func TestCapExpanded(t *testing.T) {
cap50 := CapFromCenterAngle(xAxisPt, 50.0*s1.Degree)
cap51 := CapFromCenterAngle(xAxisPt, 51.0*s1.Degree)
if !empty.Expanded(s1.Angle(fullHeight)).IsEmpty() {
t.Error("Expanding empty cap should return an empty cap")
}
if !full.Expanded(s1.Angle(fullHeight)).IsFull() {
t.Error("Expanding a full cap should return an full cap")
}
if !cap50.Expanded(0).ApproxEqual(cap50) {
t.Error("Expanding a cap by 0° should be equal to the original")
}
if !cap50.Expanded(1 * s1.Degree).ApproxEqual(cap51) {
t.Error("Expanding 50° by 1° should equal the 51° cap")
}
if cap50.Expanded(129.99 * s1.Degree).IsFull() {
t.Error("Expanding 50° by 129.99° should not give a full cap")
}
if !cap50.Expanded(130.01 * s1.Degree).IsFull() {
t.Error("Expanding 50° by 130.01° should give a full cap")
}
}
func TestCapContainsCell(t *testing.T) {
faceRadius := math.Atan(math.Sqrt2)
for face := 0; face < 6; face++ {
// The cell consisting of the entire face.
rootCell := CellFromCellID(CellIDFromFace(face))
// A leaf cell at the midpoint of the v=1 edge.
edgeCell := CellFromPoint(Point{faceUVToXYZ(face, 0, 1-eps)})
// A leaf cell at the u=1, v=1 corner
cornerCell := CellFromPoint(Point{faceUVToXYZ(face, 1-eps, 1-eps)})
// Quick check for full and empty caps.
if !full.ContainsCell(rootCell) {
t.Errorf("Cap(%v).ContainsCell(%v) = %t; want = %t", full, rootCell, false, true)
}
// Check intersections with the bounding caps of the leaf cells that are adjacent to
// cornerCell along the Hilbert curve. Because this corner is at (u=1,v=1), the curve
// stays locally within the same cube face.
first := cornerCell.id.Advance(-3)
last := cornerCell.id.Advance(4)
for id := first; id < last; id = id.Next() {
c := CellFromCellID(id).CapBound()
if got, want := c.ContainsCell(cornerCell), id == cornerCell.id; got != want {
t.Errorf("Cap(%v).ContainsCell(%v) = %t; want = %t", c, cornerCell, got, want)
}
}
for capFace := 0; capFace < 6; capFace++ {
// A cap that barely contains all of capFace.
center := unitNorm(capFace)
covering := CapFromCenterAngle(center, s1.Angle(faceRadius+eps))
if got, want := covering.ContainsCell(rootCell), capFace == face; got != want {
t.Errorf("Cap(%v).ContainsCell(%v) = %t; want = %t", covering, rootCell, got, want)
}
if got, want := covering.ContainsCell(edgeCell), center.Vector.Dot(edgeCell.id.Point().Vector) > 0.1; got != want {
t.Errorf("Cap(%v).ContainsCell(%v) = %t; want = %t", covering, edgeCell, got, want)
}
if got, want := covering.ContainsCell(edgeCell), covering.IntersectsCell(edgeCell); got != want {
t.Errorf("Cap(%v).ContainsCell(%v) = %t; want = %t", covering, edgeCell, got, want)
}
if got, want := covering.ContainsCell(cornerCell), capFace == face; got != want {
t.Errorf("Cap(%v).ContainsCell(%v) = %t; want = %t", covering, cornerCell, got, want)
}
// A cap that barely intersects the edges of capFace.
