`import "github.com/golang/geo/s2"`

Package s2 implements types and functions for working with geometry in S² (spherical geometry).

Its related packages, parallel to this one, are s1 (operates on S¹), r1 (operates on ℝ¹) and r3 (operates on ℝ³).

This package provides types and functions for the S2 cell hierarchy and coordinate systems. The S2 cell hierarchy is a hierarchical decomposition of the surface of a unit sphere (S²) into “cells”; it is highly efficient, scales from continental size to under 1 cm² and preserves spatial locality (nearby cells have close IDs).

A presentation that gives an overview of S2 is https://docs.google.com/presentation/d/1Hl4KapfAENAOf4gv-pSngKwvS_jwNVHRPZTTDzXXn6Q/view.

- Variables
- func ChordAngleBetweenPoints(x, y Point) s1.ChordAngle
- func ClipEdge(a, b r2.Point, clip r2.Rect) (aClip, bClip r2.Point, intersects bool)
- func ClipToFace(a, b Point, face int) (aUV, bUV r2.Point, intersects bool)
- func ClipToPaddedFace(a, b Point, f int, padding float64) (aUV, bUV r2.Point, intersects bool)
- func DistanceFraction(x, a, b Point) float64
- func DistanceFromSegment(x, a, b Point) s1.Angle
- func IsDistanceLess(x, a, b Point, limit s1.ChordAngle) bool
- func IsInteriorDistanceLess(x, a, b Point, limit s1.ChordAngle) bool
- func MaybeUpdateMinDistance(x, a, b Point, minDist s1.ChordAngle) (s1.ChordAngle, bool)
- func MaybeUpdateMinInteriorDistance(x, a, b Point, minDist s1.ChordAngle) (s1.ChordAngle, bool)
- func OrderedCCW(a, b, c, o Point) bool
- func PointArea(a, b, c Point) float64
- func Sign(a, b, c Point) bool
- func SimpleCrossing(a, b, c, d Point) bool
- func VertexCrossing(a, b, c, d Point) bool
- func WedgeContains(a0, ab1, a2, b0, b2 Point) bool
- func WedgeIntersects(a0, ab1, a2, b0, b2 Point) bool
- type Cap
- func CapFromCenterAngle(center Point, angle s1.Angle) Cap
- func CapFromCenterArea(center Point, area float64) Cap
- func CapFromCenterChordAngle(center Point, radius s1.ChordAngle) Cap
- func CapFromCenterHeight(center Point, height float64) Cap
- func CapFromPoint(p Point) Cap
- func EmptyCap() Cap
- func FullCap() Cap
- func (c Cap) AddCap(other Cap) Cap
- func (c Cap) AddPoint(p Point) Cap
- func (c Cap) ApproxEqual(other Cap) bool
- func (c Cap) Area() float64
- func (c Cap) CapBound() Cap
- func (c Cap) Center() Point
- func (c Cap) Centroid() Point
- func (c Cap) Complement() Cap
- func (c Cap) Contains(other Cap) bool
- func (c Cap) ContainsCell(cell Cell) bool
- func (c Cap) ContainsPoint(p Point) bool
- func (c Cap) Equal(other Cap) bool
- func (c Cap) Expanded(distance s1.Angle) Cap
- func (c Cap) Height() float64
- func (c Cap) InteriorContainsPoint(p Point) bool
- func (c Cap) InteriorIntersects(other Cap) bool
- func (c Cap) Intersects(other Cap) bool
- func (c Cap) IntersectsCell(cell Cell) bool
- func (c Cap) IsEmpty() bool
- func (c Cap) IsFull() bool
- func (c Cap) IsValid() bool
- func (c Cap) Radius() s1.Angle
- func (c Cap) RectBound() Rect
- func (c Cap) String() string
- func (c Cap) Union(other Cap) Cap
- type Cell
- func CellFromCellID(id CellID) Cell
- func CellFromLatLng(ll LatLng) Cell
- func CellFromPoint(p Point) Cell
- func (c Cell) ApproxArea() float64
- func (c Cell) AverageArea() float64
- func (c Cell) BoundUV() r2.Rect
- func (c Cell) CapBound() Cap
- func (c Cell) Center() Point
- func (c Cell) Children() ([4]Cell, bool)
- func (c Cell) ContainsCell(oc Cell) bool
- func (c Cell) ContainsPoint(p Point) bool
- func (c Cell) Edge(k int) Point
- func (c Cell) ExactArea() float64
- func (c Cell) Face() int
- func (c Cell) ID() CellID
- func (c Cell) IntersectsCell(oc Cell) bool
- func (c Cell) IsLeaf() bool
- func (c Cell) Level() int
- func (c Cell) RectBound() Rect
- func (c Cell) SizeIJ() int
- func (c Cell) Vertex(k int) Point
- type CellID
- func CellIDFromFace(face int) CellID
- func CellIDFromFacePosLevel(face int, pos uint64, level int) CellID
- func CellIDFromLatLng(ll LatLng) CellID
- func CellIDFromToken(s string) CellID
- func (ci CellID) Advance(steps int64) CellID
- func (ci CellID) AdvanceWrap(steps int64) CellID
- func (ci CellID) AllNeighbors(level int) []CellID
- func (ci CellID) ChildBegin() CellID
- func (ci CellID) ChildBeginAtLevel(level int) CellID
- func (ci CellID) ChildEnd() CellID
- func (ci CellID) ChildEndAtLevel(level int) CellID
- func (ci CellID) ChildPosition(level int) int
- func (ci CellID) Children() [4]CellID
- func (ci CellID) CommonAncestorLevel(other CellID) (level int, ok bool)
- func (ci CellID) Contains(oci CellID) bool
- func (ci CellID) EdgeNeighbors() [4]CellID
- func (ci CellID) Face() int
- func (ci CellID) Intersects(oci CellID) bool
- func (ci CellID) IsLeaf() bool
- func (ci CellID) IsValid() bool
- func (ci CellID) LatLng() LatLng
- func (ci CellID) Level() int
- func (ci CellID) MaxTile(limit CellID) CellID
- func (ci CellID) Next() CellID
- func (ci CellID) NextWrap() CellID
- func (ci CellID) Parent(level int) CellID
- func (ci CellID) Point() Point
- func (ci CellID) Pos() uint64
- func (ci CellID) Prev() CellID
- func (ci CellID) PrevWrap() CellID
- func (ci CellID) RangeMax() CellID
- func (ci CellID) RangeMin() CellID
- func (ci CellID) String() string
- func (ci CellID) ToToken() string
- func (ci CellID) VertexNeighbors(level int) []CellID
- type CellRelation
- type CellUnion
- func CellUnionFromRange(begin, end CellID) CellUnion
- func (cu *CellUnion) CapBound() Cap
- func (cu *CellUnion) ContainsCell(c Cell) bool
- func (cu *CellUnion) ContainsCellID(id CellID) bool
- func (cu *CellUnion) ContainsPoint(p Point) bool
- func (cu *CellUnion) Denormalize(minLevel, levelMod int)
- func (cu *CellUnion) IntersectsCell(c Cell) bool
- func (cu *CellUnion) IntersectsCellID(id CellID) bool
- func (cu *CellUnion) LeafCellsCovered() int64
- func (cu *CellUnion) Normalize()
- func (cu *CellUnion) RectBound() Rect
- type Crossing
- type Direction
- type EdgeCrosser
- func NewChainEdgeCrosser(a, b, c Point) *EdgeCrosser
- func NewEdgeCrosser(a, b Point) *EdgeCrosser
- func (e *EdgeCrosser) ChainCrossingSign(d Point) Crossing
- func (e *EdgeCrosser) CrossingSign(c, d Point) Crossing
- func (e *EdgeCrosser) EdgeOrVertexChainCrossing(d Point) bool
- func (e *EdgeCrosser) EdgeOrVertexCrossing(c, d Point) bool
- func (e *EdgeCrosser) RestartAt(c Point)
