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Diffstat (limited to 'vendor/github.com/golang/geo/s2/cell.go')
| -rw-r--r-- | vendor/github.com/golang/geo/s2/cell.go | 698 |
1 files changed, 698 insertions, 0 deletions
diff --git a/vendor/github.com/golang/geo/s2/cell.go b/vendor/github.com/golang/geo/s2/cell.go new file mode 100644 index 000000000..0a01a4f1f --- /dev/null +++ b/vendor/github.com/golang/geo/s2/cell.go @@ -0,0 +1,698 @@ +// Copyright 2014 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 ( + "io" + "math" + + "github.com/golang/geo/r1" + "github.com/golang/geo/r2" + "github.com/golang/geo/r3" + "github.com/golang/geo/s1" +) + +// Cell is an S2 region object that represents a cell. Unlike CellIDs, +// it supports efficient containment and intersection tests. However, it is +// also a more expensive representation. +type Cell struct { + face int8 + level int8 + orientation int8 + id CellID + uv r2.Rect +} + +// CellFromCellID constructs a Cell corresponding to the given CellID. +func CellFromCellID(id CellID) Cell { + c := Cell{} + c.id = id + f, i, j, o := c.id.faceIJOrientation() + c.face = int8(f) + c.level = int8(c.id.Level()) + c.orientation = int8(o) + c.uv = ijLevelToBoundUV(i, j, int(c.level)) + return c +} + +// CellFromPoint constructs a cell for the given Point. +func CellFromPoint(p Point) Cell { + return CellFromCellID(cellIDFromPoint(p)) +} + +// CellFromLatLng constructs a cell for the given LatLng. +func CellFromLatLng(ll LatLng) Cell { + return CellFromCellID(CellIDFromLatLng(ll)) +} + +// Face returns the face this cell is on. +func (c Cell) Face() int { + return int(c.face) +} + +// oppositeFace returns the face opposite the given face. +func oppositeFace(face int) int { + return (face + 3) % 6 +} + +// Level returns the level of this cell. +func (c Cell) Level() int { + return int(c.level) +} + +// ID returns the CellID this cell represents. +func (c Cell) ID() CellID { + return c.id +} + +// IsLeaf returns whether this Cell is a leaf or not. +func (c Cell) IsLeaf() bool { + return c.level == maxLevel +} + +// SizeIJ returns the edge length of this cell in (i,j)-space. +func (c Cell) SizeIJ() int { + return sizeIJ(int(c.level)) +} + +// SizeST returns the edge length of this cell in (s,t)-space. +func (c Cell) SizeST() float64 { + return c.id.sizeST(int(c.level)) +} + +// Vertex returns the k-th vertex of the cell (k = 0,1,2,3) in CCW order +// (lower left, lower right, upper right, upper left in the UV plane). +func (c Cell) Vertex(k int) Point { + return Point{faceUVToXYZ(int(c.face), c.uv.Vertices()[k].X, c.uv.Vertices()[k].Y).Normalize()} +} + +// Edge returns the inward-facing normal of the great circle passing through +// the CCW ordered edge from vertex k to vertex k+1 (mod 4) (for k = 0,1,2,3). +func (c Cell) Edge(k int) Point { + switch k { + case 0: + return Point{vNorm(int(c.face), c.uv.Y.Lo).Normalize()} // Bottom + case 1: + return Point{uNorm(int(c.face), c.uv.X.Hi).Normalize()} // Right + case 2: + return Point{vNorm(int(c.face), c.uv.Y.Hi).Mul(-1.0).Normalize()} // Top + default: + return Point{uNorm(int(c.face), c.uv.X.Lo).Mul(-1.0).Normalize()} // Left + } +} + +// BoundUV returns the bounds of this cell in (u,v)-space. +func (c Cell) BoundUV() r2.Rect { + return c.uv +} + +// Center returns the direction vector corresponding to the center in +// (s,t)-space of the given cell. This is the point at which the cell is +// divided into four subcells; it is not necessarily the centroid of the +// cell in (u,v)-space or (x,y,z)-space +func (c Cell) Center() Point { + return