bulging := CapFromCenterAngle(center, s1.Angle(math.Pi/4+eps))
if bulging.ContainsCell(rootCell) {
t.Errorf("Cap(%v).ContainsCell(%v) = %t; want = %t", bulging, rootCell, true, false)
}
if got, want := bulging.ContainsCell(edgeCell), capFace == face; got != want {
t.Errorf("Cap(%v).ContainsCell(%v) = %t; want = %t", bulging, edgeCell, got, want)
}
if bulging.ContainsCell(cornerCell) {
t.Errorf("Cap(%v).ContainsCell(%v) = %t; want = %t", bulging, cornerCell, true, false)
}
}
}
}
func TestCapContains(t *testing.T) {
tests := []struct {
c1, c2 Cap
want bool
}{
{empty, empty, true},
{full, empty, true},
{full, full, true},
{empty, xAxis, false},
{full, xAxis, true},
{xAxis, full, false},
{xAxis, xAxis, true},
{xAxis, empty, true},
{hemi, tiny, true},
{hemi, CapFromCenterAngle(xAxisPt, s1.Angle(math.Pi/4-epsilon)), true},
{hemi, CapFromCenterAngle(xAxisPt, s1.Angle(math.Pi/4+epsilon)), false},
{concave, hemi, true},
{concave, CapFromCenterHeight(Point{concave.center.Mul(-1.0)}, 0.1), false},
}
for _, test := range tests {
if got := test.c1.Contains(test.c2); got != test.want {
t.Errorf("%v.Contains(%v) = %t; want %t", test.c1, test.c2, got, test.want)
}
}
}
// Radius returns the cap's radius.
func (c Cap) Radius() s1.Angle {
if c.IsEmpty() {
return s1.Angle(emptyHeight)
}
// This could also be computed as acos(1 - height_), but the following
// formula is much more accurate when the cap height is small. It
// follows from the relationship h = 1 - cos(r) = 2 sin^2(r/2).
return s1.Angle(2 * math.Asin(math.Sqrt(0.5*c.height)))
}
// samplePointFromRect returns a point chosen uniformly at random (with respect
// to area on the sphere) from the given rectangle.
func samplePointFromRect(rect Rect) Point {
// First choose a latitude uniformly with respect to area on the sphere.
sinLo := math.Sin(rect.Lat.Lo)
sinHi := math.Sin(rect.Lat.Hi)
lat := math.Asin(randomUniformFloat64(sinLo, sinHi))
// Now choose longitude uniformly within the given range.
lng := rect.Lng.Lo + randomFloat64()*rect.Lng.Length()
return PointFromLatLng(LatLng{s1.Angle(lat), s1.Angle(lng)}.Normalized())
}
func TestCapAddPoint(t *testing.T) {
tests := []struct {
have Cap
p Point
want Cap
}{
// Cap plus its center equals itself.
{xAxis, xAxisPt, xAxis},
{yAxis, yAxisPt, yAxis},
// Cap plus opposite point equals full.
{xAxis, PointFromCoords(-1, 0, 0), full},
{yAxis, PointFromCoords(0, -1, 0), full},
// Cap plus orthogonal axis equals half cap.
{xAxis, PointFromCoords(0, 0, 1), CapFromCenterAngle(xAxisPt, s1.Angle(math.Pi/2.0))},
{xAxis, PointFromCoords(0, 0, -1), CapFromCenterAngle(xAxisPt, s1.Angle(math.Pi/2.0))},
// The 45 degree angled hemisphere plus some points.
{
hemi,
PointFromCoords(0, 1, -1),
CapFromCenterAngle(Point{PointFromCoords(1, 0, 1).Normalize()},
s1.Angle(120.0)*s1.Degree),
},
{
hemi,
PointFromCoords(0, -1, -1),
CapFromCenterAngle(Point{PointFromCoords(1, 0, 1).Normalize()},
s1.Angle(120.0)*s1.Degree),
},
{
// This angle between this point and the center is acos(-sqrt(2/3))
hemi,
PointFromCoords(-1, -1, -1),
CapFromCenterAngle(Point{PointFromCoords(1, 0, 1).Normalize()},
s1.Angle(2.5261129449194)),
},
{hemi, PointFromCoords(0, 1, 1), hemi},
{hemi, PointFromCoords(1, 0, 0), hemi},
}
for _, test := range tests {
got := test.have.AddPoint(test.p)
if !got.ApproxEqual(test.want) {
t.Errorf("%v.AddPoint(%v) = %v, want %v", test.have, test.p, got, test.want)
}
if !got.ContainsPoint(test.p) {
t.Errorf("%v.AddPoint(%v) did not contain added point", test.have, test.p)
}
}
}
func rectFromDegrees(latLo, lngLo, latHi, lngHi float64) Rect {
// Convenience method to construct a rectangle. This method is
// intentionally *not* in the S2LatLngRect interface because the
// argument order is ambiguous, but is fine for the test.