- type LatLng
- func LatLngFromDegrees(lat, lng float64) LatLng
- func LatLngFromPoint(p Point) LatLng
- func (ll LatLng) Distance(ll2 LatLng) s1.Angle
- func (ll LatLng) IsValid() bool
- func (ll LatLng) Normalized() LatLng
- func (ll LatLng) String() string
- type Loop
- func EmptyLoop() *Loop
- func FullLoop() *Loop
- func LoopFromCell(c Cell) *Loop
- func LoopFromPoints(pts []Point) *Loop
- func RegularLoop(center Point, radius s1.Angle, numVertices int) *Loop
- func RegularLoopForFrame(frame matrix3x3, radius s1.Angle, numVertices int) *Loop
- func (l *Loop) CapBound() Cap
- func (l *Loop) ContainsCell(target Cell) bool
- func (l *Loop) ContainsOrigin() bool
- func (l *Loop) ContainsPoint(p Point) bool
- func (l *Loop) Edge(i int) (a, b Point)
- func (l *Loop) HasInterior() bool
- func (l *Loop) IntersectsCell(target Cell) bool
- func (l *Loop) IsEmpty() bool
- func (l *Loop) IsFull() bool
- func (l *Loop) NumEdges() int
- func (l *Loop) RectBound() Rect
- func (l *Loop) Vertex(i int) Point
- func (l *Loop) Vertices() []Point
- type Metric
- func (m Metric) ClosestLevel(val float64) int
- func (m Metric) MaxLevel(val float64) int
- func (m Metric) MinLevel(val float64) int
- func (m Metric) Value(level int) float64
- type PaddedCell
- func PaddedCellFromCellID(id CellID, padding float64) *PaddedCell
- func PaddedCellFromParentIJ(parent *PaddedCell, i, j int) *PaddedCell
- func (p PaddedCell) Bound() r2.Rect
- func (p PaddedCell) CellID() CellID
- func (p PaddedCell) Center() Point
- func (p PaddedCell) ChildIJ(pos int) (i, j int)
- func (p PaddedCell) EntryVertex() Point
- func (p PaddedCell) ExitVertex() Point
- func (p PaddedCell) Level() int
- func (p *PaddedCell) Middle() r2.Rect
- func (p PaddedCell) Padding() float64
- func (p *PaddedCell) ShrinkToFit(rect r2.Rect) CellID
- type Point
- func ClosestPoint(x, a, b Point) Point
- func Interpolate(t float64, a, b Point) Point
- func InterpolateAtDistance(ax s1.Angle, a, b Point) Point
- func OriginPoint() Point
- func PlanarCentroid(a, b, c Point) Point
- func PointFromCoords(x, y, z float64) Point
- func PointFromLatLng(ll LatLng) Point
- func TrueCentroid(a, b, c Point) Point
- func (p Point) ApproxEqual(other Point) bool
- func (p Point) CapBound() Cap
- func (p Point) Contains(other Point) bool
- func (p Point) ContainsCell(c Cell) bool
- func (p Point) ContainsPoint(other Point) bool
- func (p Point) Distance(b Point) s1.Angle
- func (p Point) IntersectsCell(c Cell) bool
- func (p Point) PointCross(op Point) Point
- func (p Point) RectBound() Rect
- type Polygon
- func FullPolygon() *Polygon
- func PolygonFromLoops(loops []*Loop) *Polygon
- func (p *Polygon) CapBound() Cap
- func (p *Polygon) ContainsOrigin() bool
- func (p *Polygon) Edge(e int) (a, b Point)
- func (p *Polygon) HasInterior() bool
- func (p *Polygon) IsEmpty() bool
- func (p *Polygon) IsFull() bool
- func (p *Polygon) LastDescendant(k int) int
- func (p *Polygon) Loop(k int) *Loop
- func (p *Polygon) Loops() []*Loop
- func (p *Polygon) NumEdges() int
- func (p *Polygon) NumLoops() int
- func (p *Polygon) Parent(k int) (index int, ok bool)
- func (p *Polygon) RectBound() Rect
- type Polyline
- func PolylineFromLatLngs(points []LatLng) *Polyline
- func (p *Polyline) CapBound() Cap
- func (p *Polyline) Centroid() Point
- func (p *Polyline) ContainsCell(cell Cell) bool
- func (p *Polyline) ContainsOrigin() bool
- func (p *Polyline) ContainsPoint(point Point) bool
- func (p *Polyline) Edge(i int) (a, b Point)
- func (p *Polyline) Equals(b *Polyline) bool
- func (p *Polyline) HasInterior() bool
- func (p *Polyline) IntersectsCell(cell Cell) bool
- func (p *Polyline) Length() s1.Angle
- func (p *Polyline) NumEdges() int
- func (p *Polyline) RectBound() Rect
- func (p *Polyline) Reverse()
- type Rect
- func EmptyRect() Rect
- func ExpandForSubregions(bound Rect) Rect
- func FullRect() Rect
- func RectFromCenterSize(center, size LatLng) Rect
- func RectFromLatLng(p LatLng) Rect
- func (r Rect) AddPoint(ll LatLng) Rect
- func (r Rect) Area() float64
- func (r Rect) CapBound() Cap
- func (r Rect) Center() LatLng
- func (r Rect) Contains(other Rect) bool
- func (r Rect) ContainsCell(c Cell) bool
- func (r Rect) ContainsLatLng(ll LatLng) bool
- func (r Rect) ContainsPoint(p Point) bool
- func (r Rect) Hi() LatLng
- func (r Rect) Intersection(other Rect) Rect
- func (r Rect) Intersects(other Rect) bool
- func (r Rect) IntersectsCell(c Cell) bool
- func (r Rect) IsEmpty() bool
- func (r Rect) IsFull() bool
- func (r Rect) IsPoint() bool
- func (r Rect) IsValid() bool
- func (r Rect) Lo() LatLng
- func (r Rect) PolarClosure() Rect
- func (r Rect) RectBound() Rect
- func (r Rect) Size() LatLng
- func (r Rect) String() string
- func (r Rect) Union(other Rect) Rect
- func (r Rect) Vertex(i int) LatLng
- type RectBounder
- func NewRectBounder() *RectBounder
- func (r *RectBounder) AddPoint(b Point)
- func (r *RectBounder) RectBound() Rect
- type Region
- type RegionCoverer
- func (rc *RegionCoverer) CellUnion(region Region) CellUnion
- func (rc *RegionCoverer) Covering(region Region) CellUnion
- func (rc *RegionCoverer) FastCovering(cap Cap) CellUnion
- func (rc *RegionCoverer) InteriorCellUnion(region Region) CellUnion
- func (rc *RegionCoverer) InteriorCovering(region Region) CellUnion
- type Shape
- type ShapeIndex
- func NewShapeIndex() *ShapeIndex
- func (s *ShapeIndex) Add(shape Shape)
- func (s *ShapeIndex) Begin() *ShapeIndexIterator
- func (s *ShapeIndex) End() *ShapeIndexIterator
- func (s *ShapeIndex) IsFresh() bool
- func (s *ShapeIndex) Iterator() *ShapeIndexIterator
- func (s *ShapeIndex) Len() int
- func (s *ShapeIndex) NumEdges() int
- func (s *ShapeIndex) Remove(shape Shape)
- func (s *ShapeIndex) Reset()
- type ShapeIndexCell
- type ShapeIndexIterator
- func (s *ShapeIndexIterator) AtBegin() bool
- func (s *ShapeIndexIterator) CellID() CellID
- func (s *ShapeIndexIterator) Center() Point
- func (s *ShapeIndexIterator) Done() bool
- func (s *ShapeIndexIterator) IndexCell() *ShapeIndexCell
- func (s *ShapeIndexIterator) LocateCellID(target CellID) CellRelation
- func (s *ShapeIndexIterator) LocatePoint(p Point) bool
- func (s *ShapeIndexIterator) Next()
- func (s *ShapeIndexIterator) Prev()
- func (s *ShapeIndexIterator) Reset()
- type WedgeRel
- Bugs

