blob: a24cd71d6844531abb725c466cb5997dcfffc3fe [file] [log] [blame]
/*
* Copyright 2020 Google Inc.
*
* Use of this source code is governed by a BSD-style license that can be
* found in the LICENSE file.
*
* Initial import from skia:tests/WangsFormulaTest.cpp
*
* Copyright 2023 Rive
*/
#include "rive/math/wangs_formula.hpp"
#include <catch.hpp>
#include <functional>
namespace rive
{
constexpr static float kPrecision = 4;
constexpr static float kEpsilon = 1.f / (1 << 12);
static bool fuzzy_equal(float a, float b, float tolerance = kEpsilon)
{
assert(tolerance >= 0);
return fabsf(a - b) <= tolerance;
}
const Vec2D kSerp[4] = {{285.625f, 499.687f},
{411.625f, 808.188f},
{1064.62f, 135.688f},
{1042.63f, 585.187f}};
const Vec2D kLoop[4] = {{635.625f, 614.687f},
{171.625f, 236.188f},
{1064.62f, 135.688f},
{516.625f, 570.187f}};
const Vec2D kQuad[4] = {{460.625f, 557.187f}, {707.121f, 209.688f}, {779.628f, 577.687f}};
static void map_pts(const Mat2D& m, Vec2D out[], const Vec2D in[], int n)
{
for (int i = 0; i < n; ++i)
{
out[i] = m * in[i];
}
}
static float wangs_formula_quadratic_reference_impl(float precision, const Vec2D p[3])
{
float k = (2 * 1) / 8.f * precision;
return sqrtf(k * (p[0] - p[1] * 2 + p[2]).length());
}
static float wangs_formula_cubic_reference_impl(float precision, const Vec2D p[4])
{
float k = (3 * 2) / 8.f * precision;
return sqrtf(k *
std::max((p[0] - p[1] * 2 + p[2]).length(), (p[1] - p[2] * 2 + p[3]).length()));
}
static void chop_quad_at(const Vec2D src[3], Vec2D dst[5], float t)
{
assert(t > 0 && t < 1);
float2 p0 = simd::load2f(&src[0].x);
float2 p1 = simd::load2f(&src[1].x);
float2 p2 = simd::load2f(&src[2].x);
float2 tt(t);
float2 p01 = simd::mix(p0, p1, tt);
float2 p12 = simd::mix(p1, p2, tt);
simd::store(&dst[0].x, p0);
simd::store(&dst[1].x, p01);
simd::store(&dst[2].x, simd::mix(p01, p12, tt));
simd::store(&dst[3].x, p12);
simd::store(&dst[4].x, p2);
}
static Vec2D eval_quad_at(const Vec2D src[3], float t)
{
assert(t > 0 && t < 1);
float2 p0 = simd::load2f(&src[0].x);
float2 p1 = simd::load2f(&src[1].x);
float2 p2 = simd::load2f(&src[2].x);
float2 tt(t);
float2 p01 = simd::mix(p0, p1, tt);
float2 p12 = simd::mix(p1, p2, tt);
float2 p012 = simd::mix(p01, p12, tt);
Vec2D vec;
simd::store(&vec.x, p012);
return vec;
}
// Returns number of segments for linearized quadratic rational. This is an analogue
// to Wang's formula, taken from:
//
// J. Zheng, T. Sederberg. "Estimating Tessellation Parameter Intervals for
// Rational Curves and Surfaces." ACM Transactions on Graphics 19(1). 2000.
// See Thm 3, Corollary 1.