return Rect{
Lat: r1.Interval{
Lo: (s1.Angle(latLo) * s1.Degree).Radians(),
Hi: (s1.Angle(latHi) * s1.Degree).Radians(),
},
Lng: s1.IntervalFromEndpoints(
(s1.Angle(lngLo) * s1.Degree).Radians(),
(s1.Angle(lngHi) * s1.Degree).Radians(),
),
}
}
func TestCapAddCap(t *testing.T) {
tests := []struct {
have Cap
other Cap
want Cap
}{
// Identity cases.
{empty, empty, empty},
{full, full, full},
// Anything plus empty equals itself.
{full, empty, full},
{empty, full, full},
{xAxis, empty, xAxis},
{empty, xAxis, xAxis},
{yAxis, empty, yAxis},
{empty, yAxis, yAxis},
// Two halves make a whole.
{xAxis, xComp, full},
// Two zero-height orthogonal axis caps make a half-cap.
{xAxis, yAxis, CapFromCenterAngle(xAxisPt, s1.Angle(math.Pi/2.0))},
}
for _, test := range tests {
got := test.have.AddCap(test.other)
if !got.ApproxEqual(test.want) {
t.Errorf("%v.AddCap(%v) = %v, want %v", test.have, test.other, got, test.want)
}
}
}
// CapBound returns a cap that countains 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(PointFromCoords(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
}
// Distance returns the angle between two LatLngs.
func (ll LatLng) Distance(ll2 LatLng) s1.Angle {
// Haversine formula, as used in C++ S2LatLng::GetDistance.
lat1, lat2 := ll.Lat.Radians(), ll2.Lat.Radians()
lng1, lng2 := ll.Lng.Radians(), ll2.Lng.Radians()
dlat := math.Sin(0.5 * (lat2 - lat1))
dlng := math.Sin(0.5 * (lng2 - lng1))
x := dlat*dlat + dlng*dlng*math.Cos(lat1)*math.Cos(lat2)
return s1.Angle(2*math.Atan2(math.Sqrt(x), math.Sqrt(math.Max(0, 1-x)))) * s1.Radian
}
// Normalized returns the normalized version of the LatLng,
// with Lat clamped to [-π/2,π/2] and Lng wrapped in [-π,π].
func (ll LatLng) Normalized() LatLng {
lat := ll.Lat
if lat > northPoleLat {
lat = northPoleLat
} else if lat < southPoleLat {
lat = southPoleLat
}
lng := s1.Angle(math.Remainder(ll.Lng.Radians(), 2*math.Pi)) * s1.Radian
return LatLng{lat, lng}
}
// 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}
}
// 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.
func Interpolate(t float64, a, b Point) Point {
if t == 0 {
return a
}
if t == 1 {
return b
}
ab := a.Angle(b.Vector)
return InterpolateAtDistance(s1.Angle(t)*ab, a, b)
}
func TestInterpolateOverLongEdge(t *testing.T) {
lng := math.Pi - 1e-2
a := Point{PointFromLatLng(LatLng{0, 0}).Normalize()}
b := Point{PointFromLatLng(LatLng{0, s1.Angle(lng)}).Normalize()}
for f := 0.4; f > 1e-15; f *= 0.1 {
// Test that interpolation is accurate on a long edge (but not so long that
// the definition of the edge itself becomes too unstable).
want := Point{PointFromLatLng(LatLng{0, s1.Angle(f * lng)}).Normalize()}
if got := Interpolate(f, a, b); !pointsApproxEquals(got, want, 3e-15) {
t.Errorf("long edge Interpolate(%v, %v, %v) = %v, want %v", f, a, b, got, want)
}
// Test the remainder of the dist also matches.
wantRem := Point{PointFromLatLng(LatLng{0, s1.Angle((1 - f) * lng)}).Normalize()}
if got := Interpolate(1-f, a, b); !pointsApproxEquals(got, wantRem, 3e-15) {
t.Errorf("long edge Interpolate(%v, %v, %v) = %v, want %v", 1-f, a, b, got, wantRem)
}
}
}
func TestRadiusToHeight(t *testing.T) {
tests := []struct {
got s1.Angle
want float64
}{
// Above/below boundary checks.