cap.go cell.go cellid.go cellunion.go doc.go edgeutil.go latlng.go loop.go matrix3x3.go metric.go paddedcell.go point.go polygon.go polyline.go predicates.go rect.go region.go regioncoverer.go shapeindex.go stuv.go

❖

var ( MinAngleSpanMetric = Metric{1, 4.0 / 3} AvgAngleSpanMetric = Metric{1, math.Pi / 2} MaxAngleSpanMetric = Metric{1, 1.704897179199218452} )

Each cell is bounded by four planes passing through its four edges and the center of the sphere. These metrics relate to the angle between each pair of opposite bounding planes, or equivalently, between the planes corresponding to two different s-values or two different t-values.

❖

var ( MinWidthMetric = Metric{1, 2 * math.Sqrt2 / 3} AvgWidthMetric = Metric{1, 1.434523672886099389} MaxWidthMetric = Metric{1, MaxAngleSpanMetric.Deriv} )

The width of geometric figure is defined as the distance between two parallel bounding lines in a given direction. For cells, the minimum width is always attained between two opposite edges, and the maximum width is attained between two opposite vertices. However, for our purposes we redefine the width of a cell as the perpendicular distance between a pair of opposite edges. A cell therefore has two widths, one in each direction. The minimum width according to this definition agrees with the classic geometric one, but the maximum width is different. (The maximum geometric width corresponds to MaxDiag defined below.)

The average width in both directions for all cells at level k is approximately AvgWidthMetric.Value(k).

The width is useful for bounding the minimum or maximum distance from a point on one edge of a cell to the closest point on the opposite edge. For example, this is useful when growing regions by a fixed distance.

❖

var ( MinEdgeMetric = Metric{1, 2 * math.Sqrt2 / 3} AvgEdgeMetric = Metric{1, 1.459213746386106062} MaxEdgeMetric = Metric{1, MaxAngleSpanMetric.Deriv} // MaxEdgeAspect is the maximum edge aspect ratio over all cells at any level, // where the edge aspect ratio of a cell is defined as the ratio of its longest // edge length to its shortest edge length. MaxEdgeAspect = 1.442615274452682920 MinAreaMetric = Metric{2, 8 * math.Sqrt2 / 9} AvgAreaMetric = Metric{2, 4 * math.Pi / 6} MaxAreaMetric = Metric{2, 2.635799256963161491} )

The edge length metrics can be used to bound the minimum, maximum, or average distance from the center of one cell to the center of one of its edge neighbors. In particular, it can be used to bound the distance between adjacent cell centers along the space-filling Hilbert curve for cells at any given level.

❖

var ( MinDiagMetric = Metric{1, 8 * math.Sqrt2 / 9} AvgDiagMetric = Metric{1, 2.060422738998471683} MaxDiagMetric = Metric{1, 2.438654594434021032} // MaxDiagAspect is the maximum diagonal aspect ratio over all cells at any // level, where the diagonal aspect ratio of a cell is defined as the ratio // of its longest diagonal length to its shortest diagonal length. MaxDiagAspect = math.Sqrt(3) )

The maximum diagonal is also the maximum diameter of any cell, and also the maximum geometric width (see the comment for widths). For example, the distance from an arbitrary point to the closest cell center at a given level is at most half the maximum diagonal length.

❖

func ChordAngleBetweenPoints(x, y Point) s1.ChordAngle

ChordAngleBetweenPoints constructs a ChordAngle corresponding to the distance between the two given points. The points must be unit length.

ClipEdge returns the portion of the edge defined by AB that is contained by the given rectangle. If there is no intersection, false is returned and aClip and bClip are undefined.

ClipToFace returns the (u,v) coordinates for the portion of the edge AB that intersects the given face, or false if the edge AB does not intersect. This method guarantees that the clipped vertices lie within the [-1,1]x[-1,1] cube face rectangle and are within faceClipErrorUVDist of the line AB, but the results may differ from those produced by faceSegments.

ClipToPaddedFace returns the (u,v) coordinates for the portion of the edge AB that intersects the given face, but rather than clipping to the square [-1,1]x[-1,1] in (u,v) space, this method clips to [-R,R]x[-R,R] where R=(1+padding). Padding must be non-negative.

DistanceFraction returns the distance ratio of the point X along an edge AB. If X is on the line segment AB, this is the fraction T such that X == Interpolate(T, A, B).

This requires that A and B are distinct.

DistanceFromSegment returns the distance of point X from line segment AB. The points are expected to be normalized.

❖

func IsDistanceLess(x, a, b Point, limit s1.ChordAngle) bool

IsDistanceLess reports whether the distance from X to the edge AB is less than limit. This method is faster than DistanceFromSegment(). If you want to compare against a fixed s1.Angle, you should convert it to an s1.ChordAngle once and save the value, since this conversion is relatively expensive.

❖

func IsInteriorDistanceLess(x, a, b Point, limit s1.ChordAngle) bool

IsInteriorDistanceLess reports whether the minimum distance from X to the edge AB is attained at an interior point of AB (i.e., not an endpoint), and that distance is less than limit.

❖

func MaybeUpdateMinDistance(x, a, b Point, minDist s1.ChordAngle) (s1.ChordAngle, bool)

MaybeUpdateMinDistance checks if the distance from X to the edge AB is less then minDist, and if returns the possibly updated value and whether it was updated. The case A == B is handled correctly.

Use this method when you want to compute many distances and keep track of the minimum. It is significantly faster than using DistanceFromSegment because (1) using s1.ChordAngle is much faster than s1.Angle, and (2) it can save a lot of work by not actually computing the distance when it is obviously larger than the current minimum.

❖

func MaybeUpdateMinInteriorDistance(x, a, b Point, minDist s1.ChordAngle) (s1.ChordAngle, bool)

MaybeUpdateMinInteriorDistance reports whether the minimum distance from X to AB is attained at an interior point of AB (i.e., not an endpoint), and that distance is less than minDist. If so, the value of minDist is updated and returns true. Otherwise it is unchanged and returns false.

OrderedCCW returns true if the edges OA, OB, and OC are encountered in that order while sweeping CCW around the point O.

You can think of this as testing whether A <= B <= C with respect to the CCW ordering around O that starts at A, or equivalently, whether B is contained in the range of angles (inclusive) that starts at A and extends CCW to C. Properties:

(1) If OrderedCCW(a,b,c,o) && OrderedCCW(b,a,c,o), then a == b (2) If OrderedCCW(a,b,c,o) && OrderedCCW(a,c,b,o), then b == c (3) If OrderedCCW(a,b,c,o) && OrderedCCW(c,b,a,o), then a == b == c (4) If a == b or b == c, then OrderedCCW(a,b,c,o) is true (5) Otherwise if a == c, then OrderedCCW(a,b,c,o) is false

PointArea returns the area on the unit sphere for the triangle defined by the given points.

This method is based on l'Huilier's theorem,

tan(E/4) = sqrt(tan(s/2) tan((s-a)/2) tan((s-b)/2) tan((s-c)/2))

where E is the spherical excess of the triangle (i.e. its area),

a, b, c are the side lengths, and s is the semiperimeter (a + b + c) / 2.

The only significant source of error using l'Huilier's method is the cancellation error of the terms (s-a), (s-b), (s-c). This leads to a *relative* error of about 1e-16 * s / min(s-a, s-b, s-c). This compares to a relative error of about 1e-15 / E using Girard's formula, where E is the true area of the triangle. Girard's formula can be even worse than this for very small triangles, e.g. a triangle with a true area of 1e-30 might evaluate to 1e-5.

So, we prefer l'Huilier's formula unless dmin < s * (0.1 * E), where dmin = min(s-a, s-b, s-c). This basically includes all triangles except for extremely long and skinny ones.

Since we don't know E, we would like a conservative upper bound on the triangle area in terms of s and dmin. It's possible to show that E <= k1 * s * sqrt(s * dmin), where k1 = 2*sqrt(3)/Pi (about 1). Using this, it's easy to show that we should always use l'Huilier's method if dmin >= k2 * s^5, where k2 is about 1e-2. Furthermore, if dmin < k2 * s^5, the triangle area is at most k3 * s^4, where k3 is about 0.1. Since the best case error using Girard's formula is about 1e-15, this means that we shouldn't even consider it unless s >= 3e-4 or so.

Sign returns true if the points A, B, C are strictly counterclockwise, and returns false if the points are clockwise or collinear (i.e. if they are all contained on some great circle).

Due to numerical errors, situations may arise that are mathematically impossible, e.g. ABC may be considered strictly CCW while BCA is not. However, the implementation guarantees the following:

If Sign(a,b,c), then !Sign(c,b,a) for all a,b,c.

SimpleCrossing reports whether edge AB crosses CD at a point that is interior to both edges. Properties:

(1) SimpleCrossing(b,a,c,d) == SimpleCrossing(a,b,c,d) (2) SimpleCrossing(c,d,a,b) == SimpleCrossing(a,b,c,d)

VertexCrossing reports whether two edges "cross" in such a way that point-in-polygon containment tests can be implemented by counting the number of edge crossings.

Given two edges AB and CD where at least two vertices are identical (i.e. CrossingSign(a,b,c,d) == 0), the basic rule is that a "crossing" occurs if AB is encountered after CD during a CCW sweep around the shared vertex starting from a fixed reference point.

