staticmap/vendor/github.com/golang/geo/s2/cellid.go

/*
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 (
	"bytes"
	"fmt"
	"math"
	"strconv"
	"strings"

	"github.com/golang/geo/r1"
	"github.com/golang/geo/r2"
	"github.com/golang/geo/r3"
	"github.com/golang/geo/s1"
)

// CellID uniquely identifies a cell in the S2 cell decomposition.
// The most significant 3 bits encode the face number (0-5). The
// remaining 61 bits encode the position of the center of this cell
// along the Hilbert curve on that face. The zero value and the value
// (1<<64)-1 are invalid cell IDs. The first compares less than any
// valid cell ID, the second as greater than any valid cell ID.
//
// Sequentially increasing cell IDs follow a continuous space-filling curve
// over the entire sphere. They have the following properties:
//
//  - The ID of a cell at level k consists of a 3-bit face number followed
//    by k bit pairs that recursively select one of the four children of
//    each cell. The next bit is always 1, and all other bits are 0.
//    Therefore, the level of a cell is determined by the position of its
//    lowest-numbered bit that is turned on (for a cell at level k, this
//    position is 2 * (maxLevel - k)).
//
//  - The ID of a parent cell is at the midpoint of the range of IDs spanned
//    by its children (or by its descendants at any level).
//
// Leaf cells are often used to represent points on the unit sphere, and
// this type provides methods for converting directly between these two
// representations. For cells that represent 2D regions rather than
// discrete point, it is better to use Cells.
type CellID uint64

// TODO(dsymonds): Some of these constants should probably be exported.
const (
	faceBits = 3
	numFaces = 6
	maxLevel = 30
	// The extra position bit (61 rather than 60) lets us encode each cell as its
	// Hilbert curve position at the cell center (which is halfway along the
	// portion of the Hilbert curve that fills that cell).
	posBits    = 2*maxLevel + 1
	maxSize    = 1 << maxLevel
	wrapOffset = uint64(numFaces) << posBits
)

// CellIDFromFacePosLevel returns a cell given its face in the range
// [0,5], the 61-bit Hilbert curve position pos within that face, and
// the level in the range [0,maxLevel]. The position in the cell ID
// will be truncated to correspond to the Hilbert curve position at
// the center of the returned cell.
func CellIDFromFacePosLevel(face int, pos uint64, level int) CellID {
	return CellID(uint64(face)<<posBits + pos | 1).Parent(level)
}

// CellIDFromFace returns the cell corresponding to a given S2 cube face.
func CellIDFromFace(face int) CellID {
	return CellID((uint64(face) << posBits) + lsbForLevel(0))
}

// CellIDFromLatLng returns the leaf cell containing ll.
func CellIDFromLatLng(ll LatLng) CellID {
	return cellIDFromPoint(PointFromLatLng(ll))
}

// CellIDFromToken returns a cell given a hex-encoded string of its uint64 ID.
func CellIDFromToken(s string) CellID {
	if len(s) > 16 {
		return CellID(0)
	}
	n, err := strconv.ParseUint(s, 16, 64)
	if err != nil {
		return CellID(0)
	}
	// Equivalent to right-padding string with zeros to 16 characters.
	if len(s) < 16 {
		n = n << (4 * uint(16-len(s)))
	}
	return CellID(n)
}

// ToToken returns a hex-encoded string of the uint64 cell id, with leading
// zeros included but trailing zeros stripped.
func (ci CellID) ToToken() string {
	s := strings.TrimRight(fmt.Sprintf("%016x", uint64(ci)), "0")
	if len(s) == 0 {
		return "X"
	}
	return s
}

// IsValid reports whether ci represents a valid cell.
func (ci CellID) IsValid() bool {
	return ci.Face() < numFaces && (ci.lsb()&0x1555555555555555 != 0)
}

// Face returns the cube face for this cell ID, in the range [0,5].
func (ci CellID) Face() int { return int(uint64(ci) >> posBits) }

// Pos returns the position along the Hilbert curve of this cell ID, in the range [0,2^posBits-1].
func (ci CellID) Pos() uint64 { return uint64(ci) & (^uint64(0) >> faceBits) }

// Level returns the subdivision level of this cell ID, in the range [0, maxLevel].
func (ci CellID) Level() int {
	return maxLevel - findLSBSetNonZero64(uint64(ci))>>1
}

// IsLeaf returns whether this cell ID is at the deepest level;
// that is, the level at which the cells are smallest.
func (ci CellID) IsLeaf() bool { return uint64(ci)&1 != 0 }