{s1.Angle(-0.5), emptyHeight},
{s1.Angle(0), 0},
{s1.Angle(math.Pi), fullHeight},
{s1.Angle(2 * math.Pi), fullHeight},
// Degree tests.
{-7.0 * s1.Degree, emptyHeight},
{-0.0 * s1.Degree, 0},
{0.0 * s1.Degree, 0},
{12.0 * s1.Degree, 0.02185239926619},
{30.0 * s1.Degree, 0.13397459621556},
{45.0 * s1.Degree, 0.29289321881345},
{90.0 * s1.Degree, 1.0},
{179.99 * s1.Degree, 1.99999998476912},
{180.0 * s1.Degree, fullHeight},
{270.0 * s1.Degree, fullHeight},
// Radians tests.
{-1.0 * s1.Radian, emptyHeight},
{-0.0 * s1.Radian, 0},
{0.0 * s1.Radian, 0},
{1.0 * s1.Radian, 0.45969769413186},
{math.Pi / 2.0 * s1.Radian, 1.0},
{2.0 * s1.Radian, 1.41614683654714},
{3.0 * s1.Radian, 1.98999249660044},
{math.Pi * s1.Radian, fullHeight},
{4.0 * s1.Radian, fullHeight},
}
for _, test := range tests {
// float64Eq comes from s2latlng_test.go
if got := radiusToHeight(test.got); !float64Eq(got, test.want) {
t.Errorf("radiusToHeight(%v) = %v; want %v", test.got, got, test.want)
}
}
}
func TestContainsPointConsistentWithS2CellIDFromPoint(t *testing.T) {
// Construct many points that are nearly on a Cell edge, and verify that
// CellFromCellID(cellIDFromPoint(p)).Contains(p) is always true.
for iter := 0; iter < 1000; iter++ {
cell := CellFromCellID(randomCellID())
i1 := randomUniformInt(4)
i2 := (i1 + 1) & 3
v1 := cell.Vertex(i1)
v2 := samplePointFromCap(CapFromCenterAngle(cell.Vertex(i2), s1.Angle(epsilon)))
p := Interpolate(randomFloat64(), v1, v2)
if !CellFromCellID(cellIDFromPoint(p)).ContainsPoint(p) {
t.Errorf("For p=%v, CellFromCellID(cellIDFromPoint(p)).ContainsPoint(p) was false", p)
}
}
}
func TestRectCapBound(t *testing.T) {
tests := []struct {
r Rect
want Cap
}{
{ // Bounding cap at center is smaller.
rectFromDegrees(-45, -45, 45, 45),
CapFromCenterHeight(PointFromCoords(1, 0, 0), 0.5),
},
{ // Bounding cap at north pole is smaller.
rectFromDegrees(88, -80, 89, 80),
CapFromCenterAngle(PointFromCoords(0, 0, 1), s1.Angle(2)*s1.Degree),
},
{ // Longitude span > 180 degrees.
rectFromDegrees(-30, -150, -10, 50),
CapFromCenterAngle(PointFromCoords(0, 0, -1), s1.Angle(80)*s1.Degree),
},
}
for _, test := range tests {
if got := test.r.CapBound(); !test.want.ApproxEqual(got) {
t.Errorf("%v.CapBound() was %v, want %v", test.r, got, test.want)
}
}
}
// perturbATowardsB returns a point that has been shifted some distance towards the
// second point based on a random number.
func perturbATowardsB(a, b Point) Point {
choice := randomFloat64()
if choice < 0.1 {
return a
}
if choice < 0.3 {
// Return a point that is exactly proportional to A and that still
// satisfies IsUnitLength().