//
// Input points should be in projected space.
static float wangs_formula_conic_reference_impl(float precision, const Vec2D P[3], const float w)
{
// Compute center of bounding box in projected space
float min_x = P[0].x, max_x = min_x, min_y = P[0].y, max_y = min_y;
for (int i = 1; i < 3; i++)
{
min_x = std::min(min_x, P[i].x);
max_x = std::max(max_x, P[i].x);
min_y = std::min(min_y, P[i].y);
max_y = std::max(max_y, P[i].y);
}
const Vec2D C = Vec2D(0.5f * (min_x + max_x), 0.5f * (min_y + max_y));
// Translate control points and compute max length
Vec2D tP[3] = {P[0] - C, P[1] - C, P[2] - C};
float max_len = 0;
for (int i = 0; i < 3; i++)
{
max_len = std::max(max_len, tP[i].length());
}
assert(max_len > 0);
// Compute delta = parametric step size of linearization
const float eps = 1 / precision;
const float r_minus_eps = std::max(0.f, max_len - eps);
const float min_w = std::min(w, 1.f);
const float numer = 4 * min_w * eps;
const float denom =
(tP[2] - tP[1] * 2 * w + tP[0]).length() + r_minus_eps * std::abs(1 - 2 * w + 1);
const float delta = sqrtf(numer / denom);
// Return corresponding num segments in the interval [tmin,tmax]
constexpr float tmin = 0, tmax = 1;
assert(delta > 0);
return (tmax - tmin) / delta;
}
static float frand() { return rand() / static_cast<float>(RAND_MAX); }
static float frand_range(float min, float max) { return min + frand() * (max - min); }
static void for_random_matrices(std::function<void(const Mat2D&)> f)
{
srand(0);
Mat2D m{};
f(m);
for (int i = -10; i <= 30; ++i)
{
for (int j = -10; j <= 30; ++j)
{
m[0] = std::ldexp(1 + frand(), i);
m[1] = 0;
m[2] = 0;
m[3] = std::ldexp(1 + frand(), j);
f(m);
m[0] = std::ldexp(1 + frand(), i);
m[1] = std::ldexp(1 + frand(), (j + i) / 2);
m[2] = std::ldexp(1 + frand(), (j + i) / 2);
m[3] = std::ldexp(1 + frand(), j);
f(m);
}
}
}
static void for_random_beziers(int numPoints,
std::function<void(const Vec2D[])> f,
int maxExponent = 30)
{
srand(0);
assert(numPoints <= 4);
Vec2D pts[4];
for (int i = -10; i <= maxExponent; ++i)
{
for (int j = 0; j < numPoints; ++j)
{
pts[j] = {std::ldexp(1 + frand(), i), std::ldexp(1 + frand(), i)};
}
f(pts);
}
}
// Ensure the optimized "*_log2" versions return the same value as ceil(std::log2(f)).
TEST_CASE("wangs_formula_log2", "[wangs_formula]")
{
// Constructs a cubic such that the 'length' term in wang's formula == term.
//
// f = sqrt(k * length(max(abs(p0 - p1*2 + p2),
// abs(p1 - p2*2 + p3))));
auto setupCubicLengthTerm = [](int seed, Vec2D pts[], float term) {
memset(pts, 0, sizeof(Vec2D) * 4);
Vec2D term2d = (seed & 1) ? Vec2D(term, 0) : Vec2D(.5f, std::sqrt(3) / 2) * term;
seed >>= 1;
if (seed & 1)
{
term2d.x = -term2d.x;
}
seed >>= 1;
if (seed & 1)
{
std::swap(term2d.x, term2d.y);
}
seed >>= 1;
switch (seed % 4)
{
case 0:
pts[0] = term2d;
pts[3] = term2d * .75f;
return;
case 1:
pts[1] = term2d * -.5f;
return;
case 2:
pts[1] = term2d * -.5f;
return;
case 3:
pts[3] = term2d;
pts[0] = term2d * .75f;
return;
}
};
// Constructs a quadratic such that the 'length' term in wang's formula == term.
//
// f = sqrt(k * length(p0 - p1*2 + p2));
auto setupQuadraticLengthTerm = [](int seed, Vec2D pts[], float term) {
memset(pts, 0, sizeof(Vec2D) * 3);
Vec2D term2d = (seed & 1) ? Vec2D(term, 0) : Vec2D(.5f, std::sqrt(3) / 2) * term;
seed >>= 1;
if (seed & 1)
{
term2d.x = -term2d.x;
}
seed >>= 1;
if (seed & 1)
{
std::swap(term2d.x, term2d.y);
}
seed >>= 1;
switch (seed % 3)
{
case 0:
pts[0] = term2d;
return;
case 1:
pts[1] = term2d * -.5f;
return;
case 2:
pts[2] = term2d;
return;
}
};
// wangs_formula_cubic and wangs_formula_quadratic both use rsqrt instead of sqrt for speed.