Point{c.id.rawPoint().Normalize()} +} + +// Children returns the four direct children of this cell in traversal order +// and returns true. If this is a leaf cell, or the children could not be created, +// false is returned. +// The C++ method is called Subdivide. +func (c Cell) Children() ([4]Cell, bool) { + var children [4]Cell + + if c.id.IsLeaf() { + return children, false + } + + // Compute the cell midpoint in uv-space. + uvMid := c.id.centerUV() + + // Create four children with the appropriate bounds. + cid := c.id.ChildBegin() + for pos := 0; pos < 4; pos++ { + children[pos] = Cell{ + face: c.face, + level: c.level + 1, + orientation: c.orientation ^ int8(posToOrientation[pos]), + id: cid, + } + + // We want to split the cell in half in u and v. To decide which + // side to set equal to the midpoint value, we look at cell's (i,j) + // position within its parent. The index for i is in bit 1 of ij. + ij := posToIJ[c.orientation][pos] + i := ij >> 1 + j := ij & 1 + if i == 1 { + children[pos].uv.X.Hi = c.uv.X.Hi + children[pos].uv.X.Lo = uvMid.X + } else { + children[pos].uv.X.Lo = c.uv.X.Lo + children[pos].uv.X.Hi = uvMid.X + } + if j == 1 { + children[pos].uv.Y.Hi = c.uv.Y.Hi + children[pos].uv.Y.Lo = uvMid.Y + } else { + children[pos].uv.Y.Lo = c.uv.Y.Lo + children[pos].uv.Y.Hi = uvMid.Y + } + cid = cid.Next() + } + return children, true +} + +// ExactArea returns the area of this cell as accurately as possible. +func (c Cell) ExactArea() float64 { + v0, v1, v2, v3 := c.Vertex(0), c.Vertex(1), c.Vertex(2), c.Vertex(3) + return PointArea(v0, v1, v2) + PointArea(v0, v2, v3) +} + +// ApproxArea returns the approximate area of this cell. This method is accurate +// to within 3% percent for all cell sizes and accurate to within 0.1% for cells +// at level 5 or higher (i.e. squares 350km to a side or smaller on the Earth's +// surface). It is moderately cheap to compute. +func (c Cell) ApproxArea() float64 { + // All cells at the first two levels have the same area. + if c.level < 2 { + return c.AverageArea() + } + + // First, compute the approximate area of the cell when projected + // perpendicular to its normal. The cross product of its diagonals gives + // the normal, and the length of the normal is twice the projected area. + flatArea := 0.5 * (c.Vertex(2).Sub(c.Vertex(0).Vector). + Cross(c.Vertex(3).Sub(c.Vertex(1).Vector)).Norm()) + + // Now, compensate for the curvature of the cell surface by pretending + // that the cell is shaped like a spherical cap. The ratio of the + // area of a spherical cap to the area of its projected disc turns out + // to be 2 / (1 + sqrt(1 - r*r)) where r is the radius of the disc. + // For example, when r=0 the ratio is 1, and when r=1 the ratio is 2. + // Here we set Pi*r*r == flatArea to find the equivalent disc. + return flatArea * 2 / (1 + math.Sqrt(1-math.Min(1/math.Pi*flatArea, 1))) +} + +// AverageArea returns the average area of cells at the level of this cell. +// This is accurate to within a factor of 1.7. +func (c Cell) AverageArea() float64 { + return AvgAreaMetric.Value(int(c.level)) +} + +// IntersectsCell reports whether the intersection of this cell and the other cell is not nil. +func (c Cell) IntersectsCell(oc Cell) bool { + return c.id.Intersects(oc.id) +} + +// ContainsCell reports whether this cell contains the other cell. +func (c Cell) ContainsCell(oc Cell) bool { + return c.id.Contains(oc.id) +} + +// CellUnionBound computes a covering of the Cell. +func (c Cell) CellUnionBound() []CellID { + return c.CapBound().CellUnionBound() +} + +// latitude returns the