// ChildPosition returns the child position (0..3) of this cell's
// ancestor at the given level, relative to its parent.  The argument
// should be in the range 1..kMaxLevel.  For example,
// ChildPosition(1) returns the position of this cell's level-1
// ancestor within its top-level face cell.
func (ci CellID) ChildPosition(level int) int {
	return int(uint64(ci)>>uint64(2*(maxLevel-level)+1)) & 3
}

// lsbForLevel returns the lowest-numbered bit that is on for cells at the given level.
func lsbForLevel(level int) uint64 { return 1 << uint64(2*(maxLevel-level)) }

// Parent returns the cell at the given level, which must be no greater than the current level.
func (ci CellID) Parent(level int) CellID {
	lsb := lsbForLevel(level)
	return CellID((uint64(ci) & -lsb) | lsb)
}

// immediateParent is cheaper than Parent, but assumes !ci.isFace().
func (ci CellID) immediateParent() CellID {
	nlsb := CellID(ci.lsb() << 2)
	return (ci & -nlsb) | nlsb
}

// isFace returns whether this is a top-level (face) cell.
func (ci CellID) isFace() bool { return uint64(ci)&(lsbForLevel(0)-1) == 0 }

// lsb returns the least significant bit that is set.
func (ci CellID) lsb() uint64 { return uint64(ci) & -uint64(ci) }

// Children returns the four immediate children of this cell.
// If ci is a leaf cell, it returns four identical cells that are not the children.
func (ci CellID) Children() [4]CellID {
	var ch [4]CellID
	lsb := CellID(ci.lsb())
	ch[0] = ci - lsb + lsb>>2
	lsb >>= 1
	ch[1] = ch[0] + lsb
	ch[2] = ch[1] + lsb
	ch[3] = ch[2] + lsb
	return ch
}

func sizeIJ(level int) int {
	return 1 << uint(maxLevel-level)
}

// EdgeNeighbors returns the four cells that are adjacent across the cell's four edges.
// Edges 0, 1, 2, 3 are in the down, right, up, left directions in the face space.
// All neighbors are guaranteed to be distinct.
func (ci CellID) EdgeNeighbors() [4]CellID {
	level := ci.Level()
	size := sizeIJ(level)
	f, i, j, _ := ci.faceIJOrientation()
	return [4]CellID{
		cellIDFromFaceIJWrap(f, i, j-size).Parent(level),
		cellIDFromFaceIJWrap(f, i+size, j).Parent(level),
		cellIDFromFaceIJWrap(f, i, j+size).Parent(level),
		cellIDFromFaceIJWrap(f, i-size, j).Parent(level),
	}
}

// VertexNeighbors returns the neighboring cellIDs with vertex closest to this cell at the given level.
// (Normally there are four neighbors, but the closest vertex may only have three neighbors if it is one of
// the 8 cube vertices.)
func (ci CellID) VertexNeighbors(level int) []CellID {
	halfSize := sizeIJ(level + 1)
	size := halfSize << 1
	f, i, j, _ := ci.faceIJOrientation()

	var isame, jsame bool
	var ioffset, joffset int
	if i&halfSize != 0 {
		ioffset = size
		isame = (i + size) < maxSize
	} else {
		ioffset = -size
		isame = (i - size) >= 0
	}
	if j&halfSize != 0 {
		joffset = size
		jsame = (j + size) < maxSize
	} else {
		joffset = -size
		jsame = (j - size) >= 0
	}

	results := []CellID{
		ci.Parent(level),
		cellIDFromFaceIJSame(f, i+ioffset, j, isame).Parent(level),
		cellIDFromFaceIJSame(f, i, j+joffset, jsame).Parent(level),
	}

	if isame || jsame {
		results = append(results, cellIDFromFaceIJSame(f, i+ioffset, j+joffset, isame && jsame).Parent(level))
	}

	return results
}

// AllNeighbors returns all neighbors of this cell at the given level. Two
// cells X and Y are neighbors if their boundaries intersect but their
// interiors do not. In particular, two cells that intersect at a single
// point are neighbors. Note that for cells adjacent to a face vertex, the
// same neighbor may be returned more than once. There could be up to eight
// neighbors including the diagonal ones that share the vertex.
//
// This requires level >= ci.Level().
func (ci CellID) AllNeighbors(level int) []CellID {
	var neighbors []CellID

	face, i, j, _ := ci.faceIJOrientation()

	// Find the coordinates of the lower left-hand leaf cell. We need to
	// normalize (i,j) to a known position within the cell because level
	// may be larger than this cell's level.
	size := sizeIJ(ci.Level())
	i &= -size
	j &= -size

	nbrSize := sizeIJ(level)