// Linearization is all approximate anyway, so as long as we are within ~1/2 tessellation
// segment of the reference value we are good enough.
constexpr static float kTessellationTolerance = 1 / 128.f;
for (int level = 0; level < 30; ++level)
{
float epsilon = std::ldexp(kEpsilon, level * 2);
Vec2D pts[4];
{
// Test cubic boundaries.
// f = sqrt(k * length(max(abs(p0 - p1*2 + p2),
// abs(p1 - p2*2 + p3))));
constexpr static float k = (3 * 2) / (8 * (1.f / kPrecision));
float x = std::ldexp(1, level * 2) / k;
setupCubicLengthTerm(level << 1, pts, x - epsilon);
float referenceValue = wangs_formula_cubic_reference_impl(kPrecision, pts);
REQUIRE(std::ceil(std::log2(referenceValue)) == level);
float c = wangs_formula::cubic(pts, kPrecision);
REQUIRE(fuzzy_equal(c / referenceValue, 1, kTessellationTolerance));
REQUIRE(wangs_formula::cubic_log2(pts, kPrecision) == level);
setupCubicLengthTerm(level << 1, pts, x + epsilon);
referenceValue = wangs_formula_cubic_reference_impl(kPrecision, pts);
REQUIRE(std::ceil(std::log2(referenceValue)) == level + 1);
c = wangs_formula::cubic(pts, kPrecision);
REQUIRE(fuzzy_equal(c / referenceValue, 1, kTessellationTolerance));
REQUIRE(wangs_formula::cubic_log2(pts, kPrecision) == level + 1);
}
{
// Test quadratic boundaries.
// f = std::sqrt(k * Length(p0 - p1*2 + p2));
constexpr static float k = 2 / (8 * (1.f / kPrecision));
float x = std::ldexp(1, level * 2) / k;
setupQuadraticLengthTerm(level << 1, pts, x - epsilon);
float referenceValue = wangs_formula_quadratic_reference_impl(kPrecision, pts);
REQUIRE(std::ceil(std::log2(referenceValue)) == level);
float q = wangs_formula::quadratic(pts, kPrecision);
REQUIRE(fuzzy_equal(q / referenceValue, 1, kTessellationTolerance));
REQUIRE(wangs_formula::quadratic_log2(pts, kPrecision) == level);
setupQuadraticLengthTerm(level << 1, pts, x + epsilon);
referenceValue = wangs_formula_quadratic_reference_impl(kPrecision, pts);
REQUIRE(std::ceil(std::log2(referenceValue)) == level + 1);
q = wangs_formula::quadratic(pts, kPrecision);
REQUIRE(fuzzy_equal(q / referenceValue, 1, kTessellationTolerance));
REQUIRE(wangs_formula::quadratic_log2(pts, kPrecision) == level + 1);
}
}
auto check_cubic_log2 = [&](const Vec2D* pts) {
float f = std::max(1.f, wangs_formula_cubic_reference_impl(kPrecision, pts));
int f_log2 = wangs_formula::cubic_log2(pts, kPrecision);
REQUIRE(ceilf(std::log2(f)) == f_log2);
float c = std::max(1.f, wangs_formula::cubic(pts, kPrecision));
REQUIRE(fuzzy_equal(c / f, 1, kTessellationTolerance));
};
auto check_quadratic_log2 = [&](const Vec2D* pts) {
float f = std::max(1.f, wangs_formula_quadratic_reference_impl(kPrecision, pts));
int f_log2 = wangs_formula::quadratic_log2(pts, kPrecision);
REQUIRE(ceilf(std::log2(f)) == f_log2);
float q = std::max(1.f, wangs_formula::quadratic(pts, kPrecision));
REQUIRE(fuzzy_equal(q / f, 1, kTessellationTolerance));
};
for_random_matrices([&](const Mat2D& m) {
Vec2D pts[4 + 999];
map_pts(m, pts, kSerp, 4);
check_cubic_log2(pts);
map_pts(m, pts, kLoop, 4);
check_cubic_log2(pts);