Note that according to this rule, if AB crosses CD then in general CD does not cross AB. However, this leads to the correct result when counting polygon edge crossings. For example, suppose that A,B,C are three consecutive vertices of a CCW polygon. If we now consider the edge crossings of a segment BP as P sweeps around B, the crossing number changes parity exactly when BP crosses BA or BC.

Useful properties of VertexCrossing (VC):

(1) VC(a,a,c,d) == VC(a,b,c,c) == false (2) VC(a,b,a,b) == VC(a,b,b,a) == true (3) VC(a,b,c,d) == VC(a,b,d,c) == VC(b,a,c,d) == VC(b,a,d,c) (3) If exactly one of a,b equals one of c,d, then exactly one of VC(a,b,c,d) and VC(c,d,a,b) is true

It is an error to call this method with 4 distinct vertices.

WedgeContains reports whether non-empty wedge A=(a0, ab1, a2) contains B=(b0, ab1, b2). Equivalent to WedgeRelation == WedgeProperlyContains || WedgeEquals.

WedgeIntersects reports whether non-empty wedge A=(a0, ab1, a2) intersects B=(b0, ab1, b2). Equivalent to WedgeRelation == WedgeIsDisjoint

❖

```
type Cap struct {
// contains filtered or unexported fields
}
```

Cap represents a disc-shaped region defined by a center and radius. Technically this shape is called a "spherical cap" (rather than disc) because it is not planar; the cap represents a portion of the sphere that has been cut off by a plane. The boundary of the cap is the circle defined by the intersection of the sphere and the plane. For containment purposes, the cap is a closed set, i.e. it contains its boundary.

For the most part, you can use a spherical cap wherever you would use a disc in planar geometry. The radius of the cap is measured along the surface of the sphere (rather than the straight-line distance through the interior). Thus a cap of radius π/2 is a hemisphere, and a cap of radius π covers the entire sphere.

The center is a point on the surface of the unit sphere. (Hence the need for it to be of unit length.)

A cap can also be defined by its center point and height. The height is the distance from the center point to the cutoff plane. There is also support for "empty" and "full" caps, which contain no points and all points respectively.

Here are some useful relationships between the cap height (h), the cap radius (r), the maximum chord length from the cap's center (d), and the radius of cap's base (a).

h = 1 - cos(r) = 2 * sin^2(r/2) d^2 = 2 * h = a^2 + h^2

The zero value of Cap is an invalid cap. Use EmptyCap to get a valid empty cap.

CapFromCenterAngle constructs a cap with the given center and angle.

CapFromCenterArea constructs a cap with the given center and surface area. Note that the area can also be interpreted as the solid angle subtended by the cap (because the sphere has unit radius). A negative area yields an empty cap; an area of 4*π or more yields a full cap.

❖

func CapFromCenterChordAngle(center Point, radius s1.ChordAngle) Cap

CapFromCenterChordAngle constructs a cap where the angle is expressed as an s1.ChordAngle. This constructor is more efficient than using an s1.Angle.

CapFromCenterHeight constructs a cap with the given center and height. A negative height yields an empty cap; a height of 2 or more yields a full cap. The center should be unit length.

CapFromPoint constructs a cap containing a single point.

EmptyCap returns a cap that contains no points.

FullCap returns a cap that contains all points.

AddCap increases the cap height if necessary to include the other cap. If this cap is empty, it is set to the other cap.

AddPoint increases the cap if necessary to include the given point. If this cap is empty, then the center is set to the point with a zero height. p must be unit-length.

ApproxEqual reports whether this cap is equal to the other cap within the given tolerance.

Area returns the surface area of the Cap on the unit sphere.

CapBound returns a bounding spherical cap. This is not guaranteed to be exact.

Center returns the cap's center point.

Centroid returns the true centroid of the cap multiplied by its surface area The result lies on the ray from the origin through the cap's center, but it is not unit length. Note that if you just want the "surface centroid", i.e. the normalized result, then it is simpler to call Center.

The reason for multiplying the result by the cap area is to make it easier to compute the centroid of more complicated shapes. The centroid of a union of disjoint regions can be computed simply by adding their Centroid() results. Caveat: for caps that contain a single point (i.e., zero radius), this method always returns the origin (0, 0, 0). This is because shapes with no area don't affect the centroid of a union whose total area is positive.

Complement returns the complement of the interior of the cap. A cap and its complement have the same boundary but do not share any interior points. The complement operator is not a bijection because the complement of a singleton cap (containing a single point) is the same as the complement of an empty cap.

Contains reports whether this cap contains the other.

ContainsCell reports whether the cap contains the given cell.

ContainsPoint reports whether this cap contains the point.

Equal reports whether this cap is equal to the other cap.

Expanded returns a new cap expanded by the given angle. If the cap is empty, it returns an empty cap.

Height returns the height of the cap. This is the distance from the center point to the cutoff plane.

InteriorContainsPoint reports whether the point is within the interior of this cap.

InteriorIntersects reports whether this caps interior intersects the other cap.

Intersects reports whether this cap intersects the other cap. i.e. whether they have any points in common.

IntersectsCell reports whether the cap intersects the cell.

IsEmpty reports whether the cap is empty, i.e. it contains no points.

IsFull reports whether the cap is full, i.e. it contains all points.

IsValid reports whether the Cap is considered valid.

Radius returns the cap radius as an s1.Angle. (Note that the cap angle is stored internally as a ChordAngle, so this method requires a trigonometric operation and may yield a slightly different result than the value passed to CapFromCenterAngle).

RectBound returns a bounding latitude-longitude rectangle. The bounds are not guaranteed to be tight.

Union returns the smallest cap which encloses this cap and other.

❖

```
type Cell struct {
// contains filtered or unexported fields
}
```

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.

CellFromCellID constructs a Cell corresponding to the given CellID.

CellFromLatLng constructs a cell for the given LatLng.

CellFromPoint constructs a cell for the given Point.

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.

AverageArea returns the average area of cells at the level of this cell. This is accurate to within a factor of 1.7.

BoundUV returns the bounds of this cell in (u,v)-space.

CapBound returns the bounding cap of this cell.

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

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.

ContainsCell reports whether this cell contains the other cell.

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.

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).

ExactArea returns the area of this cell as accurately as possible.

Face returns the face this cell is on.

ID returns the CellID this cell represents.

IntersectsCell reports whether the intersection of this cell and the other cell is not nil.

IsLeaf returns whether this Cell is a leaf or not.

Level returns the level of this cell.

RectBound returns the bounding rectangle of this cell.

SizeIJ returns the CellID value for the cells 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).

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.

CellIDFromFace returns the cell corresponding to a given S2 cube face.

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.

CellIDFromLatLng returns the leaf cell containing ll.

CellIDFromToken returns a cell given a hex-encoded string of its uint64 ID.

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().

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.

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().

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() { ... }

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.

ChildEnd returns the first cell after a traversal of the children of this cell in Hilbert curve order. The returned cell may be invalid.

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.

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.

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.

CommonAncestorLevel returns the level of the common ancestor of the two S2 CellIDs.

Contains returns true iff the CellID contains oci.

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.

Face returns the cube face for this cell ID, in the range [0,5].

Intersects returns true iff the CellID intersects oci.

IsLeaf returns whether this cell ID is at the deepest level; that is, the level at which the cells are smallest.

IsValid reports whether ci represents a valid cell.

LatLng returns the center of the s2 cell on the sphere as a LatLng.

Level returns the subdivision level of this cell ID, in the range [0, maxLevel].

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.

Next returns the next cell along the Hilbert curve. This is expected to be used with ChildBegin and ChildEnd, or ChildBeginAtLevel and ChildEndAtLevel.

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.

Parent returns the cell at the given level, which must be no greater than the current level.

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).

Pos returns the position along the Hilbert curve of this cell ID, in the range [0,2^posBits-1].

Prev returns the previous cell along the Hilbert curve.

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.

RangeMax returns the maximum CellID that is contained within this cell.

RangeMin returns the minimum CellID that is contained within this cell.

String returns the string representation of the cell ID in the form "1/3210".

ToToken returns a hex-encoded string of the uint64 cell id, with leading zeros included but trailing zeros stripped.

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.)

CellRelation describes the possible relationships between a target cell and the cells of the ShapeIndex. If the target is an index cell or is contained by an index cell, it is Indexed. If the target is subdivided into one or more index cells, it is Subdivided. Otherwise it is Disjoint.

❖

const ( Indexed CellRelation = iota Subdivided Disjoint )

The possible CellRelations for a ShapeIndex.

A CellUnion is a collection of CellIDs.

