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gm_test.go
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gm_test.go
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package gm
import (
"math"
"testing"
"github.com/golang/geo/r2"
"github.com/golang/geo/r3"
"github.com/golang/geo/s2"
)
const (
pi = math.Pi
sqrt2 = math.Sqrt2
sqrt3 = 1.73205080756887729352744634150587236694280525381038062805580698 // https://oeis.org/A002194
)
func TestNew(t *testing.T) {
for _, test := range []struct {
p, n s2.LatLng
gm *GeneralizedMercator
}{
// Antipodes on the great circle of longitude ±90° (X == 0)
{
p: s2.LatLng{Lat: pi / 2, Lng: 0},
n: s2.LatLng{Lat: -pi / 2, Lng: 0},
gm: &GeneralizedMercator{
pos: r3.Vector{0, 0, 1},
neg: r3.Vector{0, 0, -1},
i: r3.Vector{1, 0, 0},
j: r3.Vector{0, 1, 0},
k: r3.Vector{0, 0, 1},
d: math.Inf(1),
},
},
{
p: s2.LatLng{Lat: -pi / 2, Lng: 0},
n: s2.LatLng{Lat: pi / 2, Lng: 0},
gm: &GeneralizedMercator{
pos: r3.Vector{0, 0, -1},
neg: r3.Vector{0, 0, 1},
i: r3.Vector{1, 0, 0},
j: r3.Vector{0, -1, 0},
k: r3.Vector{0, 0, -1},
d: math.Inf(1),
},
},
{
p: s2.LatLng{Lat: 0, Lng: pi / 2},
n: s2.LatLng{Lat: 0, Lng: -pi / 2},
gm: &GeneralizedMercator{
pos: r3.Vector{0, 1, 0},
neg: r3.Vector{0, -1, 0},
i: r3.Vector{1, 0, 0},
j: r3.Vector{0, 0, -1},
k: r3.Vector{0, 1, 0},
d: math.Inf(1),
},
},
{
p: s2.LatLng{Lat: 0, Lng: -pi / 2},
n: s2.LatLng{Lat: 0, Lng: pi / 2},
gm: &GeneralizedMercator{
pos: r3.Vector{0, -1, 0},
neg: r3.Vector{0, 1, 0},
i: r3.Vector{1, 0, 0},
j: r3.Vector{0, 0, 1},
k: r3.Vector{0, -1, 0},
d: math.Inf(1),
},
},
{
p: s2.LatLng{Lat: pi / 4, Lng: -pi / 2},
n: s2.LatLng{Lat: -pi / 4, Lng: pi / 2},
gm: &GeneralizedMercator{
pos: r3.Vector{0, -sqrt2 / 2, sqrt2 / 2},
neg: r3.Vector{0, sqrt2 / 2, -sqrt2 / 2},
i: r3.Vector{1, 0, 0},
j: r3.Vector{0, sqrt2 / 2, sqrt2 / 2},
k: r3.Vector{0, -sqrt2 / 2, sqrt2 / 2},
d: math.Inf(1),
},
},
// Antipodes elsewhere on the Equator (X != 0, Z == 0)
{
p: s2.LatLng{Lat: 0, Lng: 0},
n: s2.LatLng{Lat: 0, Lng: pi},
gm: &GeneralizedMercator{
pos: r3.Vector{1, 0, 0},
neg: r3.Vector{-1, 0, 0},
i: r3.Vector{0, 0, 1},
j: r3.Vector{0, -1, 0},
k: r3.Vector{1, 0, 0},
d: math.Inf(1),
},
},
{
p: s2.LatLng{Lat: 0, Lng: pi},
n: s2.LatLng{Lat: 0, Lng: 0},
gm: &GeneralizedMercator{
pos: r3.Vector{-1, 0, 0},
neg: r3.Vector{1, 0, 0},
i: r3.Vector{0, 0, 1},
j: r3.Vector{0, 1, 0},
k: r3.Vector{-1, 0, 0},
d: math.Inf(1),
},
},
{
p: s2.LatLng{Lat: 0, Lng: pi / 4},
n: s2.LatLng{Lat: 0, Lng: -3 * pi / 4},
gm: &GeneralizedMercator{
pos: r3.Vector{sqrt2 / 2, sqrt2 / 2, 0},