latitude of the cell vertex in radians given by (i,j), +// where i and j indicate the Hi (1) or Lo (0) corner. +func (c Cell) latitude(i, j int) float64 { + var u, v float64 + switch { + case i == 0 && j == 0: + u = c.uv.X.Lo + v = c.uv.Y.Lo + case i == 0 && j == 1: + u = c.uv.X.Lo + v = c.uv.Y.Hi + case i == 1 && j == 0: + u = c.uv.X.Hi + v = c.uv.Y.Lo + case i == 1 && j == 1: + u = c.uv.X.Hi + v = c.uv.Y.Hi + default: + panic("i and/or j is out of bounds") + } + return latitude(Point{faceUVToXYZ(int(c.face), u, v)}).Radians() +} + +// longitude returns the longitude of the cell vertex in radians given by (i,j), +// where i and j indicate the Hi (1) or Lo (0) corner. +func (c Cell) longitude(i, j int) float64 { + var u, v float64 + switch { + case i == 0 && j == 0: + u = c.uv.X.Lo + v = c.uv.Y.Lo + case i == 0 && j == 1: + u = c.uv.X.Lo + v = c.uv.Y.Hi + case i == 1 && j == 0: + u = c.uv.X.Hi + v = c.uv.Y.Lo + case i == 1 && j == 1: + u = c.uv.X.Hi + v = c.uv.Y.Hi + default: + panic("i and/or j is out of bounds") + } + return longitude(Point{faceUVToXYZ(int(c.face), u, v)}).Radians() +} + +var ( + poleMinLat = math.Asin(math.Sqrt(1.0/3)) - 0.5*dblEpsilon +) + +// RectBound returns the bounding rectangle of this cell. +func (c Cell) RectBound() Rect { + if c.level > 0 { + // Except for cells at level 0, the latitude and longitude extremes are + // attained at the vertices. Furthermore, the latitude range is + // determined by one pair of diagonally opposite vertices and the + // longitude range is determined by the other pair. + // + // We first determine which corner (i,j) of the cell has the largest + // absolute latitude. To maximize latitude, we want to find the point in + // the cell that has the largest absolute z-coordinate and the smallest + // absolute x- and y-coordinates. To do this we look at each coordinate + // (u and v), and determine whether we want to minimize or maximize that + // coordinate based on the axis direction and the cell's (u,v) quadrant. + u := c.uv.X.Lo + c.uv.X.Hi + v := c.uv.Y.Lo + c.uv.Y.Hi + var i, j int + if uAxis(int(c.face)).Z == 0 { + if u < 0 { + i = 1 + } + } else if u > 0 { + i = 1 + } + if vAxis(int(c.face)).Z == 0 { + if v < 0 { + j = 1 + } + } else if v > 0 { + j = 1 + } + lat := r1.IntervalFromPoint(c.latitude(i, j)).AddPoint(c.latitude(1-i, 1-j)) + lng := s1.EmptyInterval().AddPoint(c.longitude(i, 1-j)).AddPoint(c.longitude(1-i, j)) + + // We grow the bounds slightly to make sure that the bounding rectangle + // contains LatLngFromPoint(P) for any point P inside the loop L defined by the + // four *normalized* vertices. Note that normalization of a vector can + // change its direction by up to 0.5 * dblEpsilon radians, and it is not + // enough just to add Normalize calls to the code above because the + // latitude/longitude ranges are not necessarily determined by diagonally + // opposite vertex pairs after normalization. + // + // We would like to bound the amount by which the latitude/longitude of a + // contained point P can exceed the bounds computed above. In the case of + // longitude, the normalization error can change the direction of rounding + // leading to a maximum difference in longitude of 2 * dblEpsilon. In + // the case of latitude, the normalization error can shift the latitude by + // up to 0.5 * dblEpsilon and the other sources of error can cause the + // two latitudes to differ by up to another 1.5 * dblEpsilon, which also + // leads to a maximum difference of 2 * dblEpsilon. + return Rect{lat, lng}.expanded(LatLng{s1.Angle(2 * dblEpsilon), s1.Angle(2 * dblEpsilon)}).PolarClosure() + } + + // The 4 cells around the equator extend to +/-45 degrees latitude at the + // midpoints of their top and bottom edges. The two cells covering the + // poles extend down to +/-35.26 degrees at their vertices. The maximum + // error in this calculation is 0.5 * dblEpsilon. + var bound Rect + switch c.face { + case 0: + bound = Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{-math.Pi / 4, math.Pi / 4}} + case 1: + bound = Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{math.Pi / 4, 3 * math.Pi / 4}} + case 2: + bound = Rect{r1.Interval{poleMinLat, math.Pi / 2}, s1.FullInterval()} + case 3: + bound = Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{3 * math.Pi / 4, -3 * math.Pi / 4}} + case 4: + bound = Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{-3 * math.Pi / 4, -math.Pi / 4}} + default: + bound = Rect{r1.Interval{-math.Pi / 2, -poleMinLat}, s1.FullInterval()} + } + + // Finally, we expand the bound to account for the error when a point P is + // converted to an LatLng to test for containment. (The bound should be + // large enough so that it contains the computed LatLng of any contained + // point, not just the infinite-precision version.) We don't need to expand + // longitude because longitude is calculated via a single call to math.Atan2, + // which is guaranteed to be semi-monotonic. + return bound.expanded(LatLng{s1.Angle(dblEpsilon), s1.Angle(0)}) +} + +// CapBound returns the bounding cap of this cell. +func (c Cell) CapBound() Cap { + // We use the cell center in (u,v)-space as the cap axis. This vector is very close + // to GetCenter() and faster to compute. Neither one of these vectors yields the + // bounding cap with minimal surface area, but they are both pretty close. + cap := CapFromPoint(Point{faceUVToXYZ(int(c.face), c.uv.Center().X, c.uv.Center().Y).Normalize()}) + for k := 0; k < 4; k++ { + cap = cap.AddPoint(c.Vertex(k)) + } + return cap +} + +// ContainsPoint reports whether this cell contains the given point. Note that +// unlike Loop/Polygon, a Cell is considered to be a closed set. This means +// that a point on a Cell's edge or vertex belong to the Cell and the relevant +// adjacent Cells too. +// +// If you want every point to be contained by exactly one Cell, +// you will need to convert the Cell to a Loop. +func (c Cell) ContainsPoint(p Point) bool { + var uv r2.Point + var ok bool + if uv.X, uv.Y, ok = faceXYZToUV(int(c.face), p); !ok { + return false + } + + // Expand the (u,v) bound to ensure that + // + // CellFromPoint(p).ContainsPoint(p) + // + // is always true. To do this, we need to account for the error when + // converting from (u,v) coordinates to (s,t) coordinates. In the + // normal case the total error is at most dblEpsilon. + return c.uv.ExpandedByMargin(dblEpsilon).ContainsPoint(uv) +} + +// Encode encodes the Cell. +func (c Cell) Encode(w io.Writer) error { + e := &encoder{w: w} + c.encode(e) + return e.err +} + +func (c Cell) encode(e *encoder) { + c.id.encode(e) +} + +// Decode decodes the Cell. +func (c *Cell) Decode(r io.Reader) error { + d := &decoder{r: asByteReader(r)} + c.decode(d) + return d.err +} + +func (c *Cell) decode(d *decoder) { + c.id.decode(d) + *c = CellFromCellID(c.id) +} + +// vertexChordDist2 returns the squared chord distance from point P to the +// given corner vertex specified by the Hi or Lo values of each. +func (c Cell) vertexChordDist2(p Point, xHi, yHi bool) s1.ChordAngle { + x := c.uv.X.Lo + y := c.uv.Y.Lo + if xHi { + x = c.uv.X.Hi + } + if yHi { + y = c.uv.Y.Hi + } + + return