It is normalized if it is sorted, and does not contain redundancy. Specifically, it may not contain the same CellID twice, nor a CellID that is contained by another, nor the four sibling CellIDs that are children of a single higher level CellID.

CellUnionFromRange creates a CellUnion that covers the half-open range of leaf cells [begin, end). If begin == end the resulting union is empty. This requires that begin and end are both leaves, and begin <= end. To create a closed-ended range, pass in end.Next().

CapBound returns a Cap that bounds this entity.

ContainsCell reports whether this cell union contains the given cell.

ContainsCellID reports whether the cell union contains the given cell ID. Containment is defined with respect to regions, e.g. a cell contains its 4 children.

This method assumes that the CellUnion has been normalized.

ContainsPoint reports whether this cell union contains the given point.

Denormalize replaces this CellUnion with an expanded version of the CellUnion where any cell whose level is less than minLevel or where (level - minLevel) is not a multiple of levelMod is replaced by its children, until either both of these conditions are satisfied or the maximum level is reached.

IntersectsCell reports whether this cell union intersects the given cell.

IntersectsCellID reports whether this cell union intersects the given cell ID.

This method assumes that the CellUnion has been normalized.

LeafCellsCovered reports the number of leaf cells covered by this cell union. This will be no more than 6*2^60 for the whole sphere.

Normalize normalizes the CellUnion.

RectBound returns a Rect that bounds this entity.

A Crossing indicates how edges cross.

❖

const ( // Cross means the edges cross. Cross Crossing = iota // MaybeCross means two vertices from different edges are the same. MaybeCross // DoNotCross means the edges do not cross. DoNotCross )

Direction is an indication of the ordering of a set of points.

These are the three options for the direction of a set of points.

RobustSign returns a Direction representing the ordering of the points. CounterClockwise is returned if the points are in counter-clockwise order, Clockwise for clockwise, and Indeterminate if any two points are the same (collinear), or the sign could not completely be determined.

This function has additional logic to make sure that the above properties hold even when the three points are coplanar, and to deal with the limitations of floating-point arithmetic.

RobustSign satisfies the following conditions:

(1) RobustSign(a,b,c) == Indeterminate if and only if a == b, b == c, or c == a (2) RobustSign(b,c,a) == RobustSign(a,b,c) for all a,b,c (3) RobustSign(c,b,a) == -RobustSign(a,b,c) for all a,b,c

In other words:

(1) The result is Indeterminate if and only if two points are the same. (2) Rotating the order of the arguments does not affect the result. (3) Exchanging any two arguments inverts the result.

On the other hand, note that it is not true in general that RobustSign(-a,b,c) == -RobustSign(a,b,c), or any similar identities involving antipodal points.

❖

```
type EdgeCrosser struct {
// contains filtered or unexported fields
}
```

EdgeCrosser allows edges to be efficiently tested for intersection with a given fixed edge AB. It is especially efficient when testing for intersection with an edge chain connecting vertices v0, v1, v2, ...

❖

func NewChainEdgeCrosser(a, b, c Point) *EdgeCrosser

NewChainEdgeCrosser is a convenience constructor that uses AB as the fixed edge, and C as the first vertex of the vertex chain (equivalent to calling RestartAt(c)).

You don't need to use this or any of the chain functions unless you're trying to squeeze out every last drop of performance. Essentially all you are saving is a test whether the first vertex of the current edge is the same as the second vertex of the previous edge.

❖

func NewEdgeCrosser(a, b Point) *EdgeCrosser

NewEdgeCrosser returns an EdgeCrosser with the fixed edge AB.

❖

func (e *EdgeCrosser) ChainCrossingSign(d Point) Crossing

ChainCrossingSign is like CrossingSign, but uses the last vertex passed to one of the crossing methods (or RestartAt) as the first vertex of the current edge.

❖

func (e *EdgeCrosser) CrossingSign(c, d Point) Crossing

CrossingSign reports whether the edge AB intersects the edge CD. If any two vertices from different edges are the same, returns MaybeCross. If either edge is degenerate (A == B or C == D), returns either DoNotCross or MaybeCross.

Properties of CrossingSign:

(1) CrossingSign(b,a,c,d) == CrossingSign(a,b,c,d) (2) CrossingSign(c,d,a,b) == CrossingSign(a,b,c,d) (3) CrossingSign(a,b,c,d) == MaybeCross if a==c, a==d, b==c, b==d (3) CrossingSign(a,b,c,d) == DoNotCross or MaybeCross if a==b or c==d

Note that if you want to check an edge against a chain of other edges, it is slightly more efficient to use the single-argument version ChainCrossingSign below.

❖

func (e *EdgeCrosser) EdgeOrVertexChainCrossing(d Point) bool

EdgeOrVertexChainCrossing is like EdgeOrVertexCrossing, but uses the last vertex passed to one of the crossing methods (or RestartAt) as the first vertex of the current edge.

❖

func (e *EdgeCrosser) EdgeOrVertexCrossing(c, d Point) bool

EdgeOrVertexCrossing reports whether if CrossingSign(c, d) > 0, or AB and CD share a vertex and VertexCrossing(a, b, c, d) is true.

This method extends the concept of a "crossing" to the case where AB and CD have a vertex in common. The two edges may or may not cross, according to the rules defined in VertexCrossing above. The rules are designed so that point containment tests can be implemented simply by counting edge crossings. Similarly, determining whether one edge chain crosses another edge chain can be implemented by counting.

❖

func (e *EdgeCrosser) RestartAt(c Point)

RestartAt sets the current point of the edge crosser to be c. Call this method when your chain 'jumps' to a new place. The argument must point to a value that persists until the next call.

LatLng represents a point on the unit sphere as a pair of angles.

LatLngFromDegrees returns a LatLng for the coordinates given in degrees.

LatLngFromPoint returns an LatLng for a given Point.

Distance returns the angle between two LatLngs.

IsValid returns true iff the LatLng is normalized, with Lat ∈ [-π/2,π/2] and Lng ∈ [-π,π].

Normalized returns the normalized version of the LatLng, with Lat clamped to [-π/2,π/2] and Lng wrapped in [-π,π].

❖

```
type Loop struct {
// contains filtered or unexported fields
}
```

Loop represents a simple spherical polygon. It consists of a sequence of vertices where the first vertex is implicitly connected to the last. All loops are defined to have a CCW orientation, i.e. the interior of the loop is on the left side of the edges. This implies that a clockwise loop enclosing a small area is interpreted to be a CCW loop enclosing a very large area.

Loops are not allowed to have any duplicate vertices (whether adjacent or not), and non-adjacent edges are not allowed to intersect. Loops must have at least 3 vertices (except for the "empty" and "full" loops discussed below).

There are two special loops: the "empty" loop contains no points and the "full" loop contains all points. These loops do not have any edges, but to preserve the invariant that every loop can be represented as a vertex chain, they are defined as having exactly one vertex each (see EmptyLoop and FullLoop).

EmptyLoop returns a special "empty" loop.

FullLoop returns a special "full" loop.

LoopFromCell constructs a loop corresponding to the given cell.

Note that the loop and cell *do not* contain exactly the same set of points, because Loop and Cell have slightly different definitions of point containment. For example, a Cell vertex is contained by all four neighboring Cells, but it is contained by exactly one of four Loops constructed from those cells. As another example, the cell coverings of cell and LoopFromCell(cell) will be different, because the loop contains points on its boundary that actually belong to other cells (i.e., the covering will include a layer of neighboring cells).

LoopFromPoints constructs a loop from the given points.

RegularLoop creates a loop with the given number of vertices, all located on a circle of the specified radius around the given center.

RegularLoopForFrame creates a loop centered around the z-axis of the given coordinate frame, with the first vertex in the direction of the positive x-axis.

CapBound returns a bounding cap that may have more padding than the corresponding RectBound. The bound is conservative such that if the loop contains a point P, the bound also contains it.

ContainsCell reports whether the given Cell is contained by this Loop.

ContainsOrigin reports true if this loop contains s2.OriginPoint().

ContainsPoint returns true if the loop contains the point.

Edge returns the endpoints for the given edge index.

HasInterior returns true because all loops have an interior.

IntersectsCell reports whether this Loop intersects the given cell.

IsEmpty reports true if this is the special "empty" loop that contains no points.

IsFull reports true if this is the special "full" loop that contains all points.

NumEdges returns the number of edges in this shape.

RectBound returns a tight bounding rectangle. If the loop contains the point, the bound also contains it.