for {
b := Point{a.Mul(2 - a.Norm() + 5*(randomFloat64()-0.5)*dblEpsilon)}
if !b.ApproxEqual(a) && b.IsUnit() {
return b
}
}
}
if choice < 0.5 {
// Return a point such that the distance squared to A will underflow.
return InterpolateAtDistance(1e-300, a, b)
}
// Otherwise return a point whose distance from A is near dblEpsilon such
// that the log of the pdf is uniformly distributed.
distance := dblEpsilon * 1e-5 * math.Pow(1e6, randomFloat64())
return InterpolateAtDistance(s1.Angle(distance), a, b)
}
func latitude(p Point) s1.Angle {
return s1.Angle(math.Atan2(p.Z, math.Sqrt(p.X*p.X+p.Y*p.Y))) * s1.Radian
}
func TestExpandForSubregions(t *testing.T) {
// Test the full and empty bounds.
if !ExpandForSubregions(FullRect()).IsFull() {
t.Errorf("Subregion Bound of full rect should be full")
}
if !ExpandForSubregions(EmptyRect()).IsEmpty() {
t.Errorf("Subregion Bound of empty rect should be empty")
}
tests := []struct {
xLat, xLng, yLat, yLng float64
wantFull bool
}{
// Cases where the bound does not straddle the equator (but almost does),
// and spans nearly 180 degrees in longitude.
{3e-16, 0, 1e-14, math.Pi, true},
{9e-16, 0, 1e-14, math.Pi, false},
{1e-16, 7e-16, 1e-14, math.Pi, true},
{3e-16, 14e-16, 1e-14, math.Pi, false},
{1e-100, 14e-16, 1e-14, math.Pi, true},
{1e-100, 22e-16, 1e-14, math.Pi, false},
// Cases where the bound spans at most 90 degrees in longitude, and almost
// 180 degrees in latitude. Note that DBL_EPSILON is about 2.22e-16, which
// implies that the double-precision value just below Pi/2 can be written as
// (math.Pi/2 - 2e-16).
{-math.Pi / 2, -1e-15, math.Pi/2 - 7e-16, 0, true},
{-math.Pi / 2, -1e-15, math.Pi/2 - 30e-16, 0, false},
{-math.Pi/2 + 4e-16, 0, math.Pi/2 - 2e-16, 1e-7, true},
{-math.Pi/2 + 30e-16, 0, math.Pi / 2, 1e-7, false},
{-math.Pi/2 + 4e-16, 0, math.Pi/2 - 4e-16, math.Pi / 2, true},
{-math.Pi / 2, 0, math.Pi/2 - 30e-16, math.Pi / 2, false},
// Cases where the bound straddles the equator and spans more than 90
// degrees in longitude. These are the cases where the critical distance is
// between a corner of the bound and the opposite longitudinal edge. Unlike
// the cases above, here the bound may contain nearly-antipodal points (to
// within 3.055 * DBL_EPSILON) even though the latitude and longitude ranges
// are both significantly less than (math.Pi - 3.055 * DBL_EPSILON).
{-math.Pi / 2, 0, math.Pi/2 - 1e-8, math.Pi - 1e-7, true},
{-math.Pi / 2, 0, math.Pi/2 - 1e-7, math.Pi - 1e-7, false},
{-math.Pi/2 + 1e-12, -math.Pi + 1e-4, math.Pi / 2, 0, true},
{-math.Pi/2 + 1e-11, -math.Pi + 1e-4, math.Pi / 2, 0, true},
}
for _, tc := range tests {
in := RectFromLatLng(LatLng{s1.Angle(tc.xLat), s1.Angle(tc.xLng)})
in = in.AddPoint(LatLng{s1.Angle(tc.yLat), s1.Angle(tc.yLng)})
got := ExpandForSubregions(in)
// Test that the bound is actually expanded.
if !got.Contains(in) {
t.Errorf("Subregion bound of (%f, %f, %f, %f) should contain original rect", tc.xLat, tc.xLng, tc.yLat, tc.yLng)
}
if in.Lat == validRectLatRange && in.Lat.ContainsInterval(got.Lat) {
t.Errorf("Subregion bound of (%f, %f, %f, %f) shouldn't be contained by original rect", tc.xLat, tc.xLng, tc.yLat, tc.yLng)
}
// We check the various situations where the bound contains nearly-antipodal points. The tests are organized into pairs
// where the two bounds are similar except that the first bound meets the nearly-antipodal criteria while the second does not.
