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// 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
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// See the License for the specific language governing permissions and
// limitations under the License.
package s2
import (
// Rect represents a closed latitude-longitude rectangle.
type Rect struct {
Lat r1.Interval
Lng s1.Interval
var (
validRectLatRange = r1.Interval{-math.Pi / 2, math.Pi / 2}
validRectLngRange = s1.FullInterval()
// EmptyRect returns the empty rectangle.
func EmptyRect() Rect { return Rect{r1.EmptyInterval(), s1.EmptyInterval()} }
// FullRect returns the full rectangle.
func FullRect() Rect { return Rect{validRectLatRange, validRectLngRange} }
// RectFromLatLng constructs a rectangle containing a single point p.
func RectFromLatLng(p LatLng) Rect {
return Rect{
Lat: r1.Interval{p.Lat.Radians(), p.Lat.Radians()},
Lng: s1.Interval{p.Lng.Radians(), p.Lng.Radians()},
// 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]
func RectFromCenterSize(center, size LatLng) Rect {
half := LatLng{size.Lat / 2, size.Lng / 2}
return RectFromLatLng(center).expanded(half)
// IsValid returns true iff the rectangle is valid.
// This requires Lat ⊆ [-π/2,π/2] and Lng ⊆ [-π,π], and Lat = ∅ iff Lng = ∅
func (r Rect) IsValid() bool {
return math.Abs(r.Lat.Lo) <= math.Pi/2 &&
math.Abs(r.Lat.Hi) <= math.Pi/2 &&
r.Lng.IsValid() &&
r.Lat.IsEmpty() == r.Lng.IsEmpty()
// IsEmpty reports whether the rectangle is empty.
func (r Rect) IsEmpty() bool { return r.Lat.IsEmpty() }
// IsFull reports whether the rectangle is full.
func (r Rect) IsFull() bool { return r.Lat.Equal(validRectLatRange) && r.Lng.IsFull() }
// IsPoint reports whether the rectangle is a single point.
func (r Rect) IsPoint() bool { return r.Lat.Lo == r.Lat.Hi && r.Lng.Lo == r.Lng.Hi }
// 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).
func (r Rect) Vertex(i int) LatLng {
var lat, lng float64
switch i {
case 0:
lat = r.Lat.Lo
lng = r.Lng.Lo
case 1:
lat = r.Lat.Lo
lng = r.Lng.Hi
case 2:
lat = r.Lat.Hi
lng = r.Lng.Hi
case 3:
lat = r.Lat.Hi
lng = r.Lng.Lo
return LatLng{s1.Angle(lat) * s1.Radian, s1.Angle(lng) * s1.Radian}
// Lo returns one corner of the rectangle.
func (r Rect) Lo() LatLng {
return LatLng{s1.Angle(r.Lat.Lo) * s1.Radian, s1.Angle(r.Lng.Lo) * s1.Radian}
// Hi returns the other corner of the rectangle.
func (r Rect) Hi() LatLng {
return LatLng{s1.Angle(r.Lat.Hi) * s1.Radian, s1.Angle(r.Lng.Hi) * s1.Radian}
// Center returns the center of the rectangle.
func (r Rect) Center() LatLng {
return LatLng{s1.Angle(r.Lat.Center()) * s1.Radian, s1.Angle(r.Lng.Center()) * s1.Radian}
// Size returns the size of the Rect.
func (r Rect) Size() LatLng {
return LatLng{s1.Angle(r.Lat.Length()) * s1.Radian, s1.Angle(r.Lng.Length()) * s1.Radian}
// Area returns the surface area of the Rect.
func (r Rect) Area() float64 {
if r.IsEmpty() {
return 0
capDiff := math.Abs(math.Sin(r.Lat.Hi) - math.Sin(r.Lat.Lo))
return r.Lng.Length() * capDiff
// AddPoint increases the size of the rectangle to include the given point.
func (r Rect) AddPoint(ll LatLng) Rect {
if !ll.IsValid() {
return r
return Rect{
Lat: r.Lat.AddPoint(ll.Lat.Radians()),
Lng: r.Lng.AddPoint(ll.Lng.Radians()),
// expanded returns a rectangle that has been expanded by margin.Lat on each side
// in the latitude direction, and by margin.Lng on each side in the longitude
// direction. If either margin is negative, then it shrinks the rectangle on
// the corresponding sides instead. The resulting rectangle may be empty.
// The latitude-longitude space has the topology of a cylinder. Longitudes
// "wrap around" at +/-180 degrees, while latitudes are clamped to range [-90, 90].
