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Diffstat (limited to 'vendor/github.com/golang/geo/s2/rect_bounder.go')
| -rw-r--r-- | vendor/github.com/golang/geo/s2/rect_bounder.go | 352 |
1 files changed, 0 insertions, 352 deletions
diff --git a/vendor/github.com/golang/geo/s2/rect_bounder.go b/vendor/github.com/golang/geo/s2/rect_bounder.go deleted file mode 100644 index 419dea0c1..000000000 --- a/vendor/github.com/golang/geo/s2/rect_bounder.go +++ /dev/null @@ -1,352 +0,0 @@ -// 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() -} |