if got.IsFull() != tc.wantFull {
t.Errorf("Subregion Bound of (%f, %f, %f, %f).IsFull should be %t", tc.xLat, tc.xLng, tc.yLat, tc.yLng, tc.wantFull)
}
}
rectTests := []struct {
xLat, xLng, yLat, yLng float64
wantRect Rect
}{
{1.5, -math.Pi / 2, 1.5, math.Pi/2 - 2e-16, Rect{r1.Interval{1.5, 1.5}, s1.FullInterval()}},
{1.5, -math.Pi / 2, 1.5, math.Pi/2 - 7e-16, Rect{r1.Interval{1.5, 1.5}, s1.Interval{-math.Pi / 2, math.Pi/2 - 7e-16}}},
// Check for cases where the bound is expanded to include one of the poles
{-math.Pi/2 + 1e-15, 0, -math.Pi/2 + 1e-15, 0, Rect{r1.Interval{-math.Pi / 2, -math.Pi/2 + 1e-15}, s1.FullInterval()}},
{math.Pi/2 - 1e-15, 0, math.Pi/2 - 1e-15, 0, Rect{r1.Interval{math.Pi/2 - 1e-15, math.Pi / 2}, s1.FullInterval()}},
}
for _, tc := range rectTests {
// Now we test cases where the bound does not contain nearly-antipodal
// points, but it does contain points that are approximately 180 degrees
// apart in latitude.
in := RectFromLatLng(LatLng{s1.Angle(tc.xLat), s1.Angle(tc.xLng)})
in = in.AddPoint(LatLng{s1.Angle(tc.yLat), s1.Angle(tc.yLng)})
got := ExpandForSubregions(in)
if !rectsApproxEqual(got, tc.wantRect, rectErrorLat, rectErrorLng) {
t.Errorf("Subregion Bound of (%f, %f, %f, %f) = (%v) should be %v", tc.xLat, tc.xLng, tc.yLat, tc.yLng, got, tc.wantRect)
}
}
}
// 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}
}
// EarthAngle converts a to distance on earth in meters to an angle
func EarthAngle(dist float64) s1.Angle {
return s1.Angle(dist / EarthRadiusMeters)
}
// kmToAngle converts a distance on the Earth's surface to an angle.
func kmToAngle(km float64) s1.Angle {
// The Earth's mean radius in kilometers (according to NASA).
const earthRadiusKm = 6371.01
return s1.Angle(km / earthRadiusKm)
}
// RectBound returns the bounding rectangle of this cell.
func (c Cell) RectBound() Rect {
if c.level > 0 {
// Except for cells at level 0, the latitude and longitude extremes are
// attained at the vertices. Furthermore, the latitude range is
// determined by one pair of diagonally opposite vertices and the
// longitude range is determined by the other pair.
//
// We first determine which corner (i,j) of the cell has the largest
// absolute latitude. To maximize latitude, we want to find the point in
// the cell that has the largest absolute z-coordinate and the smallest
// absolute x- and y-coordinates. To do this we look at each coordinate
// (u and v), and determine whether we want to minimize or maximize that
// coordinate based on the axis direction and the cell's (u,v) quadrant.
u := c.uv.X.Lo + c.uv.X.Hi
v := c.uv.Y.Lo + c.uv.Y.Hi
var i, j int
if uAxis(int(c.face)).Z == 0 {
if u < 0 {
i = 1
}
} else if u > 0 {
i = 1
}
if vAxis(int(c.face)).Z == 0 {
if v < 0 {
j = 1
}
} else if v > 0 {
j = 1
}
lat := r1.IntervalFromPoint(c.latitude(i, j)).AddPoint(c.latitude(1-i, 1-j))
lng := s1.EmptyInterval().AddPoint(c.longitude(i, 1-j)).AddPoint(c.longitude(1-i, j))
// We grow the bounds slightly to make sure that the bounding rectangle
// contains LatLngFromPoint(P) for any point P inside the loop L defined by the
// four *normalized* vertices. Note that normalization of a vector can
// change its direction by up to 0.5 * dblEpsilon radians, and it is not
// enough just to add Normalize calls to the code above because the
// latitude/longitude ranges are not necessarily determined by diagonally
// opposite vertex pairs after normalization.