// This means that any expansion (positive or negative) of the full longitude range
// remains full (since the "rectangle" is actually a continuous band around the
// cylinder), while expansion of the full latitude range remains full only if the
// margin is positive.
// If either the latitude or longitude interval becomes empty after
// expansion by a negative margin, the result is empty.
// Note that if an expanded rectangle contains a pole, it may not contain
// all possible lat/lng representations of that pole, e.g., both points [π/2,0]
// and [π/2,1] represent the same pole, but they might not be contained by the
// same Rect.
// If you are trying to grow a rectangle by a certain distance on the
// sphere (e.g. 5km), refer to the ExpandedByDistance() C++ method implementation
// instead.
func (r Rect) expanded(margin LatLng) Rect {
lat := r.Lat.Expanded(margin.Lat.Radians())
lng := r.Lng.Expanded(margin.Lng.Radians())
if lat.IsEmpty() || lng.IsEmpty() {
return EmptyRect()
return Rect{
Lat: lat.Intersection(validRectLatRange),
Lng: lng,
func (r Rect) String() string { return fmt.Sprintf("[Lo%v, Hi%v]", r.Lo(), r.Hi()) }
// 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.
func (r Rect) PolarClosure() Rect {
if r.Lat.Lo == -math.Pi/2 || r.Lat.Hi == math.Pi/2 {
return Rect{r.Lat, s1.FullInterval()}
return r
// Union returns the smallest Rect containing the union of this rectangle and the given rectangle.
func (r Rect) Union(other Rect) Rect {
return Rect{
Lat: r.Lat.Union(other.Lat),
Lng: r.Lng.Union(other.Lng),
// 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.
func (r Rect) Intersection(other Rect) Rect {
lat := r.Lat.Intersection(other.Lat)
lng := r.Lng.Intersection(other.Lng)
if lat.IsEmpty() || lng.IsEmpty() {
return EmptyRect()
return Rect{lat, lng}
// Intersects reports whether this rectangle and the other have any points in common.
func (r Rect) Intersects(other Rect) bool {
return r.Lat.Intersects(other.Lat) && r.Lng.Intersects(other.Lng)
// CapBound returns a cap that contains Rect.
func (r Rect) CapBound() Cap {
// We consider two possible bounding caps, one whose axis passes
// through the center of the lat-long rectangle and one whose axis
// is the north or south pole. We return the smaller of the two caps.
if r.IsEmpty() {
return EmptyCap()
var poleZ, poleAngle float64
if r.Lat.Hi+r.Lat.Lo < 0 {
// South pole axis yields smaller cap.
poleZ = -1
poleAngle = math.Pi/2 + r.Lat.Hi
} else {
poleZ = 1
poleAngle = math.Pi/2 - r.Lat.Lo
poleCap := CapFromCenterAngle(Point{r3.Vector{0, 0, poleZ}}, s1.Angle(poleAngle)*s1.Radian)
// For bounding rectangles that span 180 degrees or less in longitude, the
// maximum cap size is achieved at one of the rectangle vertices. For
// rectangles that are larger than 180 degrees, we punt and always return a
// bounding cap centered at one of the two poles.
if math.Remainder(r.Lng.Hi-r.Lng.Lo, 2*math.Pi) >= 0 && r.Lng.Hi-r.Lng.Lo < 2*math.Pi {
midCap := CapFromPoint(PointFromLatLng(r.Center())).AddPoint(PointFromLatLng(r.Lo())).AddPoint(PointFromLatLng(r.Hi()))
if midCap.Height() < poleCap.Height() {
return midCap
return poleCap
// RectBound returns itself.
func (r Rect) RectBound() Rect {
return r
// Contains reports whether this Rect contains the other Rect.
func (r Rect) Contains(other Rect) bool {
return r.Lat.ContainsInterval(other.Lat) && r.Lng.ContainsInterval(other.Lng)
// ContainsCell reports whether the given Cell is contained by this Rect.
func (r Rect) ContainsCell(c Cell) bool {
// A latitude-longitude rectangle contains a cell if and only if it contains
// the cell's bounding rectangle. This test is exact from a mathematical
// point of view, assuming that the bounds returned by Cell.RectBound()
// are tight. However, note that there can be a loss of precision when
// converting between representations -- for example, if an s2.Cell is
// converted to a polygon, the polygon's bounding rectangle may not contain
// the cell's bounding rectangle. This has some slightly unexpected side
// effects; for instance, if one creates an s2.Polygon from an s2.Cell, the
// polygon will contain the cell, but the polygon's bounding box will not.