	// We compute the top-bottom, left-right, and diagonal neighbors in one
	// pass. The loop test is at the end of the loop to avoid 32-bit overflow.
	for k := -nbrSize; ; k += nbrSize {
		var sameFace bool
		if k < 0 {
			sameFace = (j+k >= 0)
		} else if k >= size {
			sameFace = (j+k < maxSize)
		} else {
			sameFace = true
			// Top and bottom neighbors.
			neighbors = append(neighbors, cellIDFromFaceIJSame(face, i+k, j-nbrSize,
				j-size >= 0).Parent(level))
			neighbors = append(neighbors, cellIDFromFaceIJSame(face, i+k, j+size,
				j+size < maxSize).Parent(level))
		}

		// Left, right, and diagonal neighbors.
		neighbors = append(neighbors, cellIDFromFaceIJSame(face, i-nbrSize, j+k,
			sameFace && i-size >= 0).Parent(level))
		neighbors = append(neighbors, cellIDFromFaceIJSame(face, i+size, j+k,
			sameFace && i+size < maxSize).Parent(level))

		if k >= size {
			break
		}
	}

	return neighbors
}

// RangeMin returns the minimum CellID that is contained within this cell.
func (ci CellID) RangeMin() CellID { return CellID(uint64(ci) - (ci.lsb() - 1)) }

// RangeMax returns the maximum CellID that is contained within this cell.
func (ci CellID) RangeMax() CellID { return CellID(uint64(ci) + (ci.lsb() - 1)) }

// Contains returns true iff the CellID contains oci.
func (ci CellID) Contains(oci CellID) bool {
	return uint64(ci.RangeMin()) <= uint64(oci) && uint64(oci) <= uint64(ci.RangeMax())
}

// Intersects returns true iff the CellID intersects oci.
func (ci CellID) Intersects(oci CellID) bool {
	return uint64(oci.RangeMin()) <= uint64(ci.RangeMax()) && uint64(oci.RangeMax()) >= uint64(ci.RangeMin())
}

// String returns the string representation of the cell ID in the form "1/3210".
func (ci CellID) String() string {
	if !ci.IsValid() {
		return "Invalid: " + strconv.FormatInt(int64(ci), 16)
	}
	var b bytes.Buffer
	b.WriteByte("012345"[ci.Face()]) // values > 5 will have been picked off by !IsValid above
	b.WriteByte('/')
	for level := 1; level <= ci.Level(); level++ {
		b.WriteByte("0123"[ci.ChildPosition(level)])
	}
	return b.String()
}

// Point returns the center of the s2 cell on the sphere as a Point.
// The maximum directional error in Point (compared to the exact
// mathematical result) is 1.5 * dblEpsilon radians, and the maximum length
// error is 2 * dblEpsilon (the same as Normalize).
func (ci CellID) Point() Point { return Point{ci.rawPoint().Normalize()} }

// LatLng returns the center of the s2 cell on the sphere as a LatLng.
func (ci CellID) LatLng() LatLng { return LatLngFromPoint(Point{ci.rawPoint()}) }

// ChildBegin returns the first child in a traversal of the children of this cell, in Hilbert curve order.
//
//    for ci := c.ChildBegin(); ci != c.ChildEnd(); ci = ci.Next() {
//        ...
//    }
func (ci CellID) ChildBegin() CellID {
	ol := ci.lsb()
	return CellID(uint64(ci) - ol + ol>>2)
}

// ChildBeginAtLevel returns the first cell in a traversal of children a given level deeper than this cell, in
// Hilbert curve order. The given level must be no smaller than the cell's level.
// See ChildBegin for example use.
func (ci CellID) ChildBeginAtLevel(level int) CellID {
	return CellID(uint64(ci) - ci.lsb() + lsbForLevel(level))
}

// ChildEnd returns the first cell after a traversal of the children of this cell in Hilbert curve order.
// The returned cell may be invalid.
func (ci CellID) ChildEnd() CellID {
	ol := ci.lsb()
	return CellID(uint64(ci) + ol + ol>>2)
}

// ChildEndAtLevel returns the first cell after the last child in a traversal of children a given level deeper
// than this cell, in Hilbert curve order.
// The given level must be no smaller than the cell's level.
// The returned cell may be invalid.
func (ci CellID) ChildEndAtLevel(level int) CellID {
	return CellID(uint64(ci) + ci.lsb() + lsbForLevel(level))
}

