gtsocial-umbx

cell.go (24880B)

---

 // 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
 }