return r.Contains(c.RectBound())
// ContainsLatLng reports whether the given LatLng is within the Rect.
func (r Rect) ContainsLatLng(ll LatLng) bool {
if !ll.IsValid() {
return false
return r.Lat.Contains(ll.Lat.Radians()) && r.Lng.Contains(ll.Lng.Radians())
// ContainsPoint reports whether the given Point is within the Rect.
func (r Rect) ContainsPoint(p Point) bool {
return r.ContainsLatLng(LatLngFromPoint(p))
// CellUnionBound computes a covering of the Rect.
func (r Rect) CellUnionBound() []CellID {
return r.CapBound().CellUnionBound()
// intersectsLatEdge reports whether the edge AB intersects the given edge of constant
// latitude. Requires the points to have unit length.
func intersectsLatEdge(a, b Point, lat s1.Angle, lng s1.Interval) bool {
// Unfortunately, lines of constant latitude are curves on
// the sphere. They can intersect a straight edge in 0, 1, or 2 points.
// First, compute the normal to the plane AB that points vaguely north.
z := Point{a.PointCross(b).Normalize()}
if z.Z < 0 {
z = Point{z.Mul(-1)}
// Extend this to an orthonormal frame (x,y,z) where x is the direction
// where the great circle through AB achieves its maximium latitude.
y := Point{z.PointCross(PointFromCoords(0, 0, 1)).Normalize()}
x := y.Cross(z.Vector)
// Compute the angle "theta" from the x-axis (in the x-y plane defined
// above) where the great circle intersects the given line of latitude.
sinLat := math.Sin(float64(lat))
if math.Abs(sinLat) >= x.Z {
// The great circle does not reach the given latitude.
return false
cosTheta := sinLat / x.Z
sinTheta := math.Sqrt(1 - cosTheta*cosTheta)
theta := math.Atan2(sinTheta, cosTheta)
// The candidate intersection points are located +/- theta in the x-y
// plane. For an intersection to be valid, we need to check that the
// intersection point is contained in the interior of the edge AB and
// also that it is contained within the given longitude interval "lng".
// Compute the range of theta values spanned by the edge AB.
abTheta := s1.IntervalFromPointPair(
math.Atan2(a.Dot(y.Vector), a.Dot(x)),
math.Atan2(b.Dot(y.Vector), b.Dot(x)))
if abTheta.Contains(theta) {
// Check if the intersection point is also in the given lng interval.
isect := x.Mul(cosTheta).Add(y.Mul(sinTheta))
if lng.Contains(math.Atan2(isect.Y, isect.X)) {
return true
if abTheta.Contains(-theta) {
// Check if the other intersection point is also in the given lng interval.
isect := x.Mul(cosTheta).Sub(y.Mul(sinTheta))
if lng.Contains(math.Atan2(isect.Y, isect.X)) {
return true
return false
// intersectsLngEdge reports whether the edge AB intersects the given edge of constant
// longitude. Requires the points to have unit length.
func intersectsLngEdge(a, b Point, lat r1.Interval, lng s1.Angle) bool {
// The nice thing about edges of constant longitude is that
// they are straight lines on the sphere (geodesics).
return CrossingSign(a, b, PointFromLatLng(LatLng{s1.Angle(lat.Lo), lng}),
PointFromLatLng(LatLng{s1.Angle(lat.Hi), lng})) == Cross
// IntersectsCell reports whether this rectangle intersects the given cell. This is an
// exact test and may be fairly expensive.
func (r Rect) IntersectsCell(c Cell) bool {
// First we eliminate the cases where one region completely contains the
// other. Once these are disposed of, then the regions will intersect
// if and only if their boundaries intersect.
if r.IsEmpty() {
return false
if r.ContainsPoint(Point{}) {
return true
if c.ContainsPoint(PointFromLatLng(r.Center())) {
return true
// Quick rejection test (not required for correctness).
if !r.Intersects(c.RectBound()) {
return false
// Precompute the cell vertices as points and latitude-longitudes. We also
// check whether the Cell contains any corner of the rectangle, or
// vice-versa, since the edge-crossing tests only check the edge interiors.
vertices := [4]Point{}
latlngs := [4]LatLng{}
for i := range vertices {
vertices[i] = c.Vertex(i)
latlngs[i] = LatLngFromPoint(vertices[i])
if r.ContainsLatLng(latlngs[i]) {
return true
if c.ContainsPoint(PointFromLatLng(r.Vertex(i))) {
return true
// Now check whether the boundaries intersect. Unfortunately, a
// latitude-longitude rectangle does not have straight edges: two edges
// are curved, and at least one of them is concave.
