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package curve
import (
"fmt"
"math"
"testing"
"github.com/google/go-cmp/cmp"
"github.com/google/go-cmp/cmp/cmpopts"
)
func TestCubicBezDeriv(t *testing.T) {
// y = x^2
c := CubicBez{
Pt(0.0, 0.0),
Pt(1.0/3.0, 0.0),
Pt(2.0/3.0, 1.0/3.0),
Pt(1.0, 1.0),
}
deriv := c.Differentiate()
const n = 10
const delta = 1e-6
for i := range n + 1 {
ts := float64(i) / float64(n)
p := c.Eval(ts)
p1 := c.Eval(ts + delta)
dApprox := p1.Sub(p).Mul(1.0 / delta)
d := Vec2(deriv.Eval(ts))
if l := d.Sub(dApprox).Hypot(); l >= delta*2 {
t.Errorf("got difference of %g, want at most %g", l, delta*2)
}
}
}
func TestCubicBezToQuadratics(t *testing.T) {
// y = x^3
c := CubicBez{
Pt(0.0, 0.0),
Pt(1.0/3.0, 0.0),
Pt(2.0/3.0, 0.0),
Pt(1.0, 1.0),
}
for i := range 10 {
accuracy := math.Pow(0.1, float64(i))
worst := 0.0
for seq := range c.Quadratics(accuracy) {
t0, t1, q := seq.Start, seq.End, seq.Segment
const epsilon = 1e-12
if delta := q.Start().Sub(c.Eval(t0)).Hypot(); delta > epsilon {
t.Fatalf("%g > %g", delta, epsilon)
}
if delta := q.End().Sub(c.Eval(t1)).Hypot(); delta > epsilon {
t.Fatalf("%g > %g", delta, epsilon)
}
const n = 4
for j := range n + 1 {
ts := float64(j) / float64(n)
p := q.Eval(ts)
error := math.Abs(p.Y - math.Pow(p.X, 3))
if worst > error {
error = worst
}
if error > accuracy {
t.Fatalf("got error %g for desired accuracy of %g", error, accuracy)
}
}
}
}
}
func TestCubicBezToQuadraticsDegenerate(t *testing.T) {
// ensure ToQuadratics returns something given colinear points
c := CubicBez{
Pt(0.0, 9.0),
Pt(6.0, 6.0),
Pt(12.0, 3.0),
Pt(18.0, 0.0),
}
var n int
for range c.Quadratics(1e-6) {
n++
}
if n != 1 {
t.Errorf("got %d quadratics, expected 1", n)
}
}
func BenchmarkCubicBezToQuadratics(b *testing.B) {
shape := CubicBez{
P0: Pt(20, 40),
P1: Pt(40, 80),
P2: Pt(-40, 40),
P3: Pt(42, 62),
}
for i := range 11 {
acc := 1.0 / math.Pow(10, float64(2*i))
b.Run(fmt.Sprintf("1e-%d", 2*i), func(b *testing.B) {
for range b.N {
for range shape.Quadratics(acc) {
}
}
})
}
}
func TestIntersectCubic(t *testing.T) {
c := CubicBez{Pt(0.0, -10.0), Pt(10.0, 20.0), Pt(20.0, -20.0), Pt(30.0, 10.0)}
vLine := Line{Pt(10.0, -10.0), Pt(10.0, 10.0)}
xs, n := c.IntersectLine(vLine)
want := []LineIntersection{{16.0 / 27.0, 1.0 / 3.0}}
diff(t, want, xs[:n], cmpopts.EquateApprox(0, 1e-8))
hLine := Line{Pt(0.0, 0.0), Pt(100.0, 0.0)}
if _, n := c.IntersectLine(hLine); n != 3 {
t.Errorf("got %d intersections, want 3", n)
}
}
func TestCubicBezExtrema(t *testing.T) {
// y = x^2
q := CubicBez{Pt(0.0, 0.0), Pt(0.0, 1.0), Pt(1.0, 1.0), Pt(1.0, 0.0)}
extrema, n := q.Extrema()
if n != 1 {
t.Fatalf("got %d extrema, expected 1", n)
}