Vertex returns the vertex for the given index. For convenience, the vertex indices wrap automatically for methods that do index math such as Edge. i.e., Vertex(NumEdges() + n) is the same as Vertex(n).

Vertices returns the vertices in the loop.

❖

type Metric struct { // Dim is either 1 or 2, for a 1D or 2D metric respectively. Dim int // Deriv is the scaling factor for the metric. Deriv float64 }

A Metric is a measure for cells. It is used to describe the shape and size of cells. They are useful for deciding which cell level to use in order to satisfy a given condition (e.g. that cell vertices must be no further than "x" apart). You can use the Value(level) method to compute the corresponding length or area on the unit sphere for cells at a given level. The minimum and maximum bounds are valid for cells at all levels, but they may be somewhat conservative for very large cells (e.g. face cells).

ClosestLevel returns the level at which the metric has approximately the given value. The return value is always a valid level. For example, AvgEdgeMetric.ClosestLevel(0.1) returns the level at which the average cell edge length is approximately 0.1.

MaxLevel returns the maximum level such that the metric is at least the given value, or zero if there is no such level.

For example, MaxLevel(0.1) returns the maximum level such that all cells have a minimum width of 0.1 or larger. The returned value is always a valid level.

In C++, this is called GetLevelForMinValue.

MinLevel returns the minimum level such that the metric is at most the given value, or maxLevel (30) if there is no such level.

For example, MinLevel(0.1) returns the minimum level such that all cell diagonal lengths are 0.1 or smaller. The returned value is always a valid level.

In C++, this is called GetLevelForMaxValue.

Value returns the value of the metric at the given level.

❖

```
type PaddedCell struct {
// contains filtered or unexported fields
}
```

PaddedCell represents a Cell whose (u,v)-range has been expanded on all sides by a given amount of "padding". Unlike Cell, its methods and representation are optimized for clipping edges against Cell boundaries to determine which cells are intersected by a given set of edges.

❖

func PaddedCellFromCellID(id CellID, padding float64) *PaddedCell

PaddedCellFromCellID constructs a padded cell with the given padding.

❖

func PaddedCellFromParentIJ(parent *PaddedCell, i, j int) *PaddedCell

PaddedCellFromParentIJ constructs the child of parent with the given (i,j) index. The four child cells have indices of (0,0), (0,1), (1,0), (1,1), where the i and j indices correspond to increasing u- and v-values respectively.

❖

func (p PaddedCell) Bound() r2.Rect

Bound returns the bounds for this cell in (u,v)-space including padding.

❖

func (p PaddedCell) CellID() CellID

CellID returns the CellID this padded cell represents.

❖

func (p PaddedCell) Center() Point

Center returns the center of this cell.

❖

func (p PaddedCell) ChildIJ(pos int) (i, j int)

ChildIJ returns the (i,j) coordinates for the child cell at the given traversal position. The traversal position corresponds to the order in which child cells are visited by the Hilbert curve.

❖

func (p PaddedCell) EntryVertex() Point

EntryVertex return the vertex where the space-filling curve enters this cell.

❖

func (p PaddedCell) ExitVertex() Point

ExitVertex returns the vertex where the space-filling curve exits this cell.

❖

func (p PaddedCell) Level() int

Level returns the level this cell is at.

❖

func (p *PaddedCell) Middle() r2.Rect

Middle returns the rectangle in the middle of this cell that belongs to all four of its children in (u,v)-space.

❖

func (p PaddedCell) Padding() float64

Padding returns the amount of padding on this cell.

❖

func (p *PaddedCell) ShrinkToFit(rect r2.Rect) CellID

ShrinkToFit returns the smallest CellID that contains all descendants of this padded cell whose bounds intersect the given rect. For algorithms that use recursive subdivision to find the cells that intersect a particular object, this method can be used to skip all of the initial subdivision steps where only one child needs to be expanded.

Note that this method is not the same as returning the smallest cell that contains the intersection of this cell with rect. Because of the padding, even if one child completely contains rect it is still possible that a neighboring child may also intersect the given rect.

The provided Rect must intersect the bounds of this cell.

Point represents a point on the unit sphere as a normalized 3D vector. Fields should be treated as read-only. Use one of the factory methods for creation.

ClosestPoint returns the point along the edge AB that is closest to the point X. The fractional distance of this point along the edge AB can be obtained using DistanceFraction.

This requires that all points are unit length.

Interpolate returns the point X along the line segment AB whose distance from A is the given fraction "t" of the distance AB. Does NOT require that "t" be between 0 and 1. Note that all distances are measured on the surface of the sphere, so this is more complicated than just computing (1-t)*a + t*b and normalizing the result.

InterpolateAtDistance returns the point X along the line segment AB whose distance from A is the angle ax.

OriginPoint returns a unique "origin" on the sphere for operations that need a fixed reference point. In particular, this is the "point at infinity" used for point-in-polygon testing (by counting the number of edge crossings).

It should *not* be a point that is commonly used in edge tests in order to avoid triggering code to handle degenerate cases (this rules out the north and south poles). It should also not be on the boundary of any low-level S2Cell for the same reason.

PlanarCentroid returns the centroid of the planar triangle ABC, which is not normalized. It can be normalized to unit length to obtain the "surface centroid" of the corresponding spherical triangle, i.e. the intersection of the three medians. However, note that for large spherical triangles the surface centroid may be nowhere near the intuitive "center" (see example in TrueCentroid comments).

Note that the surface centroid may be nowhere near the intuitive "center" of a spherical triangle. For example, consider the triangle with vertices A=(1,eps,0), B=(0,0,1), C=(-1,eps,0) (a quarter-sphere). The surface centroid of this triangle is at S=(0, 2*eps, 1), which is within a distance of 2*eps of the vertex B. Note that the median from A (the segment connecting A to the midpoint of BC) passes through S, since this is the shortest path connecting the two endpoints. On the other hand, the true centroid is at M=(0, 0.5, 0.5), which when projected onto the surface is a much more reasonable interpretation of the "center" of this triangle.

PointFromCoords creates a new normalized point from coordinates.

This always returns a valid point. If the given coordinates can not be normalized the origin point will be returned.

This behavior is different from the C++ construction of a S2Point from coordinates (i.e. S2Point(x, y, z)) in that in C++ they do not Normalize.

PointFromLatLng returns an Point for the given LatLng. The maximum error in the result is 1.5 * dblEpsilon. (This does not include the error of converting degrees, E5, E6, or E7 into radians.)

TrueCentroid returns the true centroid of the spherical triangle ABC multiplied by the signed area of spherical triangle ABC. The result is not normalized. The reasons for multiplying by the signed area are (1) this is the quantity that needs to be summed to compute the centroid of a union or difference of triangles, and (2) it's actually easier to calculate this way. All points must have unit length.

The true centroid (mass centroid) is defined as the surface integral over the spherical triangle of (x,y,z) divided by the triangle area. This is the point that the triangle would rotate around if it was spinning in empty space.

The best centroid for most purposes is the true centroid. Unlike the planar and surface centroids, the true centroid behaves linearly as regions are added or subtracted. That is, if you split a triangle into pieces and compute the average of their centroids (weighted by triangle area), the result equals the centroid of the original triangle. This is not true of the other centroids.

ApproxEqual reports whether the two points are similar enough to be equal.

CapBound returns a bounding cap for this point.

Contains reports if this Point contains the other Point. (This method matches all other s2 types where the reflexive Contains method does not contain the type's name.)

ContainsCell returns false as Points do not contain any other S2 types.

ContainsPoint reports if this Point contains the other Point. (This method is named to satisfy the Region interface.)

Distance returns the angle between two points.

IntersectsCell reports whether this Point intersects the given cell.

PointCross returns a Point that is orthogonal to both p and op. This is similar to p.Cross(op) (the true cross product) except that it does a better job of ensuring orthogonality when the Point is nearly parallel to op, it returns a non-zero result even when p == op or p == -op and the result is a Point.

It satisfies the following properties (f == PointCross):

(1) f(p, op) != 0 for all p, op (2) f(op,p) == -f(p,op) unless p == op or p == -op (3) f(-p,op) == -f(p,op) unless p == op or p == -op (4) f(p,-op) == -f(p,op) unless p == op or p == -op

RectBound returns a bounding latitude-longitude rectangle from this point.

❖

```
type Polygon struct {
// contains filtered or unexported fields
}
```

Polygon represents a sequence of zero or more loops; recall that the interior of a loop is defined to be its left-hand side (see Loop).

