2) * 4.309 + 1.5) * dblEpsilon
// ~= 1.687e-15
if latGapSouth+latGapNorth < 1.687e-15 {
return FullRect()
}
} 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(maxLatGap) / 1 = sin(dMin) / sin(lngGap)
// sin(dMin) = sin(maxLatGap) * sin(lngGap)
//
// where "maxLatGap" = max(latGapSouth, latGapNorth) and "dMin" 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 "maxLatGap" or lngGap is extremely small (in which
// case sin(t) ~= t), and recalling that "maxLatGap" has an error of up
// to 0.75 * dblEpsilon, we want to test whether
//
// maxLatGap * lngGap < (4.309 + 0.75) * (Pi/2) * dblEpsilon
// ~= 1.765e-15
if math.Max(latGapSouth, latGapNorth)*lngGap < 1.765e-15 {
return FullRect()
}
}
// Next we need to check whether the subregion might contain any edges that
// span (math.Pi - 2 * dblEpsilon) radians or more in longitude, since AddPoint
// sets the longitude bound to Full in that case. This corresponds to
// testing whether (lngGap <= 0) in lngExpansion below.
// Otherwise, the maximum latitude error in AddPoint is 4.8 * dblEpsilon.
// 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
// dblEpsilon.)
//
// For longitude, we rely on the fact that atan2 is correctly rounded and
// therefore no additional bounds expansion is necessary.
latExpansion := 9 * dblEpsilon
lngExpansion := 0.0
if lngGap <= 0 {
lngExpansion = math.Pi
}
return bound.expanded(LatLng{s1.Angle(latExpansion), s1.Angle(lngExpansion)}).PolarClosure()
}
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