if want := 0.5; math.Abs(extrema[0]-want) > 1e-6 {
t.Errorf("got extrema %v, want %v", extrema[0], want)
}
q = CubicBez{Pt(0.4, 0.5), Pt(0.0, 1.0), Pt(1.0, 0.0), Pt(0.5, 0.4)}
extrema, n = q.Extrema()
if n != 4 {
t.Fatalf("got %d extrema, expected 4", n)
}
}
func TestCubicNearest(t *testing.T) {
verify := func(c CubicBez, pt Point, want float64) {
t.Helper()
_, got := c.Nearest(pt, 1e-6)
if math.Abs(got-want) > 1e-6 {
t.Errorf("got %v, want %v", got, want)
}
}
// y = x^3
c := CubicBez{Pt(0.0, 0.0), Pt(1.0/3.0, 0.0), Pt(2.0/3.0, 0.0), Pt(1.0, 1.0)}
verify(c, Pt(0.1, 0.001), 0.1)
verify(c, Pt(0.2, 0.008), 0.2)
verify(c, Pt(0.3, 0.027), 0.3)
verify(c, Pt(0.4, 0.064), 0.4)
verify(c, Pt(0.5, 0.125), 0.5)
verify(c, Pt(0.6, 0.216), 0.6)
verify(c, Pt(0.7, 0.343), 0.7)
verify(c, Pt(0.8, 0.512), 0.8)
verify(c, Pt(0.9, 0.729), 0.9)
verify(c, Pt(1.0, 1.0), 1.0)
verify(c, Pt(1.1, 1.1), 1.0)
verify(c, Pt(-0.1, 0.0), 0.0)
a := Rotate(0.5)
verify(c.Transform(a), Pt(0.1, 0.001).Transform(a), 0.1)
}
func TestCubicBezInflections(t *testing.T) {
approx := cmpopts.EquateApprox(0, 1e-6)
c := CubicBez{
Pt(0.0, 0.0),
Pt(0.8, 1.0),
Pt(0.2, 1.0),
Pt(1.0, 0.0),
}
inflections, n := c.Inflections()
want := []float64{
0.311018,
0.688982,
}
diff(t, want, inflections[:n], approx)
c = CubicBez{Pt(0.0, 0.0), Pt(1.0, 1.0), Pt(2.0, -1.0), Pt(3.0, 0.0)}
inflections, n = c.Inflections()
want = []float64{0.5}
diff(t, want, inflections[:n])
c = CubicBez{Pt(0.0, 0.0), Pt(1.0, 1.0), Pt(2.0, 1.0), Pt(3.0, 0.0)}
inflections, n = c.Inflections()
diff(t, []float64{}, inflections[:n])
}
var pointComparer = cmp.Comparer(func(p1, p2 Point) bool {
return p1.Distance(p2) <= 1e-12
})
func TestCubicBezApproxSpline(t *testing.T) {
c1 := CubicBez{
Pt(550.0, 258.0),
Pt(1044.0, 482.0),
Pt(2029.0, 1841.0),
Pt(1934.0, 1554.0),
}
{
quad, _ := c1.tryApproxQuadratic(344.0)
expected := QuadBez{
Pt(550.0, 258.0),
Pt(1673.665720592873, 767.5164401068898),
Pt(1934.0, 1554.0),
}
diff(t, expected, quad)
}
{
_, ok := c1.tryApproxQuadratic(343.0)
if ok {
t.Errorf("expected approximation to fail")
}
}
{
spline, _ := c1.approxQuadSplineN(2, 343.0)
expected := QuadBSpline{
Pt(550.0, 258.0),
Pt(920.5, 426.0),
Pt(2005.25, 1769.25),
Pt(1934.0, 1554.0),
}
diff(t, expected, spline, pointComparer)
}
{
spline, _ := c1.ApproxQuadSpline(5.0)
expected := QuadBSpline{
Pt(550.0, 258.0),
Pt(673.5, 314.0),
Pt(984.8777777777776, 584.2666666666667),
Pt(1312.6305555555557, 927.825),
Pt(1613.1194444444443, 1267.425),
Pt(1842.7055555555555, 1525.8166666666666),
Pt(1957.75, 1625.75),
Pt(1934.0, 1554.0),
}
diff(t, expected, spline, pointComparer)
}
}
func TestCubicBezCubicsToQuadraticSplines(t *testing.T) {
curves := []CubicBez{
CubicBez{
Pt(550.0, 258.0),