for i := range vertices {
edgeLng := s1.IntervalFromEndpoints(latlngs[i].Lng.Radians(), latlngs[(i+1)&3].Lng.Radians())
if !r.Lng.Intersects(edgeLng) {
a := vertices[i]
b := vertices[(i+1)&3]
if edgeLng.Contains(r.Lng.Lo) && intersectsLngEdge(a, b, r.Lat, s1.Angle(r.Lng.Lo)) {
return true
if edgeLng.Contains(r.Lng.Hi) && intersectsLngEdge(a, b, r.Lat, s1.Angle(r.Lng.Hi)) {
return true
if intersectsLatEdge(a, b, s1.Angle(r.Lat.Lo), r.Lng) {
return true
if intersectsLatEdge(a, b, s1.Angle(r.Lat.Hi), r.Lng) {
return true
return false
// Encode encodes the Rect.
func (r Rect) Encode(w io.Writer) error {
e := &encoder{w: w}
return e.err
func (r Rect) encode(e *encoder) {
// Decode decodes a rectangle.
func (r *Rect) Decode(rd io.Reader) error {
d := &decoder{r: asByteReader(rd)}
return d.err
func (r *Rect) decode(d *decoder) {
if version := d.readUint8(); int(version) != int(encodingVersion) && d.err == nil {
d.err = fmt.Errorf("can't decode version %d; my version: %d", version, encodingVersion)
r.Lat.Lo = d.readFloat64()
r.Lat.Hi = d.readFloat64()
r.Lng.Lo = d.readFloat64()
r.Lng.Hi = d.readFloat64()
// DistanceToLatLng returns the minimum distance (measured along the surface of the sphere)
// from a given point to the rectangle (both its boundary and its interior).
// If r is empty, the result is meaningless.
// The latlng must be valid.
func (r Rect) DistanceToLatLng(ll LatLng) s1.Angle {
if r.Lng.Contains(float64(ll.Lng)) {
return maxAngle(0, ll.Lat-s1.Angle(r.Lat.Hi), s1.Angle(r.Lat.Lo)-ll.Lat)
i := s1.IntervalFromEndpoints(r.Lng.Hi, r.Lng.ComplementCenter())
rectLng := r.Lng.Lo
if i.Contains(float64(ll.Lng)) {
rectLng = r.Lng.Hi
lo := LatLng{s1.Angle(r.Lat.Lo) * s1.Radian, s1.Angle(rectLng) * s1.Radian}
hi := LatLng{s1.Angle(r.Lat.Hi) * s1.Radian, s1.Angle(rectLng) * s1.Radian}
return DistanceFromSegment(PointFromLatLng(ll), PointFromLatLng(lo), PointFromLatLng(hi))
// DirectedHausdorffDistance returns the directed Hausdorff distance (measured along the
// surface of the sphere) to the given Rect. The directed Hausdorff
// distance from rectangle A to rectangle B is given by
// h(A, B) = max_{p in A} min_{q in B} d(p, q).
func (r Rect) DirectedHausdorffDistance(other Rect) s1.Angle {
if r.IsEmpty() {
return 0 * s1.Radian
if other.IsEmpty() {
return math.Pi * s1.Radian
lng := r.Lng.DirectedHausdorffDistance(other.Lng)
return directedHausdorffDistance(lng, r.Lat, other.Lat)
// HausdorffDistance returns the undirected Hausdorff distance (measured along the
// surface of the sphere) to the given Rect.
// The Hausdorff distance between rectangle A and rectangle B is given by
// H(A, B) = max{h(A, B), h(B, A)}.
func (r Rect) HausdorffDistance(other Rect) s1.Angle {
return maxAngle(r.DirectedHausdorffDistance(other),
// directedHausdorffDistance returns the directed Hausdorff distance
// from one longitudinal edge spanning latitude range 'a' to the other
// longitudinal edge spanning latitude range 'b', with their longitudinal
// difference given by 'lngDiff'.
func directedHausdorffDistance(lngDiff s1.Angle, a, b r1.Interval) s1.Angle {
// By symmetry, we can assume a's longitude is 0 and b's longitude is
// lngDiff. Call b's two endpoints bLo and bHi. Let H be the hemisphere
// containing a and delimited by the longitude line of b. The Voronoi diagram
// of b on H has three edges (portions of great circles) all orthogonal to b
// and meeting at bLo cross bHi.