// Next returns the next cell along the Hilbert curve.
// This is expected to be used with ChildBegin and ChildEnd,
// or ChildBeginAtLevel and ChildEndAtLevel.
func (ci CellID) Next() CellID {
	return CellID(uint64(ci) + ci.lsb()<<1)
}

// Prev returns the previous cell along the Hilbert curve.
func (ci CellID) Prev() CellID {
	return CellID(uint64(ci) - ci.lsb()<<1)
}

// NextWrap returns the next cell along the Hilbert curve, wrapping from last to
// first as necessary. This should not be used with ChildBegin and ChildEnd.
func (ci CellID) NextWrap() CellID {
	n := ci.Next()
	if uint64(n) < wrapOffset {
		return n
	}
	return CellID(uint64(n) - wrapOffset)
}

// PrevWrap returns the previous cell along the Hilbert curve, wrapping around from
// first to last as necessary. This should not be used with ChildBegin and ChildEnd.
func (ci CellID) PrevWrap() CellID {
	p := ci.Prev()
	if uint64(p) < wrapOffset {
		return p
	}
	return CellID(uint64(p) + wrapOffset)
}

// AdvanceWrap advances or retreats the indicated number of steps along the
// Hilbert curve at the current level and returns the new position. The
// position wraps between the first and last faces as necessary.
func (ci CellID) AdvanceWrap(steps int64) CellID {
	if steps == 0 {
		return ci
	}

	// We clamp the number of steps if necessary to ensure that we do not
	// advance past the End() or before the Begin() of this level.
	shift := uint(2*(maxLevel-ci.Level()) + 1)
	if steps < 0 {
		if min := -int64(uint64(ci) >> shift); steps < min {
			wrap := int64(wrapOffset >> shift)
			steps %= wrap
			if steps < min {
				steps += wrap
			}
		}
	} else {
		// Unlike Advance(), we don't want to return End(level).
		if max := int64((wrapOffset - uint64(ci)) >> shift); steps > max {
			wrap := int64(wrapOffset >> shift)
			steps %= wrap
			if steps > max {
				steps -= wrap
			}
		}
	}

	// If steps is negative, then shifting it left has undefined behavior.
	// Cast to uint64 for a 2's complement answer.
	return CellID(uint64(ci) + (uint64(steps) << shift))
}

// TODO: the methods below are not exported yet.  Settle on the entire API design
// before doing this.  Do we want to mirror the C++ one as closely as possible?

// distanceFromBegin returns the number of steps that this cell is from the first
// node in the S2 heirarchy at our level. (i.e., FromFace(0).ChildBeginAtLevel(ci.Level())).
// The return value is always non-negative.
func (ci CellID) distanceFromBegin() int64 {
	return int64(ci >> uint64(2*(maxLevel-ci.Level())+1))
}

// rawPoint returns an unnormalized r3 vector from the origin through the center
// of the s2 cell on the sphere.
func (ci CellID) rawPoint() r3.Vector {
	face, si, ti := ci.faceSiTi()
	return faceUVToXYZ(face, stToUV((0.5/maxSize)*float64(si)), stToUV((0.5/maxSize)*float64(ti)))
}

// faceSiTi returns the Face/Si/Ti coordinates of the center of the cell.
func (ci CellID) faceSiTi() (face, si, ti int) {
	face, i, j, _ := ci.faceIJOrientation()
	delta := 0
	if ci.IsLeaf() {
		delta = 1
	} else {
		if (i^(int(ci)>>2))&1 != 0 {
			delta = 2
		}
	}
	return face, 2*i + delta, 2*j + delta
}

// faceIJOrientation uses the global lookupIJ table to unfiddle the bits of ci.
func (ci CellID) faceIJOrientation() (f, i, j, orientation int) {
	f = ci.Face()
	orientation = f & swapMask
	nbits := maxLevel - 7*lookupBits // first iteration

	for k := 7; k >= 0; k-- {
		orientation += (int(uint64(ci)>>uint64(k*2*lookupBits+1)) & ((1 << uint((2 * nbits))) - 1)) << 2
		orientation = lookupIJ[orientation]
		i += (orientation >> (lookupBits + 2)) << uint(k*lookupBits)
		j += ((orientation >> 2) & ((1 << lookupBits) - 1)) << uint(k*lookupBits)
		orientation &= (swapMask | invertMask)
		nbits = lookupBits // following iterations
	}