neg: r3.Vector{-sqrt2 / 2, -sqrt2 / 2, 0},
i: r3.Vector{0, 0, 1},
j: r3.Vector{sqrt2 / 2, -sqrt2 / 2, 0},
k: r3.Vector{sqrt2 / 2, sqrt2 / 2, 0},
d: math.Inf(1),
},
},
// Other antipodes (X != 0, Z != 0)
{
p: s2.LatLng{Lat: pi / 4, Lng: 0},
n: s2.LatLng{Lat: -pi / 4, Lng: pi},
gm: &GeneralizedMercator{
pos: r3.Vector{sqrt2 / 2, 0, sqrt2 / 2},
neg: r3.Vector{-sqrt2 / 2, 0, -sqrt2 / 2},
i: r3.Vector{sqrt2 / 2, 0, -sqrt2 / 2},
j: r3.Vector{0, 1, 0},
k: r3.Vector{sqrt2 / 2, 0, sqrt2 / 2},
d: math.Inf(1),
},
},
{
p: s2.LatLng{Lat: -pi / 4, Lng: 0},
n: s2.LatLng{Lat: pi / 4, Lng: pi},
gm: &GeneralizedMercator{
pos: r3.Vector{sqrt2 / 2, 0, -sqrt2 / 2},
neg: r3.Vector{-sqrt2 / 2, 0, sqrt2 / 2},
i: r3.Vector{sqrt2 / 2, 0, sqrt2 / 2},
j: r3.Vector{0, -1, 0},
k: r3.Vector{sqrt2 / 2, 0, -sqrt2 / 2},
d: math.Inf(1),
},
},
{
p: s2.LatLng{Lat: pi / 4, Lng: pi / 4},
n: s2.LatLng{Lat: -pi / 4, Lng: -3 * pi / 4},
gm: &GeneralizedMercator{
pos: r3.Vector{0.5, 0.5, sqrt2 / 2},
neg: r3.Vector{-0.5, -0.5, -sqrt2 / 2},
i: r3.Vector{sqrt2 / sqrt3, 0, -1 / sqrt3},
j: r3.Vector{-1 / (2 * sqrt3), sqrt3 / 2, -1 / (sqrt2 * sqrt3)},
k: r3.Vector{0.5, 0.5, sqrt2 / 2},
d: math.Inf(1),
},
},
// Non-antipodes
{
p: s2.LatLng{Lat: pi / 3, Lng: 0},
n: s2.LatLng{Lat: -pi / 3, Lng: 0},
gm: &GeneralizedMercator{
pos: r3.Vector{0.5, 0, sqrt3 / 2},
neg: r3.Vector{0.5, 0, -sqrt3 / 2},
i: r3.Vector{1, 0, 0},
j: r3.Vector{0, 1, 0},
k: r3.Vector{0, 0, 1},
d: 2,
},
},
{
p: s2.LatLng{Lat: pi / 3, Lng: pi / 4},
n: s2.LatLng{Lat: -pi / 3, Lng: pi / 4},
gm: &GeneralizedMercator{
pos: r3.Vector{sqrt2 / 4, sqrt2 / 4, sqrt3 / 2},
neg: r3.Vector{sqrt2 / 4, sqrt2 / 4, -sqrt3 / 2},
i: r3.Vector{sqrt2 / 2, sqrt2 / 2, 0},
j: r3.Vector{-sqrt2 / 2, sqrt2 / 2, 0},
k: r3.Vector{0, 0, 1},
d: 2,
},
},
{
p: s2.LatLng{Lat: pi / 2, Lng: 0},
n: s2.LatLng{Lat: -pi / 6, Lng: 2 * pi / 3},
gm: &GeneralizedMercator{
pos: r3.Vector{0, 0, 1},
neg: r3.Vector{-sqrt3 / 4, 0.75, -0.5},
i: r3.Vector{-sqrt3 / 4, 0.75, 0.5},
j: r3.Vector{-sqrt3 / 2, -0.5, 0},
k: r3.Vector{0.25, -sqrt3 / 4, sqrt3 / 2},
d: 2,
},
},
{
p: s2.LatLng{Lat: -pi / 4, Lng: 0},
n: s2.LatLng{Lat: -pi / 4, Lng: pi},
gm: &GeneralizedMercator{
pos: r3.Vector{sqrt2 / 2, 0, -sqrt2 / 2},
neg: r3.Vector{-sqrt2 / 2, 0, -sqrt2 / 2},
i: r3.Vector{0, 0, -1},
j: r3.Vector{0, 1, 0},
k: r3.Vector{1, 0, 0},
d: sqrt2,
},
},
} {
if gm := New(test.p, test.n); !gmApproxEqual(gm, test.gm) {
t.Errorf("New(%v, %v): got %+v, want %+v", test.p, test.n, gm, test.gm)
}
}
}
type proj struct {
s s2.LatLng
r r2.Point
}
var projTests = []struct {
gm *GeneralizedMercator