When the polygon is initialized, the given loops are automatically converted into a canonical form consisting of "shells" and "holes". Shells and holes are both oriented CCW, and are nested hierarchically. The loops are reordered to correspond to a preorder traversal of the nesting hierarchy.

Polygons may represent any region of the sphere with a polygonal boundary, including the entire sphere (known as the "full" polygon). The full polygon consists of a single full loop (see Loop), whereas the empty polygon has no loops at all.

Use FullPolygon() to construct a full polygon. The zero value of Polygon is treated as the empty polygon.

Polygons have the following restrictions:

- Loops may not cross, i.e. the boundary of a loop may not intersect both the interior and exterior of any other loop. - Loops may not share edges, i.e. if a loop contains an edge AB, then no other loop may contain AB or BA. - Loops may share vertices, however no vertex may appear twice in a single loop (see Loop). - No loop may be empty. The full loop may appear only in the full polygon.

FullPolygon returns a special "full" polygon.

PolygonFromLoops constructs a polygon from the given hierarchically nested loops. The polygon interior consists of the points contained by an odd number of loops. (Recall that a loop contains the set of points on its left-hand side.)

This method figures out the loop nesting hierarchy and assigns every loop a depth. Shells have even depths, and holes have odd depths.

NOTE: this function is NOT YET IMPLEMENTED for more than one loop and will panic if given a slice of length > 1.

CapBound returns a bounding spherical cap.

ContainsOrigin returns whether this shape contains the origin.

Edge returns endpoints for the given edge index.

HasInterior reports whether this Polygon has an interior.

IsEmpty reports whether this is the special "empty" polygon (consisting of no loops).

IsFull reports whether this is the special "full" polygon (consisting of a single loop that encompasses the entire sphere).

LastDescendant returns the index of the last loop that is contained within loop k. If k is negative, it returns the last loop in the polygon. Note that loops are indexed according to a preorder traversal of the nesting hierarchy, so the immediate children of loop k can be found by iterating over the loops (k+1)..LastDescendant(k) and selecting those whose depth is equal to Loop(k).depth+1.

Loop returns the loop at the given index. Note that during initialization, the given loops are reordered according to a preorder traversal of the loop nesting hierarchy. This implies that every loop is immediately followed by its descendants. This hierarchy can be traversed using the methods Parent, LastDescendant, and Loop.depth.

Loops returns the loops in this polygon.

NumEdges returns the number of edges in this shape.

NumLoops returns the number of loops in this polygon.

Parent returns the index of the parent of loop k. If the loop does not have a parent, ok=false is returned.

RectBound returns a bounding latitude-longitude rectangle.

Polyline represents a sequence of zero or more vertices connected by straight edges (geodesics). Edges of length 0 and 180 degrees are not allowed, i.e. adjacent vertices should not be identical or antipodal.

PolylineFromLatLngs creates a new Polyline from the given LatLngs.

CapBound returns the bounding Cap for this Polyline.

Centroid returns the true centroid of the polyline multiplied by the length of the polyline. The result is not unit length, so you may wish to normalize it.

Scaling by the Polyline length makes it easy to compute the centroid of several Polylines (by simply adding up their centroids).

ContainsCell reports whether this Polyline contains the given Cell. Always returns false because "containment" is not numerically well-defined except at the Polyline vertices.

ContainsOrigin returns false because there is no interior to contain s2.Origin.

ContainsPoint returns false since Polylines are not closed.

Edge returns endpoints for the given edge index.

Equals reports whether the given Polyline is exactly the same as this one.

HasInterior returns false as Polylines are not closed.

IntersectsCell reports whether this Polyline intersects the given Cell.

Length returns the length of this Polyline.

NumEdges returns the number of edges in this shape.

RectBound returns the bounding Rect for this Polyline.

Reverse reverses the order of the Polyline vertices.

Rect represents a closed latitude-longitude rectangle.

EmptyRect returns the empty rectangle.

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).

FullRect returns the full rectangle.

RectFromCenterSize constructs a rectangle with the given size and center. center needs to be normalized, but size does not. The latitude interval of the result is clamped to [-90,90] degrees, and the longitude interval of the result is FullRect() if and only if the longitude size is 360 degrees or more.

Examples of clamping (in degrees):

center=(80,170), size=(40,60) -> lat=[60,90], lng=[140,-160] center=(10,40), size=(210,400) -> lat=[-90,90], lng=[-180,180] center=(-90,180), size=(20,50) -> lat=[-90,-80], lng=[155,-155]

RectFromLatLng constructs a rectangle containing a single point p.

AddPoint increases the size of the rectangle to include the given point.

Area returns the surface area of the Rect.

CapBound returns a cap that countains Rect.

Center returns the center of the rectangle.

Contains reports whether this Rect contains the other Rect.

ContainsCell reports whether the given Cell is contained by this Rect.

ContainsLatLng reports whether the given LatLng is within the Rect.

ContainsPoint reports whether the given Point is within the Rect.

Hi returns the other corner of the rectangle.

Intersection returns the smallest rectangle containing the intersection of this rectangle and the given rectangle. Note that the region of intersection may consist of two disjoint rectangles, in which case a single rectangle spanning both of them is returned.

Intersects reports whether this rectangle and the other have any points in common.

IntersectsCell reports whether this rectangle intersects the given cell. This is an exact test and may be fairly expensive.

IsEmpty reports whether the rectangle is empty.

IsFull reports whether the rectangle is full.

IsPoint reports whether the rectangle is a single point.

IsValid returns true iff the rectangle is valid. This requires Lat ⊆ [-π/2,π/2] and Lng ⊆ [-π,π], and Lat = ∅ iff Lng = ∅

Lo returns one corner of the rectangle.

PolarClosure returns the rectangle unmodified if it does not include either pole. If it includes either pole, PolarClosure returns an expansion of the rectangle along the longitudinal range to include all possible representations of the contained poles.

RectBound returns itself.

Size returns the size of the Rect.

Union returns the smallest Rect containing the union of this rectangle and the given rectangle.

Vertex returns the i-th vertex of the rectangle (i = 0,1,2,3) in CCW order (lower left, lower right, upper right, upper left).

❖

```
type RectBounder struct {
// contains filtered or unexported fields
}
```

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)).

❖

func NewRectBounder() *RectBounder

NewRectBounder returns a new instance of a RectBounder.

❖

func (r *RectBounder) AddPoint(b Point)

AddPoint adds the given point to the chain. The Point must be unit length.

❖

func (r *RectBounder) RectBound() Rect

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.

❖

type Region interface { // CapBound returns a bounding spherical cap. This is not guaranteed to be exact. CapBound() Cap // RectBound returns a bounding latitude-longitude rectangle that contains // the region. The bounds are not guaranteed to be tight. RectBound() Rect // ContainsCell reports whether the region completely contains the given region. // It returns false if containment could not be determined. ContainsCell(c Cell) bool // IntersectsCell reports whether the region intersects the given cell or // if intersection could not be determined. It returns false if the region // does not intersect. IntersectsCell(c Cell) bool // ContainsPoint reports whether the region contains the given point or not. // The point should be unit length, although some implementations may relax this restriction. ContainsPoint(p Point) bool }

A Region represents a two-dimensional region on the unit sphere.

The purpose of this interface is to allow complex regions to be approximated as simpler regions. The interface is restricted to methods that are useful for computing approximations.

❖

type RegionCoverer struct { MinLevel int // the minimum cell level to be used. MaxLevel int // the maximum cell level to be used. LevelMod int // the LevelMod to be used. MaxCells int // the maximum desired number of cells in the approximation. }

RegionCoverer allows arbitrary regions to be approximated as unions of cells (CellUnion). This is useful for implementing various sorts of search and precomputation operations.

Typical usage:

rc := &s2.RegionCoverer{MaxLevel: 30, MaxCells: 5} r := s2.Region(CapFromCenterArea(center, area)) covering := rc.Covering(r)

This yields a CellUnion of at most 5 cells that is guaranteed to cover the given region (a disc-shaped region on the sphere).

For covering, only cells where (level - MinLevel) is a multiple of LevelMod will be used. This effectively allows the branching factor of the S2 CellID hierarchy to be increased. Currently the only parameter values allowed are 0/1, 2, or 3, corresponding to branching factors of 4, 16, and 64 respectively.