ChordAngleBetweenPoints(p, PointFromCoords(x, y, 1)) +} + +// uEdgeIsClosest reports whether a point P is closer to the interior of the specified +// Cell edge (either the lower or upper edge of the Cell) or to the endpoints. +func (c Cell) uEdgeIsClosest(p Point, vHi bool) bool { + u0 := c.uv.X.Lo + u1 := c.uv.X.Hi + v := c.uv.Y.Lo + if vHi { + v = c.uv.Y.Hi + } + // These are the normals to the planes that are perpendicular to the edge + // and pass through one of its two endpoints. + dir0 := r3.Vector{v*v + 1, -u0 * v, -u0} + dir1 := r3.Vector{v*v + 1, -u1 * v, -u1} + return p.Dot(dir0) > 0 && p.Dot(dir1) < 0 +} + +// vEdgeIsClosest reports whether a point P is closer to the interior of the specified +// Cell edge (either the right or left edge of the Cell) or to the endpoints. +func (c Cell) vEdgeIsClosest(p Point, uHi bool) bool { + v0 := c.uv.Y.Lo + v1 := c.uv.Y.Hi + u := c.uv.X.Lo + if uHi { + u = c.uv.X.Hi + } + dir0 := r3.Vector{-u * v0, u*u + 1, -v0} + dir1 := r3.Vector{-u * v1, u*u + 1, -v1} + return p.Dot(dir0) > 0 && p.Dot(dir1) < 0 +} + +// edgeDistance reports the distance from a Point P to a given Cell edge. The point +// P is given by its dot product, and the uv edge by its normal in the +// given coordinate value. +func edgeDistance(ij, uv float64) s1.ChordAngle { + // Let P by the target point and let R be the closest point on the given + // edge AB. The desired distance PR can be expressed as PR^2 = PQ^2 + QR^2 + // where Q is the point P projected onto the plane through the great circle + // through AB. We can compute the distance PQ^2 perpendicular to the plane + // from "dirIJ" (the dot product of the target point P with the edge + // normal) and the squared length the edge normal (1 + uv**2). + pq2 := (ij * ij) / (1 + uv*uv) + + // We can compute the distance QR as (1 - OQ) where O is the sphere origin, + // and we can compute OQ^2 = 1 - PQ^2 using the Pythagorean theorem. + // (This calculation loses accuracy as angle POQ approaches Pi/2.) + qr := 1 - math.Sqrt(1-pq2) + return s1.ChordAngleFromSquaredLength(pq2 + qr*qr) +} + +// distanceInternal reports the distance from the given point to the interior of +// the cell if toInterior is true or to the boundary of the cell otherwise. +func (c Cell) distanceInternal(targetXYZ Point, toInterior bool) s1.ChordAngle { + // All calculations are done in the (u,v,w) coordinates of this cell's face. + target := faceXYZtoUVW(int(c.face), targetXYZ) + + // Compute dot products with all four upward or rightward-facing edge + // normals. dirIJ is the dot product for the edge corresponding to axis + // I, endpoint J. For example, dir01 is the right edge of the Cell + // (corresponding to the upper endpoint of the u-axis). + dir00 := target.X - target.Z*c.uv.X.Lo + dir01 := target.X - target.Z*c.uv.X.Hi + dir10 := target.Y - target.Z*c.uv.Y.Lo + dir11 := target.Y - target.Z*c.uv.Y.Hi + inside := true + if dir00 < 0 { + inside = false // Target is to the left of the cell + if c.vEdgeIsClosest(target, false) { + return edgeDistance(-dir00, c.uv.X.Lo) + } + } + if dir01 > 0 { + inside = false // Target is to the right of the cell + if c.vEdgeIsClosest(target, true) { + return edgeDistance(dir01, c.uv.X.Hi) + } + } + if dir10 < 0 { + inside = false // Target is below the cell + if c.uEdgeIsClosest(target, false) { + return edgeDistance(-dir10, c.uv.Y.Lo) + } + } + if dir11 > 0 { + inside = false // Target is above the cell + if c.uEdgeIsClosest(target, true) { + return edgeDistance(dir11, c.uv.Y.Hi) + } + } + if inside { + if toInterior { + return s1.ChordAngle(0) + } + // Although you might