	if ci.lsb()&0x1111111111111110 != 0 {
		orientation ^= swapMask
	}

	return
}

// cellIDFromFaceIJ returns a leaf cell given its cube face (range 0..5) and IJ coordinates.
func cellIDFromFaceIJ(f, i, j int) CellID {
	// Note that this value gets shifted one bit to the left at the end
	// of the function.
	n := uint64(f) << (posBits - 1)
	// Alternating faces have opposite Hilbert curve orientations; this
	// is necessary in order for all faces to have a right-handed
	// coordinate system.
	bits := f & swapMask
	// Each iteration maps 4 bits of "i" and "j" into 8 bits of the Hilbert
	// curve position.  The lookup table transforms a 10-bit key of the form
	// "iiiijjjjoo" to a 10-bit value of the form "ppppppppoo", where the
	// letters [ijpo] denote bits of "i", "j", Hilbert curve position, and
	// Hilbert curve orientation respectively.
	for k := 7; k >= 0; k-- {
		mask := (1 << lookupBits) - 1
		bits += int((i>>uint(k*lookupBits))&mask) << (lookupBits + 2)
		bits += int((j>>uint(k*lookupBits))&mask) << 2
		bits = lookupPos[bits]
		n |= uint64(bits>>2) << (uint(k) * 2 * lookupBits)
		bits &= (swapMask | invertMask)
	}
	return CellID(n*2 + 1)
}

func cellIDFromFaceIJWrap(f, i, j int) CellID {
	// Convert i and j to the coordinates of a leaf cell just beyond the
	// boundary of this face.  This prevents 32-bit overflow in the case
	// of finding the neighbors of a face cell.
	i = clamp(i, -1, maxSize)
	j = clamp(j, -1, maxSize)

	// We want to wrap these coordinates onto the appropriate adjacent face.
	// The easiest way to do this is to convert the (i,j) coordinates to (x,y,z)
	// (which yields a point outside the normal face boundary), and then call
	// xyzToFaceUV to project back onto the correct face.
	//
	// The code below converts (i,j) to (si,ti), and then (si,ti) to (u,v) using
	// the linear projection (u=2*s-1 and v=2*t-1).  (The code further below
	// converts back using the inverse projection, s=0.5*(u+1) and t=0.5*(v+1).
	// Any projection would work here, so we use the simplest.)  We also clamp
	// the (u,v) coordinates so that the point is barely outside the
	// [-1,1]x[-1,1] face rectangle, since otherwise the reprojection step
	// (which divides by the new z coordinate) might change the other
	// coordinates enough so that we end up in the wrong leaf cell.
	const scale = 1.0 / maxSize
	limit := math.Nextafter(1, 2)
	u := math.Max(-limit, math.Min(limit, scale*float64((i<<1)+1-maxSize)))
	v := math.Max(-limit, math.Min(limit, scale*float64((j<<1)+1-maxSize)))

	// Find the leaf cell coordinates on the adjacent face, and convert
	// them to a cell id at the appropriate level.
	f, u, v = xyzToFaceUV(faceUVToXYZ(f, u, v))
	return cellIDFromFaceIJ(f, stToIJ(0.5*(u+1)), stToIJ(0.5*(v+1)))
}

func cellIDFromFaceIJSame(f, i, j int, sameFace bool) CellID {
	if sameFace {
		return cellIDFromFaceIJ(f, i, j)
	}
	return cellIDFromFaceIJWrap(f, i, j)
}

// clamp returns number closest to x within the range min..max.
func clamp(x, min, max int) int {
	if x < min {
		return min
	}
	if x > max {
		return max
	}
	return x
}

// ijToSTMin converts the i- or j-index of a leaf cell to the minimum corresponding
// s- or t-value contained by that cell. The argument must be in the range
// [0..2**30], i.e. up to one position beyond the normal range of valid leaf
// cell indices.
func ijToSTMin(i int) float64 {
	return float64(i) / float64(maxSize)
}

// stToIJ converts value in ST coordinates to a value in IJ coordinates.
func stToIJ(s float64) int {
	return clamp(int(math.Floor(maxSize*s)), 0, maxSize-1)
}

// cellIDFromPoint returns a leaf cell containing point p. Usually there is
// exactly one such cell, but for points along the edge of a cell, any
// adjacent cell may be (deterministically) chosen. This is because
// s2.CellIDs are considered to be closed sets. The returned cell will
// always contain the given point, i.e.
//
//   CellFromPoint(p).ContainsPoint(p)
//
// is always true.
func cellIDFromPoint(p Point) CellID {
	f, u, v := xyzToFaceUV(r3.Vector{p.X, p.Y, p.Z})
	i := stToIJ(uvToST(u))
	j := stToIJ(uvToST(v))
	return cellIDFromFaceIJ(f, i, j)
}