ps []proj
}{
{
gm: &GeneralizedMercator{
pos: r3.Vector{0, 0, 1},
neg: r3.Vector{0, 0, -1},
i: r3.Vector{1, 0, 0},
j: r3.Vector{0, 1, 0},
k: r3.Vector{0, 0, 1},
d: math.Inf(1),
},
ps: []proj{
{s2.LatLng{Lat: pi / 2}, r2.Point{Y: math.Inf(1)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p0(0.5, math.Inf(1)).X, Y: p0(0.5, math.Inf(1)).Y, Z: p0(0.5, math.Inf(1)).Z}}), r2.Point{X: 0, Y: math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p1(0.5, math.Inf(1)).X, Y: p1(0.5, math.Inf(1)).Y, Z: p1(0.5, math.Inf(1)).Z}}), r2.Point{X: pi / 2, Y: math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p2(0.5, math.Inf(1)).X, Y: p2(0.5, math.Inf(1)).Y, Z: p2(0.5, math.Inf(1)).Z}}), r2.Point{X: pi, Y: math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p3(0.5, math.Inf(1)).X, Y: p3(0.5, math.Inf(1)).Y, Z: p3(0.5, math.Inf(1)).Z}}), r2.Point{X: -pi / 2, Y: math.Log(sqrt3)}},
{s2.LatLng{Lat: 0, Lng: 0}, r2.Point{X: 0, Y: 0}},
{s2.LatLng{Lat: 0, Lng: pi / 2}, r2.Point{X: pi / 2, Y: 0}},
{s2.LatLng{Lat: 0, Lng: pi}, r2.Point{X: pi, Y: 0}},
{s2.LatLng{Lat: 0, Lng: -pi / 2}, r2.Point{X: -pi / 2, Y: 0}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p0(-0.5, math.Inf(1)).X, Y: p0(-0.5, math.Inf(1)).Y, Z: p0(-0.5, math.Inf(1)).Z}}), r2.Point{X: 0, Y: -math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p1(-0.5, math.Inf(1)).X, Y: p1(-0.5, math.Inf(1)).Y, Z: p1(-0.5, math.Inf(1)).Z}}), r2.Point{X: pi / 2, Y: -math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p2(-0.5, math.Inf(1)).X, Y: p2(-0.5, math.Inf(1)).Y, Z: p2(-0.5, math.Inf(1)).Z}}), r2.Point{X: pi, Y: -math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p3(-0.5, math.Inf(1)).X, Y: p3(-0.5, math.Inf(1)).Y, Z: p3(-0.5, math.Inf(1)).Z}}), r2.Point{X: -pi / 2, Y: -math.Log(sqrt3)}},
{s2.LatLng{Lat: -pi / 2}, r2.Point{Y: math.Inf(-1)}},
},
},
{
gm: &GeneralizedMercator{
pos: r3.Vector{0.5, -sqrt3 / 2, 0},
neg: r3.Vector{0.5, sqrt3 / 2, 0},
i: r3.Vector{1, 0, 0},
j: r3.Vector{0, 0, 1},
k: r3.Vector{0, -1, 0},
d: 2,
},
ps: []proj{
{s2.LatLng{Lat: 0, Lng: -pi / 3}, r2.Point{Y: math.Inf(1)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p0(0.5, 2).X, Y: -p0(0.5, 2).Z, Z: p0(0.5, 2).Y}}), r2.Point{X: 0, Y: math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p1(0.5, 2).X, Y: -p1(0.5, 2).Z, Z: p1(0.5, 2).Y}}), r2.Point{X: pi / 2, Y: math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p2(0.5, 2).X, Y: -p2(0.5, 2).Z, Z: p2(0.5, 2).Y}}), r2.Point{X: pi, Y: math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p3(0.5, 2).X, Y: -p3(0.5, 2).Z, Z: p3(0.5, 2).Y}}), r2.Point{X: -pi / 2, Y: math.Log(sqrt3)}},