think of Cells as rectangles, they are actually + // arbitrary quadrilaterals after they are projected onto the sphere. + // Therefore the simplest approach is just to find the minimum distance to + // any of the four edges. + return minChordAngle(edgeDistance(-dir00, c.uv.X.Lo), + edgeDistance(dir01, c.uv.X.Hi), + edgeDistance(-dir10, c.uv.Y.Lo), + edgeDistance(dir11, c.uv.Y.Hi)) + } + + // Otherwise, the closest point is one of the four cell vertices. Note that + // it is *not* trivial to narrow down the candidates based on the edge sign + // tests above, because (1) the edges don't meet at right angles and (2) + // there are points on the far side of the sphere that are both above *and* + // below the cell, etc. + return minChordAngle(c.vertexChordDist2(target, false, false), + c.vertexChordDist2(target, true, false), + c.vertexChordDist2(target, false, true), + c.vertexChordDist2(target, true, true)) +} + +// Distance reports the distance from the cell to the given point. Returns zero if +// the point is inside the cell. +func (c Cell) Distance(target Point) s1.ChordAngle { + return c.distanceInternal(target, true) +} + +// MaxDistance reports the maximum distance from the cell (including its interior) to the +// given point. +func (c Cell) MaxDistance(target Point) s1.ChordAngle { + // First check the 4 cell vertices. If all are within the hemisphere + // centered around target, the max distance will be to one of these vertices. + targetUVW := faceXYZtoUVW(int(c.face), target) + maxDist := maxChordAngle(c.vertexChordDist2(targetUVW, false, false), + c.vertexChordDist2(targetUVW, true, false), + c.vertexChordDist2(targetUVW, false, true), + c.vertexChordDist2(targetUVW, true, true)) + + if maxDist <= s1.RightChordAngle { + return maxDist + } + + // Otherwise, find the minimum distance dMin to the antipodal point and the + // maximum distance will be pi - dMin. + return s1.StraightChordAngle - c.BoundaryDistance(Point{target.Mul(-1)}) +} + +// BoundaryDistance reports the distance from the cell boundary to the given point. +func (c Cell) BoundaryDistance(target Point) s1.ChordAngle { + return c.distanceInternal(target, false) +} + +// DistanceToEdge returns the minimum distance from the cell to the given edge AB. Returns +// zero if the edge intersects the cell interior. +func (c Cell) DistanceToEdge(a, b Point) s1.ChordAngle { + // Possible optimizations: + // - Currently the (cell vertex, edge endpoint) distances are computed + // twice each, and the length of AB is computed 4 times. + // - To fix this, refactor GetDistance(target) so that it skips calculating + // the distance to each cell vertex. Instead, compute the cell vertices + // and distances in this function, and add a low-level UpdateMinDistance + // that allows the XA, XB, and AB distances to be passed in. + // - It might also be more efficient to do all calculations in UVW-space, + // since this would involve transforming 2 points rather than 4. + + // First, check the minimum distance to the edge endpoints A and B. + // (This also detects whether either endpoint is inside the cell.) + minDist := minChordAngle(c.Distance(a), c.Distance(b)) + if minDist == 0 { + return minDist + } + + // Otherwise, check whether the edge crosses the cell boundary. + crosser := NewChainEdgeCrosser(a, b, c.Vertex(3)) + for i := 0; i < 4; i++ { + if crosser.ChainCrossingSign(c.Vertex(i)) != DoNotCross { + return 0 + } + } + + // Finally, check whether the minimum distance occurs between a cell vertex + // and the interior of the edge AB. (Some of this work is redundant, since + // it