Pt(1044.0, 482.0),
Pt(2029.0, 1841.0),
Pt(1934.0, 1554.0),
},
CubicBez{
Pt(859.0, 384.0),
Pt(1998.0, 116.0),
Pt(1596.0, 1772.0),
Pt(8.0, 1824.0),
},
CubicBez{
Pt(1090.0, 937.0),
Pt(418.0, 1300.0),
Pt(125.0, 91.0),
Pt(104.0, 37.0),
},
}
converted, ok := CubicsToQuadraticSplines(curves, 5.0)
if !ok {
t.Fatal("could not convert cubics to b-splines")
}
if len(converted) != 3 {
t.Fatalf("got %d splines, want 3", len(converted))
}
for i, s := range converted {
if len(s) != 8 {
t.Fatalf("got %d points in spline %d, want 8", len(s), i)
}
}
diff(t, Pt(673.5, 314.0), converted[0][1], pointComparer)
diff(t, Pt(88639.0/90.0, 52584.0/90.0), converted[0][2], pointComparer)
}
func TestCubicBezApproxSplineMatchesPython(t *testing.T) {
// Ensure rounding behavior for division matches fonttools
// cu2qu.
// See https://github.com/linebender/kurbo/issues/272
cubic := CubicBez{
Pt(408.0, 321.0),
Pt(408.0, 452.0),
Pt(342.0, 560.0),
Pt(260.0, 560.0),
}
spline, ok := cubic.ApproxQuadSpline(1.0)
if !ok {
t.Fatal("could not convert cubic to b-spline")
}
diff(
t,
QuadBSpline{
Pt(408.0, 321.0),
// Previous behavior produced 386.49999999999994 for the
// y coordinate leading to inconsistent rounding.
Pt(408.0, 386.5),
Pt(368.16666666666663, 495.0833333333333),
Pt(301.0, 560.0),
Pt(260.0, 560.0),
},
spline,
/* not using pointComparer, we want an exact match */
)
}
func TestCubicToQuadraticMatchesPython(t *testing.T) {
// from https://github.com/googlefonts/fontmake-rs/issues/217
cubic := CubicBez{
Pt(796.0, 319.0),
Pt(727.0, 314.0),
Pt(242.0, 303.0),
Pt(106.0, 303.0),
}
// FontTools can approximate this curve successfully in 7 splits, we can too
if _, ok := cubic.approxQuadSplineN(7, 1.0); !ok {
t.Error("could not approximate curve in 7 splits")
}
// FontTools can solve this with accuracy 0.001, we can too
if _, ok := CubicsToQuadraticSplines([]CubicBez{cubic}, 0.001); !ok {
t.Error("could not approximate curve with 0.001 accuracy")
}
}
func TestCubicsToQuadraticSplinesMatchesPython(t *testing.T) {
// https://github.com/linebender/kurbo/pull/273
light := CubicBez{
Pt(378.0, 608.0),
Pt(378.0, 524.0),
Pt(355.0, 455.0),
Pt(266.0, 455.0),
}
regular := CubicBez{
Pt(367.0, 607.0),
Pt(367.0, 511.0),
Pt(338.0, 472.0),
Pt(243.0, 472.0)}
bold := CubicBez{
Pt(372.425, 593.05),
Pt(372.425, 524.95),
Pt(355.05, 485.95),
Pt(274.0, 485.95),
}
qsplines, _ := CubicsToQuadraticSplines([]CubicBez{light, regular, bold}, 1.0)
diff(
t,
[]QuadBSpline{
QuadBSpline{
Pt(378.0, 608.0),
Pt(378.0, 566.0),
Pt(359.0833333333333, 496.5),
Pt(310.5, 455.0),
Pt(266.0, 455.0),
},
QuadBSpline{
Pt(367.0, 607.0),
Pt(367.0, 559.0),
// Previous behavior produced 496.5 for the y coordinate
Pt(344.5833333333333, 499.49999999999994),