// ijLevelToBoundUV returns the bounds in (u,v)-space for the cell at the given
// level containing the leaf cell with the given (i,j)-coordinates.
func ijLevelToBoundUV(i, j, level int) r2.Rect {
	cellSize := sizeIJ(level)
	xLo := i & -cellSize
	yLo := j & -cellSize

	return r2.Rect{
		X: r1.Interval{
			Lo: stToUV(ijToSTMin(xLo)),
			Hi: stToUV(ijToSTMin(xLo + cellSize)),
		},
		Y: r1.Interval{
			Lo: stToUV(ijToSTMin(yLo)),
			Hi: stToUV(ijToSTMin(yLo + cellSize)),
		},
	}
}

// Constants related to the bit mangling in the Cell ID.
const (
	lookupBits = 4
	swapMask   = 0x01
	invertMask = 0x02
)

var (
	ijToPos = [4][4]int{
		{0, 1, 3, 2}, // canonical order
		{0, 3, 1, 2}, // axes swapped
		{2, 3, 1, 0}, // bits inverted
		{2, 1, 3, 0}, // swapped & inverted
	}
	posToIJ = [4][4]int{
		{0, 1, 3, 2}, // canonical order:    (0,0), (0,1), (1,1), (1,0)
		{0, 2, 3, 1}, // axes swapped:       (0,0), (1,0), (1,1), (0,1)
		{3, 2, 0, 1}, // bits inverted:      (1,1), (1,0), (0,0), (0,1)
		{3, 1, 0, 2}, // swapped & inverted: (1,1), (0,1), (0,0), (1,0)
	}
	posToOrientation = [4]int{swapMask, 0, 0, invertMask | swapMask}
	lookupIJ         [1 << (2*lookupBits + 2)]int
	lookupPos        [1 << (2*lookupBits + 2)]int
)

func init() {
	initLookupCell(0, 0, 0, 0, 0, 0)
	initLookupCell(0, 0, 0, swapMask, 0, swapMask)
	initLookupCell(0, 0, 0, invertMask, 0, invertMask)
	initLookupCell(0, 0, 0, swapMask|invertMask, 0, swapMask|invertMask)
}

// initLookupCell initializes the lookupIJ table at init time.
func initLookupCell(level, i, j, origOrientation, pos, orientation int) {
	if level == lookupBits {
		ij := (i << lookupBits) + j
		lookupPos[(ij<<2)+origOrientation] = (pos << 2) + orientation
		lookupIJ[(pos<<2)+origOrientation] = (ij << 2) + orientation
		return
	}

	level++
	i <<= 1
	j <<= 1
	pos <<= 2
	r := posToIJ[orientation]
	initLookupCell(level, i+(r[0]>>1), j+(r[0]&1), origOrientation, pos, orientation^posToOrientation[0])
	initLookupCell(level, i+(r[1]>>1), j+(r[1]&1), origOrientation, pos+1, orientation^posToOrientation[1])
	initLookupCell(level, i+(r[2]>>1), j+(r[2]&1), origOrientation, pos+2, orientation^posToOrientation[2])
	initLookupCell(level, i+(r[3]>>1), j+(r[3]&1), origOrientation, pos+3, orientation^posToOrientation[3])
}

// CommonAncestorLevel returns the level of the common ancestor of the two S2 CellIDs.
func (ci CellID) CommonAncestorLevel(other CellID) (level int, ok bool) {
	bits := uint64(ci ^ other)
	if bits < ci.lsb() {
		bits = ci.lsb()
	}
	if bits < other.lsb() {
		bits = other.lsb()
	}

	msbPos := findMSBSetNonZero64(bits)
	if msbPos > 60 {
		return 0, false
	}
	return (60 - msbPos) >> 1, true
}

// findMSBSetNonZero64 returns the index (between 0 and 63) of the most
// significant set bit. Passing zero to this function has undefined behavior.
func findMSBSetNonZero64(bits uint64) int {
	val := []uint64{0x2, 0xC, 0xF0, 0xFF00, 0xFFFF0000, 0xFFFFFFFF00000000}
	shift := []uint64{1, 2, 4, 8, 16, 32}
	var msbPos uint64
	for i := 5; i >= 0; i-- {
		if bits&val[i] != 0 {
			bits >>= shift[i]
			msbPos |= shift[i]
		}
	}
	return int(msbPos)
}

const deBruijn64 = 0x03f79d71b4ca8b09
const digitMask = uint64(1<<64 - 1)

var deBruijn64Lookup = []byte{
	0, 1, 56, 2, 57, 49, 28, 3, 61, 58, 42, 50, 38, 29, 17, 4,
	62, 47, 59, 36, 45, 43, 51, 22, 53, 39, 33, 30, 24, 18, 12, 5,
	63, 55, 48, 27, 60, 41, 37, 16, 46, 35, 44, 21, 52, 32, 23, 11,
	54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6,
}