// E1: (bLo, bLo cross bHi)
// E2: (bHi, bLo cross bHi)
// E3: (-bMid, bLo cross bHi), where bMid is the midpoint of b
// They subdivide H into three Voronoi regions. Depending on how longitude 0
// (which contains edge a) intersects these regions, we distinguish two cases:
// Case 1: it intersects three regions. This occurs when lngDiff <= π/2.
// Case 2: it intersects only two regions. This occurs when lngDiff > π/2.
// In the first case, the directed Hausdorff distance to edge b can only be
// realized by the following points on a:
// A1: two endpoints of a.
// A2: intersection of a with the equator, if b also intersects the equator.
// In the second case, the directed Hausdorff distance to edge b can only be
// realized by the following points on a:
// B1: two endpoints of a.
// B2: intersection of a with E3
// B3: farthest point from bLo to the interior of D, and farthest point from
// bHi to the interior of U, if any, where D (resp. U) is the portion
// of edge a below (resp. above) the intersection point from B2.
if lngDiff < 0 {
panic("impossible: negative lngDiff")
if lngDiff > math.Pi {
panic("impossible: lngDiff > Pi")
if lngDiff == 0 {
return s1.Angle(a.DirectedHausdorffDistance(b))
// Assumed longitude of b.
bLng := lngDiff
// Two endpoints of b.
bLo := PointFromLatLng(LatLng{s1.Angle(b.Lo), bLng})
bHi := PointFromLatLng(LatLng{s1.Angle(b.Hi), bLng})
// Cases A1 and B1.
aLo := PointFromLatLng(LatLng{s1.Angle(a.Lo), 0})
aHi := PointFromLatLng(LatLng{s1.Angle(a.Hi), 0})
maxDistance := maxAngle(
DistanceFromSegment(aLo, bLo, bHi),
DistanceFromSegment(aHi, bLo, bHi))
if lngDiff <= math.Pi/2 {
// Case A2.
if a.Contains(0) && b.Contains(0) {
maxDistance = maxAngle(maxDistance, lngDiff)
return maxDistance
// Case B2.
p := bisectorIntersection(b, bLng)
pLat := LatLngFromPoint(p).Lat
if a.Contains(float64(pLat)) {
maxDistance = maxAngle(maxDistance, p.Angle(bLo.Vector))
// Case B3.
if pLat > s1.Angle(a.Lo) {
intDist, ok := interiorMaxDistance(r1.Interval{a.Lo, math.Min(float64(pLat), a.Hi)}, bLo)
if ok {
maxDistance = maxAngle(maxDistance, intDist)
if pLat < s1.Angle(a.Hi) {
intDist, ok := interiorMaxDistance(r1.Interval{math.Max(float64(pLat), a.Lo), a.Hi}, bHi)
if ok {
maxDistance = maxAngle(maxDistance, intDist)
return maxDistance
// interiorMaxDistance returns the max distance from a point b to the segment spanning latitude range
// aLat on longitude 0 if the max occurs in the interior of aLat. Otherwise, returns (0, false).
func interiorMaxDistance(aLat r1.Interval, b Point) (a s1.Angle, ok bool) {
// Longitude 0 is in the y=0 plane. b.X >= 0 implies that the maximum
// does not occur in the interior of aLat.
if aLat.IsEmpty() || b.X >= 0 {
return 0, false
// Project b to the y=0 plane. The antipodal of the normalized projection is
// the point at which the maxium distance from b occurs, if it is contained
// in aLat.
intersectionPoint := PointFromCoords(-b.X, 0, -b.Z)
if !aLat.InteriorContains(float64(LatLngFromPoint(intersectionPoint).Lat)) {
return 0, false
return b.Angle(intersectionPoint.Vector), true
// bisectorIntersection return the intersection of longitude 0 with the bisector of an edge
// on longitude 'lng' and spanning latitude range 'lat'.
func bisectorIntersection(lat r1.Interval, lng s1.Angle) Point {
lng = s1.Angle(math.Abs(float64(lng)))
latCenter := s1.Angle(lat.Center())
// A vector orthogonal to the bisector of the given longitudinal edge.
orthoBisector := LatLng{latCenter - math.Pi/2, lng}
if latCenter < 0 {
orthoBisector = LatLng{-latCenter - math.Pi/2, lng - math.Pi}
// A vector orthogonal to longitude 0.
orthoLng := Point{r3.Vector{0, -1, 0}}
return orthoLng.PointCross(PointFromLatLng(orthoBisector))
// BUG: The major differences from the C++ version are:
// - GetCentroid, Get*Distance, Vertex, InteriorContains(LatLng|Rect|Point)
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