Pt(290.5, 472.0),
Pt(243.0, 472.0),
},
QuadBSpline{
Pt(372.425, 593.05),
Pt(372.425, 559.0),
Pt(356.98333333333335, 511.125),
Pt(314.525, 485.95),
Pt(274.0, 485.95),
},
},
qsplines,
)
}
func TestCubicBezArclen(t *testing.T) {
c := CubicBez{
Pt(0.0, 0.0),
Pt(1.0/3.0, 0.0),
Pt(2.0/3.0, 1.0/3.0),
Pt(1.0, 1.0),
}
trueArclen := 0.5*math.Sqrt(5.0) + 0.25*math.Log(2.0+math.Sqrt(5.0))
for i := range 12 {
accuracy := math.Pow(0.1, float64(i))
diff(t, trueArclen, c.Arclen(accuracy), cmpopts.EquateApprox(0, accuracy))
}
}
func TestCubicBezInvArclen(t *testing.T) {
// y = x^2 / 100
c := CubicBez{
Pt(0.0, 0.0),
Pt(100.0/3.0, 0.0),
Pt(200.0/3.0, 100.0/3.0),
Pt(100.0, 100.0),
}
trueArclen := 100.0 * (0.5*math.Sqrt(5.0) + 0.25*math.Log(2.0+math.Sqrt(5.0)))
for i := range 12 {
accuracy := math.Pow(0.1, float64(i))
n := 10
for j := range n + 1 {
arc := float64(j) * (1.0 / float64(n) * trueArclen)
tt := SolveForArclen(c, arc, accuracy*0.5)
actualArc := c.Subsegment(0.0, tt).Arclen(accuracy * 0.5)
diff(t, arc, actualArc, cmpopts.EquateApprox(0, accuracy))
}
}
// corner case: user passes accuracy larger than total arc length
accuracy := trueArclen * 1.1
arc := trueArclen * 0.5
tt := SolveForArclen(c, arc, accuracy)
actualArc := c.Subsegment(0.0, tt).Arclen(accuracy)
diff(t, arc, actualArc, cmpopts.EquateApprox(0, 2*accuracy))
}
func TestCubicBezInvArclenAccuracy(t *testing.T) {
c := CubicBez{
Pt(0.2, 0.73),
Pt(0.35, 1.08),
Pt(0.85, 1.08),
Pt(1.0, 0.73),
}
trueT := SolveForArclen(c, 0.5, 1e-12)
for i := 1; i < 12; i++ {
accuracy := math.Pow(0.1, float64(i))
approxT := SolveForArclen(c, 0.5, accuracy)
diff(t, trueT, approxT, cmpopts.EquateApprox(0, accuracy))
}
}
func TestCubicBezSignedAreaLinear(t *testing.T) {
// y = 1 - x
c := CubicBez{
Pt(1.0, 0.0),
Pt(2.0/3.0, 1.0/3.0),
Pt(1.0/3.0, 2.0/3.0),
Pt(0.0, 1.0),
}
const epsilon = 1e-12
diff(t, 0.5, c.SignedArea())
diff(t, 0.5, c.Transform(Rotate(0.5)).SignedArea(), cmpopts.EquateApprox(0, epsilon))
diff(t, 1.0, c.Transform(Translate(Vec(0.0, 1.0))).SignedArea(), cmpopts.EquateApprox(0, epsilon))
diff(t, 1.0, c.Transform(Translate(Vec(1.0, 0.0))).SignedArea(), cmpopts.EquateApprox(0, epsilon))
}
func TestCubicBezSignedArea(t *testing.T) {
// y = 1 - x^3
c := CubicBez{
Pt(1.0, 0.0),
Pt(2.0/3.0, 1.0),
Pt(1.0/3.0, 1.0),
Pt(0.0, 1.0),
}
const epsilon = 1e-12
diff(t, 0.75, c.SignedArea(), cmpopts.EquateApprox(0, epsilon))
diff(t, 0.75, c.Transform(Rotate(0.5)).SignedArea(), cmpopts.EquateApprox(0, epsilon))
diff(t, 1.25, c.Transform(Translate(Vec(0.0, 1.0))).SignedArea(), cmpopts.EquateApprox(0, epsilon))
diff(t, 1.25, c.Transform(Translate(Vec(1.0, 0.0))).SignedArea(), cmpopts.EquateApprox(0, epsilon))
}