{s2.LatLng{Lat: 0, Lng: 0}, r2.Point{X: 0, Y: 0}},
{s2.LatLng{Lat: pi / 2}, r2.Point{X: pi / 2, Y: 0}},
{s2.LatLng{Lat: 0, Lng: pi}, r2.Point{X: pi, Y: 0}},
{s2.LatLng{Lat: -pi / 2}, r2.Point{X: -pi / 2, Y: 0}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p0(-0.5, 2).X, Y: -p0(-0.5, 2).Z, Z: p0(-0.5, 2).Y}}), r2.Point{X: 0, Y: -math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p1(-0.5, 2).X, Y: -p1(-0.5, 2).Z, Z: p1(-0.5, 2).Y}}), r2.Point{X: pi / 2, Y: -math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p2(-0.5, 2).X, Y: -p2(-0.5, 2).Z, Z: p2(-0.5, 2).Y}}), r2.Point{X: pi, Y: -math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p3(-0.5, 2).X, Y: -p3(-0.5, 2).Z, Z: p3(-0.5, 2).Y}}), r2.Point{X: -pi / 2, Y: -math.Log(sqrt3)}},
{s2.LatLng{Lat: 0, Lng: pi / 3}, r2.Point{Y: math.Inf(-1)}},
},
},
{
gm: &GeneralizedMercator{
pos: r3.Vector{sqrt2 / 2, 0, -sqrt2 / 2},
neg: r3.Vector{-sqrt2 / 2, 0, -sqrt2 / 2},
i: r3.Vector{0, 0, -1},
j: r3.Vector{0, 1, 0},
k: r3.Vector{1, 0, 0},
d: sqrt2,
},
ps: []proj{
{s2.LatLng{Lat: -pi / 4, Lng: 0}, r2.Point{Y: math.Inf(1)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p0(0.5, sqrt2).Z, Y: p0(0.5, sqrt2).Y, Z: -p0(0.5, sqrt2).X}}), r2.Point{X: 0, Y: math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p1(0.5, sqrt2).Z, Y: p1(0.5, sqrt2).Y, Z: -p1(0.5, sqrt2).X}}), r2.Point{X: pi / 2, Y: math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p2(0.5, sqrt2).Z, Y: p2(0.5, sqrt2).Y, Z: -p2(0.5, sqrt2).X}}), r2.Point{X: pi, Y: math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p3(0.5, sqrt2).Z, Y: p3(0.5, sqrt2).Y, Z: -p3(0.5, sqrt2).X}}), r2.Point{X: -pi / 2, Y: math.Log(sqrt3)}},
{s2.LatLng{Lat: -pi / 2, Lng: 0}, r2.Point{X: 0, Y: 0}},
{s2.LatLng{Lat: 0, Lng: pi / 2}, r2.Point{X: pi / 2, Y: 0}},
{s2.LatLng{Lat: pi / 2, Lng: 0}, r2.Point{X: pi, Y: 0}},
{s2.LatLng{Lat: 0, Lng: -pi / 2}, r2.Point{X: -pi / 2, Y: 0}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p0(-0.5, sqrt2).Z, Y: p0(-0.5, sqrt2).Y, Z: -p0(-0.5, sqrt2).X}}), r2.Point{X: 0, Y: -math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p1(-0.5, sqrt2).Z, Y: p1(-0.5, sqrt2).Y, Z: -p1(-0.5, sqrt2).X}}), r2.Point{X: pi / 2, Y: -math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p2(-0.5, sqrt2).Z, Y: p2(-0.5, sqrt2).Y, Z: -p2(-0.5, sqrt2).X}}), r2.Point{X: pi, Y: -math.Log(sqrt3)}},
{s2.LatLngFromPoint(s2.Point{r3.Vector{X: p3(-0.5, sqrt2).Z, Y: p3(-0.5, sqrt2).Y, Z: -p3(-0.5, sqrt2).X}}), r2.Point{X: -pi / 2, Y: -math.Log(sqrt3)}},