// findLSBSetNonZero64 returns the index (between 0 and 63) of the least
// significant set bit. Passing zero to this function has undefined behavior.
//
// This code comes from trailingZeroBits in https://golang.org/src/math/big/nat.go
// which references (Knuth, volume 4, section 7.3.1).
func findLSBSetNonZero64(bits uint64) int {
	return int(deBruijn64Lookup[((bits&-bits)*(deBruijn64&digitMask))>>58])
}

// Advance advances or retreats the indicated number of steps along the
// Hilbert curve at the current level, and returns the new position. The
// position is never advanced past End() or before Begin().
func (ci CellID) Advance(steps int64) CellID {
	if steps == 0 {
		return ci
	}

	// We clamp the number of steps if necessary to ensure that we do not
	// advance past the End() or before the Begin() of this level. Note that
	// minSteps and maxSteps always fit in a signed 64-bit integer.
	stepShift := uint(2*(maxLevel-ci.Level()) + 1)
	if steps < 0 {
		minSteps := -int64(uint64(ci) >> stepShift)
		if steps < minSteps {
			steps = minSteps
		}
	} else {
		maxSteps := int64((wrapOffset + ci.lsb() - uint64(ci)) >> stepShift)
		if steps > maxSteps {
			steps = maxSteps
		}
	}
	return ci + CellID(steps)<<stepShift
}

// centerST return the center of the CellID in (s,t)-space.
func (ci CellID) centerST() r2.Point {
	_, si, ti := ci.faceSiTi()
	return r2.Point{siTiToST(uint64(si)), siTiToST(uint64(ti))}
}

// sizeST returns the edge length of this CellID in (s,t)-space at the given level.
func (ci CellID) sizeST(level int) float64 {
	return ijToSTMin(sizeIJ(level))
}

// boundST returns the bound of this CellID in (s,t)-space.
func (ci CellID) boundST() r2.Rect {
	s := ci.sizeST(ci.Level())
	return r2.RectFromCenterSize(ci.centerST(), r2.Point{s, s})
}

// centerUV returns the center of this CellID in (u,v)-space. Note that
// the center of the cell is defined as the point at which it is recursively
// subdivided into four children; in general, it is not at the midpoint of
// the (u,v) rectangle covered by the cell.
func (ci CellID) centerUV() r2.Point {
	_, si, ti := ci.faceSiTi()
	return r2.Point{stToUV(siTiToST(uint64(si))), stToUV(siTiToST(uint64(ti)))}
}

// boundUV returns the bound of this CellID in (u,v)-space.
func (ci CellID) boundUV() r2.Rect {
	_, i, j, _ := ci.faceIJOrientation()
	return ijLevelToBoundUV(i, j, ci.Level())
}

// expandEndpoint returns a new u-coordinate u' such that the distance from the
// line u=u' to the given edge (u,v0)-(u,v1) is exactly the given distance
// (which is specified as the sine of the angle corresponding to the distance).
func expandEndpoint(u, maxV, sinDist float64) float64 {
	// This is based on solving a spherical right triangle, similar to the
	// calculation in Cap.RectBound.
	// Given an edge of the form (u,v0)-(u,v1), let maxV = max(abs(v0), abs(v1)).
	sinUShift := sinDist * math.Sqrt((1+u*u+maxV*maxV)/(1+u*u))
	cosUShift := math.Sqrt(1 - sinUShift*sinUShift)
	// The following is an expansion of tan(atan(u) + asin(sinUShift)).
	return (cosUShift*u + sinUShift) / (cosUShift - sinUShift*u)
}