//
// We would like to bound the amount by which the latitude/longitude of a
// contained point P can exceed the bounds computed above. In the case of
// longitude, the normalization error can change the direction of rounding
// leading to a maximum difference in longitude of 2 * dblEpsilon. In
// the case of latitude, the normalization error can shift the latitude by
// up to 0.5 * dblEpsilon and the other sources of error can cause the
// two latitudes to differ by up to another 1.5 * dblEpsilon, which also
// leads to a maximum difference of 2 * dblEpsilon.
return Rect{lat, lng}.expanded(LatLng{s1.Angle(2 * dblEpsilon), s1.Angle(2 * dblEpsilon)}).PolarClosure()
}
// The 4 cells around the equator extend to +/-45 degrees latitude at the
// midpoints of their top and bottom edges. The two cells covering the
// poles extend down to +/-35.26 degrees at their vertices. The maximum
// error in this calculation is 0.5 * dblEpsilon.
var bound Rect
switch c.face {
case 0:
bound = Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{-math.Pi / 4, math.Pi / 4}}
case 1:
bound = Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{math.Pi / 4, 3 * math.Pi / 4}}
case 2:
bound = Rect{r1.Interval{poleMinLat, math.Pi / 2}, s1.FullInterval()}
case 3:
bound = Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{3 * math.Pi / 4, -3 * math.Pi / 4}}
case 4:
bound = Rect{r1.Interval{-math.Pi / 4, math.Pi / 4}, s1.Interval{-3 * math.Pi / 4, -math.Pi / 4}}
default:
bound = Rect{r1.Interval{-math.Pi / 2, -poleMinLat}, s1.FullInterval()}
}
// Finally, we expand the bound to account for the error when a point P is
// converted to an LatLng to test for containment. (The bound should be
// large enough so that it contains the computed LatLng of any contained
// point, not just the infinite-precision version.) We don't need to expand
// longitude because longitude is calculated via a single call to math.Atan2,
// which is guaranteed to be semi-monotonic.
return bound.expanded(LatLng{s1.Angle(dblEpsilon), s1.Angle(0)})
}
func TestCapGetRectBounds(t *testing.T) {
const epsilon = 1e-13
var tests = []struct {
desc string
have Cap
latLoDeg float64
latHiDeg float64
lngLoDeg float64
lngHiDeg float64
isFull bool
}{
{
"Cap that includes South Pole.",
CapFromCenterAngle(PointFromLatLng(LatLngFromDegrees(-45, 57)), s1.Degree*50),
-90, 5, -180, 180, true,
},
{
"Cap that is tangent to the North Pole.",
CapFromCenterAngle(PointFromCoords(1, 0, 1), s1.Radian*(math.Pi/4.0+1e-16)),
0, 90, -180, 180, true,
},
{
"Cap that at 45 degree center that goes from equator to the pole.",
CapFromCenterAngle(PointFromCoords(1, 0, 1), s1.Degree*(45+5e-15)),
0, 90, -180, 180, true,
},
{
"The eastern hemisphere.",
CapFromCenterAngle(PointFromCoords(0, 1, 0), s1.Radian*(math.Pi/2+2e-16)),
-90, 90, -180, 180, true,
},
{
"A cap centered on the equator.",
CapFromCenterAngle(PointFromLatLng(LatLngFromDegrees(0, 50)), s1.Degree*20),
-20, 20, 30, 70, false,
},
{
"A cap centered on the North Pole.",
CapFromCenterAngle(PointFromLatLng(LatLngFromDegrees(90, 123)), s1.Degree*10),
80, 90, -180, 180, true,
},
}
for _, test := range tests {