also checks the distance to the endpoints A and B again.) + // + // Note that we don't need to check the distance from the interior of AB to + // the interior of a cell edge, because the only way that this distance can + // be minimal is if the two edges cross (already checked above). + for i := 0; i < 4; i++ { + minDist, _ = UpdateMinDistance(c.Vertex(i), a, b, minDist) + } + return minDist +} + +// MaxDistanceToEdge returns the maximum distance from the cell (including its interior) +// to the given edge AB. +func (c Cell) MaxDistanceToEdge(a, b Point) s1.ChordAngle { + // If the maximum distance from both endpoints to the cell is less than π/2 + // then the maximum distance from the edge to the cell is the maximum of the + // two endpoint distances. + maxDist := maxChordAngle(c.MaxDistance(a), c.MaxDistance(b)) + if maxDist <= s1.RightChordAngle { + return maxDist + } + + return s1.StraightChordAngle - c.DistanceToEdge(Point{a.Mul(-1)}, Point{b.Mul(-1)}) +} + +// DistanceToCell returns the minimum distance from this cell to the given cell. +// It returns zero if one cell contains the other. +func (c Cell) DistanceToCell(target Cell) s1.ChordAngle { + // If the cells intersect, the distance is zero. We use the (u,v) ranges + // rather than CellID intersects so that cells that share a partial edge or + // corner are considered to intersect. + if c.face == target.face && c.uv.Intersects(target.uv) { + return 0 + } + + // Otherwise, the minimum distance always occurs between a vertex of one + // cell and an edge of the other cell (including the edge endpoints). This + // represents a total of 32 possible (vertex, edge) pairs. + // + // TODO(roberts): This could be optimized to be at least 5x faster by pruning + // the set of possible closest vertex/edge pairs using the faces and (u,v) + // ranges of both cells. + var va, vb [4]Point + for i := 0; i < 4; i++ { + va[i] = c.Vertex(i) + vb[i] = target.Vertex(i) + } + minDist := s1.InfChordAngle() + for i := 0; i < 4; i++ { + for j := 0; j < 4; j++ { + minDist, _ = UpdateMinDistance(va[i], vb[j], vb[(j+1)&3], minDist) + minDist, _ = UpdateMinDistance(vb[i], va[j], va[(j+1)&3], minDist) + } + } + return minDist +} + +// MaxDistanceToCell returns the maximum distance from the cell (including its +// interior) to the given target cell. +func (c Cell) MaxDistanceToCell(target Cell) s1.ChordAngle { + // Need to check the antipodal target for intersection with the cell. If it + // intersects, the distance is the straight ChordAngle. + // antipodalUV is the transpose of the original UV, interpreted within the opposite face. + antipodalUV := r2.Rect{target.uv.Y, target.uv.X} + if int(c.face) == oppositeFace(int(target.face)) && c.uv.Intersects(antipodalUV) { + return s1.StraightChordAngle + } + + // Otherwise, the maximum distance always occurs between a vertex of one + // cell and an edge of the other cell (including the edge endpoints). This + // represents a total of 32 possible (vertex, edge) pairs. + // + // TODO(roberts): When the maximum distance is at most π/2, the maximum is + // always attained between a pair of vertices, and this could be made much + // faster by testing each vertex pair once rather than the current 4 times. + var va, vb [4]Point + for i := 0; i < 4; i++ { + va[i] = c.Vertex(i) + vb[i] = target.Vertex(i) + } + maxDist := s1.NegativeChordAngle + for i := 0; i < 4; i++ { + for j := 0; j < 4; j++ { + maxDist, _ = UpdateMaxDistance(va[i], vb[j], vb[(j+1)&3], maxDist) + maxDist, _ = UpdateMaxDistance(vb[i], va[j], va[(j+1)&3], maxDist) + } + } + return maxDist +} |