// expandedByDistanceUV returns a rectangle expanded in (u,v)-space so that it
// contains all points within the given distance of the boundary, and return the
// smallest such rectangle. If the distance is negative, then instead shrink this
// rectangle so that it excludes all points within the given absolute distance
// of the boundary.
//
// Distances are measured *on the sphere*, not in (u,v)-space. For example,
// you can use this method to expand the (u,v)-bound of an CellID so that
// it contains all points within 5km of the original cell. You can then
// test whether a point lies within the expanded bounds like this:
//
//   if u, v, ok := faceXYZtoUV(face, point); ok && bound.ContainsPoint(r2.Point{u,v}) { ... }
//
// Limitations:
//
//  - Because the rectangle is drawn on one of the six cube-face planes
//    (i.e., {x,y,z} = +/-1), it can cover at most one hemisphere. This
//    limits the maximum amount that a rectangle can be expanded. For
//    example, CellID bounds can be expanded safely by at most 45 degrees
//    (about 5000 km on the Earth's surface).
//
//  - The implementation is not exact for negative distances. The resulting
//    rectangle will exclude all points within the given distance of the
//    boundary but may be slightly smaller than necessary.
func expandedByDistanceUV(uv r2.Rect, distance s1.Angle) r2.Rect {
	// Expand each of the four sides of the rectangle just enough to include all
	// points within the given distance of that side. (The rectangle may be
	// expanded by a different amount in (u,v)-space on each side.)
	maxU := math.Max(math.Abs(uv.X.Lo), math.Abs(uv.X.Hi))
	maxV := math.Max(math.Abs(uv.Y.Lo), math.Abs(uv.Y.Hi))
	sinDist := math.Sin(float64(distance))
	return r2.Rect{
		X: r1.Interval{expandEndpoint(uv.X.Lo, maxV, -sinDist),
			expandEndpoint(uv.X.Hi, maxV, sinDist)},
		Y: r1.Interval{expandEndpoint(uv.Y.Lo, maxU, -sinDist),
			expandEndpoint(uv.Y.Hi, maxU, sinDist)}}
}

// MaxTile returns the largest cell with the same RangeMin such that
// RangeMax < limit.RangeMin. It returns limit if no such cell exists.
// This method can be used to generate a small set of CellIDs that covers
// a given range (a tiling). This example shows how to generate a tiling
// for a semi-open range of leaf cells [start, limit):
//
//   for id := start.MaxTile(limit); id != limit; id = id.Next().MaxTile(limit)) { ... }
//
// Note that in general the cells in the tiling will be of different sizes;
// they gradually get larger (near the middle of the range) and then
// gradually get smaller as limit is approached.
func (ci CellID) MaxTile(limit CellID) CellID {
	start := ci.RangeMin()
	if start >= limit.RangeMin() {
		return limit
	}

	if ci.RangeMax() >= limit {
		// The cell is too large, shrink it. Note that when generating coverings
		// of CellID ranges, this loop usually executes only once. Also because
		// ci.RangeMin() < limit.RangeMin(), we will always exit the loop by the
		// time we reach a leaf cell.
		for {
			ci = ci.Children()[0]
			if ci.RangeMax() < limit {
				break
			}
		}
		return ci
	}

	// The cell may be too small. Grow it if necessary. Note that generally
	// this loop only iterates once.
	for !ci.isFace() {
		parent := ci.immediateParent()
		if parent.RangeMin() != start || parent.RangeMax() >= limit {
			break
		}
		ci = parent
	}
	return ci
}

// centerFaceSiTi returns the (face, si, ti) coordinates of the center of the cell.
// Note that although (si,ti) coordinates span the range [0,2**31] in general,
// the cell center coordinates are always in the range [1,2**31-1] and
// therefore can be represented using a signed 32-bit integer.
func (ci CellID) centerFaceSiTi() (face, si, ti int) {
	// First we compute the discrete (i,j) coordinates of a leaf cell contained
	// within the given cell. Given that cells are represented by the Hilbert
	// curve position corresponding at their center, it turns out that the cell
	// returned by faceIJOrientation is always one of two leaf cells closest
	// to the center of the cell (unless the given cell is a leaf cell itself,
	// in which case there is only one possibility).
	//
	// Given a cell of size s >= 2 (i.e. not a leaf cell), and letting (imin,
	// jmin) be the coordinates of its lower left-hand corner, the leaf cell
	// returned by faceIJOrientation is either (imin + s/2, jmin + s/2)
	// (imin + s/2 - 1, jmin + s/2 - 1). The first case is the one we want.
	// We can distinguish these two cases by looking at the low bit of i or
	// j. In the second case the low bit is one, unless s == 2 (i.e. the
	// level just above leaf cells) in which case the low bit is zero.
	//
	// In the code below, the expression ((i ^ (int(id) >> 2)) & 1) is true
	// if we are in the second case described above.
	face, i, j, _ := ci.faceIJOrientation()
	delta := 0
	if ci.IsLeaf() {
		delta = 1
	} else if (int64(i)^(int64(ci)>>2))&1 == 1 {
		delta = 2
	}

	// Note that (2 * {i,j} + delta) will never overflow a 32-bit integer.
	return face, 2*i + delta, 2*j + delta
}