{s2.LatLng{Lat: -pi / 4, Lng: pi}, r2.Point{Y: math.Inf(-1)}},
},
},
}
func TestProject(t *testing.T) {
for _, test := range projTests {
for _, p := range test.ps {
if got := test.gm.Project(p.s); !ptApproxEqual(got, p.r) {
t.Errorf("Project(%+v, %+v): got %+v, want %+v", test.gm, p.s, got, p.r)
}
}
}
}
func TestUnproject(t *testing.T) {
for _, test := range projTests {
for _, p := range test.ps {
if got := test.gm.Unproject(p.r); !llApproxEqual(got, p.s) {
t.Errorf("Unproject(%+v, %+v): got %+v, want %+v", test.gm, p.r, got, p.s)
}
}
}
}
func ptApproxEqual(a, b r2.Point) bool {
return (a.X == b.X || math.Abs(a.X-b.X) < 1e-15) && (a.Y == b.Y || math.Abs(a.Y-b.Y) < 1e-15)
}
func llApproxEqual(a, b s2.LatLng) bool {
return approxEqual(s2.PointFromLatLng(a).Vector, s2.PointFromLatLng(b).Vector)
}
func gmApproxEqual(a, b *GeneralizedMercator) bool {
return approxEqual(a.pos, b.pos) &&
approxEqual(a.neg, b.neg) &&
approxEqual(a.i, b.i) &&
approxEqual(a.j, b.j) &&
approxEqual(a.k, b.k) &&
(a.d == b.d || math.Abs(a.d-b.d) < 1e-15)
}
// p0, p1, p2, and p3 describe points on the circular intersection of the unit sphere
// with the plane containing x == d that lies at a signed distance of c from the origin:
// point extremum projective longitude
// p0 max x 0
// p1 max y pi/2
// p2 min x pi
// p3 min y -pi/2
func p0(c, d float64) r3.Vector {
var psi, beta = math.Asin(c), math.Asin(c / d)
return r3.Vector{
X: math.Sin(psi)*math.Sin(beta) + math.Cos(psi)*math.Cos(beta),
Y: 0,
Z: math.Sin(psi)*math.Cos(beta) - math.Cos(psi)*math.Sin(beta),
}
}
func p1(c, d float64) r3.Vector {
var psi, beta = math.Asin(c), math.Asin(c / d)
return r3.Vector{
X: math.Sin(psi) * math.Sin(beta),
Y: math.Cos(psi),
Z: math.Sin(psi) * math.Cos(beta),
}
}
func p2(c, d float64) r3.Vector {
var psi, beta = math.Asin(c), math.Asin(c / d)
return r3.Vector{
X: math.Sin(psi)*math.Sin(beta) - math.Cos(psi)*math.Cos(beta),
Y: 0,
Z: math.Sin(psi)*math.Cos(beta) + math.Cos(psi)*math.Sin(beta),
}
}
func p3(c, d float64) r3.Vector {
var psi, beta = math.Asin(c), math.Asin(c / d)
return r3.Vector{
X: math.Sin(psi) * math.Sin(beta),
Y: -math.Cos(psi),
Z: math.Sin(psi) * math.Cos(beta),
}
}
func BenchmarkProject(b *testing.B) {
gm := New(s2.LatLng{Lat: math.Pi / 3}, s2.LatLng{Lat: -math.Pi / 3})
for i := 0; i < b.N; i++ {
gm.Project(s2.LatLng{Lat: math.Pi / 4, Lng: 3 * math.Pi / 4})
}
}
func BenchmarkUnproject(b *testing.B) {
gm := New(s2.LatLng{Lat: math.Pi / 3}, s2.LatLng{Lat: -math.Pi / 3})
for i := 0; i < b.N; i++ {
gm.Unproject(r2.Point{1, 1})
}
}