Note the following:

- MinLevel takes priority over MaxCells, i.e. cells below the given level will never be used even if this causes a large number of cells to be returned. - For any setting of MaxCells, up to 6 cells may be returned if that is the minimum number of cells required (e.g. if the region intersects all six face cells). Up to 3 cells may be returned even for very tiny convex regions if they happen to be located at the intersection of three cube faces. - For any setting of MaxCells, an arbitrary number of cells may be returned if MinLevel is too high for the region being approximated. - If MaxCells is less than 4, the area of the covering may be arbitrarily large compared to the area of the original region even if the region is convex (e.g. a Cap or Rect).

The approximation algorithm is not optimal but does a pretty good job in practice. The output does not always use the maximum number of cells allowed, both because this would not always yield a better approximation, and because MaxCells is a limit on how much work is done exploring the possible covering as well as a limit on the final output size.

Because it is an approximation algorithm, one should not rely on the stability of the output. In particular, the output of the covering algorithm may change across different versions of the library.

One can also generate interior coverings, which are sets of cells which are entirely contained within a region. Interior coverings can be empty, even for non-empty regions, if there are no cells that satisfy the provided constraints and are contained by the region. Note that for performance reasons, it is wise to specify a MaxLevel when computing interior coverings - otherwise for regions with small or zero area, the algorithm may spend a lot of time subdividing cells all the way to leaf level to try to find contained cells.

❖

func (rc *RegionCoverer) CellUnion(region Region) CellUnion

CellUnion returns a normalized CellUnion that covers the given region and satisfies the restrictions except for minLevel and levelMod. These criteria cannot be satisfied using a cell union because cell unions are automatically normalized by replacing four child cells with their parent whenever possible. (Note that the list of cell ids passed to the CellUnion constructor does in fact satisfy all the given restrictions.)

❖

func (rc *RegionCoverer) Covering(region Region) CellUnion

Covering returns a CellUnion that covers the given region and satisfies the various restrictions.

❖

func (rc *RegionCoverer) FastCovering(cap Cap) CellUnion

FastCovering returns a CellUnion that covers the given region similar to Covering, except that this method is much faster and the coverings are not as tight. All of the usual parameters are respected (MaxCells, MinLevel, MaxLevel, and LevelMod), except that the implementation makes no attempt to take advantage of large values of MaxCells. (A small number of cells will always be returned.)

This function is useful as a starting point for algorithms that recursively subdivide cells.

❖

func (rc *RegionCoverer) InteriorCellUnion(region Region) CellUnion

InteriorCellUnion returns a normalized CellUnion that is contained within the given region and satisfies the restrictions except for minLevel and levelMod. These criteria cannot be satisfied using a cell union because cell unions are automatically normalized by replacing four child cells with their parent whenever possible. (Note that the list of cell ids passed to the CellUnion constructor does in fact satisfy all the given restrictions.)

❖

func (rc *RegionCoverer) InteriorCovering(region Region) CellUnion

InteriorCovering returns a CellUnion that is contained within the given region and satisfies the various restrictions.

❖

type Shape interface { // NumEdges returns the number of edges in this shape. NumEdges() int // Edge returns endpoints for the given edge index. // Zero-length edges are allowed, and can be used to represent points. Edge(i int) (a, b Point) // HasInterior reports whether this shape has an interior. If so, it must be possible // to assemble the edges into a collection of non-crossing loops. Edges may // be returned in any order, and edges may be oriented arbitrarily with // respect to the shape interior. (However, note that some Shape types // may have stronger requirements.) HasInterior() bool // ContainsOrigin returns true if this shape contains s2.Origin. // Shapes that do not have an interior will return false. ContainsOrigin() bool // contains filtered or unexported methods }

Shape defines an interface for any S2 type that needs to be indexable. A shape is a collection of edges that optionally defines an interior. It can be used to represent a set of points, a set of polylines, or a set of polygons.

❖

```
type ShapeIndex struct {
// contains filtered or unexported fields
}
```

ShapeIndex indexes a set of Shapes, where a Shape is some collection of edges that optionally defines an interior. It can be used to represent a set of points, a set of polylines, or a set of polygons. For Shapes that have interiors, the index makes it very fast to determine which Shape(s) contain a given point or region.

❖

func NewShapeIndex() *ShapeIndex

NewShapeIndex creates a new ShapeIndex.

❖

func (s *ShapeIndex) Add(shape Shape)

Add adds the given shape to the index and assign an ID to it.

❖

func (s *ShapeIndex) Begin() *ShapeIndexIterator

Begin positions the iterator at the first cell in the index.

❖

func (s *ShapeIndex) End() *ShapeIndexIterator

End positions the iterator at the last cell in the index.

❖

func (s *ShapeIndex) IsFresh() bool

IsFresh reports if there are no pending updates that need to be applied. This can be useful to avoid building the index unnecessarily, or for choosing between two different algorithms depending on whether the index is available.

The returned index status may be slightly out of date if the index was built in a different thread. This is fine for the intended use (as an efficiency hint), but it should not be used by internal methods.

❖

func (s *ShapeIndex) Iterator() *ShapeIndexIterator

Iterator returns an iterator for this index.

❖

func (s *ShapeIndex) Len() int

Len reports the number of Shapes in this index.

❖

func (s *ShapeIndex) NumEdges() int

NumEdges returns the number of edges in this index.

❖

func (s *ShapeIndex) Remove(shape Shape)

Remove removes the given shape from the index.

❖

func (s *ShapeIndex) Reset()

Reset resets the index to its original state.

❖

```
type ShapeIndexCell struct {
// contains filtered or unexported fields
}
```

ShapeIndexCell stores the index contents for a particular CellID.

❖

func NewShapeIndexCell(numShapes int) *ShapeIndexCell

NewShapeIndexCell creates a new cell that is sized to hold the given number of shapes.

❖

```
type ShapeIndexIterator struct {
// contains filtered or unexported fields
}
```

ShapeIndexIterator is an iterator that provides low-level access to the cells of the index. Cells are returned in increasing order of CellID.

for it := index.Iterator(); !it.Done(); it.Next() { fmt.Print(it.CellID()) }

❖

func (s *ShapeIndexIterator) AtBegin() bool

AtBegin reports if the iterator is positioned at the first index cell.

❖

func (s *ShapeIndexIterator) CellID() CellID

CellID returns the CellID of the cell at the current position of the iterator.

❖

func (s *ShapeIndexIterator) Center() Point

Center returns the Point at the center of the current position of the iterator.

❖

func (s *ShapeIndexIterator) Done() bool

Done reports if the iterator is positioned at or after the last index cell.

❖

func (s *ShapeIndexIterator) IndexCell() *ShapeIndexCell

IndexCell returns the ShapeIndexCell at the current position of the iterator.

❖

func (s *ShapeIndexIterator) LocateCellID(target CellID) CellRelation

LocateCellID attempts to position the iterator at the first matching indexCell in the index that has some relation to the given CellID. Let T be the target CellID. If T is contained by (or equal to) some index cell I, then the iterator is positioned at I and returns Indexed. Otherwise if T contains one or more (smaller) index cells, then position the iterator at the first such cell I and return Subdivided. Otherwise Disjoint is returned and the iterator position is undefined.

❖

func (s *ShapeIndexIterator) LocatePoint(p Point) bool

LocatePoint positions the iterator at the cell that contains the given Point. If no such cell exists, the iterator position is unspecified, and false is returned. The cell at the matched position is guaranteed to contain all edges that might intersect the line segment between target and the cell's center.

❖

func (s *ShapeIndexIterator) Next()

Next advances the iterator to the next cell in the index.

❖

func (s *ShapeIndexIterator) Prev()

Prev advances the iterator to the previous cell in the index. If the iterator is at the first cell the call does nothing.

❖

func (s *ShapeIndexIterator) Reset()

Reset the iterator to be positioned at the first cell in the index.

WedgeRel enumerates the possible relation between two wedges A and B.

❖

const ( WedgeEquals WedgeRel = iota // A and B are equal. WedgeProperlyContains // A is a strict superset of B. WedgeIsProperlyContained // A is a strict subset of B. WedgeProperlyOverlaps // A-B, B-A, and A intersect B are non-empty. WedgeIsDisjoint // A and B are disjoint. )

Define the different possible relationships between two wedges.

WedgeRelation reports the relation between two non-empty wedges A=(a0, ab1, a2) and B=(b0, ab1, b2).

☞ Differences from C++: Subdivide BoundUV Distance/DistanceToEdge VertexChordDistance

☞ The differences from the C++ version FloodFill, SimpleCovering

Package s2 imports 11 packages (graph) and is imported by 18 packages. Updated 2017-06-19. Refresh now. Tools for package owners.