r := test.have.RectBound()
if !float64Near(s1.Angle(r.Lat.Lo).Degrees(), test.latLoDeg, epsilon) {
t.Errorf("%s: %v.RectBound(), Lat.Lo not close enough, got %0.20f, want %0.20f",
test.desc, test.have, s1.Angle(r.Lat.Lo).Degrees(), test.latLoDeg)
}
if !float64Near(s1.Angle(r.Lat.Hi).Degrees(), test.latHiDeg, epsilon) {
t.Errorf("%s: %v.RectBound(), Lat.Hi not close enough, got %0.20f, want %0.20f",
test.desc, test.have, s1.Angle(r.Lat.Hi).Degrees(), test.latHiDeg)
}
if !float64Near(s1.Angle(r.Lng.Lo).Degrees(), test.lngLoDeg, epsilon) {
t.Errorf("%s: %v.RectBound(), Lng.Lo not close enough, got %0.20f, want %0.20f",
test.desc, test.have, s1.Angle(r.Lng.Lo).Degrees(), test.lngLoDeg)
}
if !float64Near(s1.Angle(r.Lng.Hi).Degrees(), test.lngHiDeg, epsilon) {
t.Errorf("%s: %v.RectBound(), Lng.Hi not close enough, got %0.20f, want %0.20f",
test.desc, test.have, s1.Angle(r.Lng.Hi).Degrees(), test.lngHiDeg)
}
if got := r.Lng.IsFull(); got != test.isFull {
t.Errorf("%s: RectBound(%v).isFull() = %t, want %t", test.desc, test.have, got, test.isFull)
}
}
// Empty and full caps.
if !EmptyCap().RectBound().IsEmpty() {
t.Errorf("RectBound() on EmptyCap should be empty.")
}
if !FullCap().RectBound().IsFull() {
t.Errorf("RectBound() on FullCap should be full.")
}
}
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
package s2
import (
"fmt"
"math"
"github.com/golang/geo/s1"
)
const (
northPoleLat = s1.Angle(math.Pi/2) * s1.Radian
southPoleLat = -northPoleLat
)
// LatLng represents a point on the unit sphere as a pair of angles.
type LatLng struct {
Lat, Lng s1.Angle
}
// LatLngFromDegrees returns a LatLng for the coordinates given in degrees.
func LatLngFromDegrees(lat, lng float64) LatLng {
return LatLng{s1.Angle(lat) * s1.Degree, s1.Angle(lng) * s1.Degree}
}
// IsValid returns true iff the LatLng is normalized, with Lat ∈ [-π/2,π/2] and Lng ∈ [-π,π].
func (ll LatLng) IsValid() bool {
// LatLngFromDegrees returns a LatLng for the coordinates given in degrees.
func LatLngFromDegrees(lat, lng float64) LatLng {
return LatLng{s1.Angle(lat) * s1.Degree, s1.Angle(lng) * s1.Degree}
}
var (
empty = EmptyCap()
full = FullCap()
defaultCap = EmptyCap()
xAxisPt = PointFromCoords(1, 0, 0)
yAxisPt = PointFromCoords(0, 1, 0)
xAxis = CapFromPoint(xAxisPt)
yAxis = CapFromPoint(yAxisPt)
xComp = xAxis.Complement()
hemi = CapFromCenterHeight(Point{PointFromCoords(1, 0, 1).Normalize()}, 1)
concave = CapFromCenterAngle(PointFromLatLng(LatLngFromDegrees(80, 10)),
s1.Angle(150.0)*s1.Degree)
tiny = CapFromCenterAngle(Point{PointFromCoords(1, 2, 3).Normalize()},
s1.Angle(tinyRad))
)
func TestCapBasicEmptyFullValid(t *testing.T) {
tests := []struct {
got Cap
empty, full, valid bool
}{
{Cap{}, false, false, false},
{empty, true, false, true},
{empty.Complement(), false, true, true},
{full, false, true, true},
{full.Complement(), true, false, true},
// Angle returns the angle between v and ov.
func (v Vector) Angle(ov Vector) s1.Angle {
return s1.Angle(math.Atan2(v.Cross(ov).Norm(), v.Dot(ov))) * s1.Radian
}
func longitude(p Point) s1.Angle {
return s1.Angle(math.Atan2(p.Y, p.X)) * s1.Radian
}