map_pts(m, pts, kQuad, 3);
check_quadratic_log2(pts);
});
for_random_beziers(4, [&](const Vec2D pts[]) { check_cubic_log2(pts); });
for_random_beziers(3, [&](const Vec2D pts[]) { check_quadratic_log2(pts); });
}
static void check_cubic_log2_with_transform(const Vec2D* pts, const Mat2D& m)
{
Vec2D ptsXformed[4];
map_pts(m, ptsXformed, pts, 4);
int expected = wangs_formula::cubic_log2(ptsXformed, kPrecision);
int actual = wangs_formula::cubic_log2(pts, kPrecision, wangs_formula::VectorXform(m));
REQUIRE(actual == expected);
};
static void check_quadratic_log2_with_transform(const Vec2D* pts, const Mat2D& m)
{
Vec2D ptsXformed[3];
map_pts(m, ptsXformed, pts, 3);
int expected = wangs_formula::quadratic_log2(ptsXformed, kPrecision);
int actual = wangs_formula::quadratic_log2(pts, kPrecision, wangs_formula::VectorXform(m));
REQUIRE(actual == expected);
};
// Ensure using transformations gives the same result as pre-transforming all points.
TEST_CASE("wangs_formula_vectorXforms", "[wangs_formula]")
{
for_random_matrices([&](const Mat2D& m) {
check_cubic_log2_with_transform(kSerp, m);
check_cubic_log2_with_transform(kLoop, m);
check_quadratic_log2_with_transform(kQuad, m);
for_random_beziers(4, [&](const Vec2D pts[]) { check_cubic_log2_with_transform(pts, m); });
for_random_beziers(3,
[&](const Vec2D pts[]) { check_quadratic_log2_with_transform(pts, m); });
});
}
TEST_CASE("wangs_formula_worst_case_cubic", "[wangs_formula]")
{
{
Vec2D worstP[] = {{0, 0}, {100, 100}, {0, 0}, {0, 0}};
REQUIRE(wangs_formula::worst_case_cubic(100, 100, kPrecision) ==
wangs_formula_cubic_reference_impl(kPrecision, worstP));
REQUIRE(wangs_formula::worst_case_cubic_log2(100, 100, kPrecision) ==
wangs_formula::cubic_log2(worstP, kPrecision));
}
{
Vec2D worstP[] = {{100, 100}, {100, 100}, {200, 200}, {100, 100}};
REQUIRE(wangs_formula::worst_case_cubic(100, 100, kPrecision) ==
wangs_formula_cubic_reference_impl(kPrecision, worstP));
REQUIRE(wangs_formula::worst_case_cubic_log2(100, 100, kPrecision) ==
wangs_formula::cubic_log2(worstP, kPrecision));
}
auto check_worst_case_cubic = [&](const Vec2D* pts) {
float2 min = simd::load2f(&pts[0].x), max = simd::load2f(&pts[0].x);
for (int i = 1; i < 4; ++i)
{
min = simd::min(min, simd::load2f(&pts[i].x));
max = simd::max(max, simd::load2f(&pts[i].x));
}
float2 size = max - min;
float worst = wangs_formula::worst_case_cubic(size.x, size.y, kPrecision);
int worst_log2 = wangs_formula::worst_case_cubic_log2(size.x, size.y, kPrecision);
float actual = wangs_formula_cubic_reference_impl(kPrecision, pts);
REQUIRE(worst >= actual);
REQUIRE(std::ceil(std::log2(std::max(1.f, worst))) == worst_log2);
};
for (int i = 0; i < 100; ++i)
{
for_random_beziers(4, [&](const Vec2D pts[]) { check_worst_case_cubic(pts); });
}
// Make sure overflow saturates at infinity (not NaN).
constexpr static float inf = std::numeric_limits<float>::infinity();
REQUIRE(wangs_formula::worst_case_cubic_pow4(inf, inf, kPrecision) == inf);
REQUIRE(wangs_formula::worst_case_cubic(inf, inf, kPrecision) == inf);
}
// Ensure Wang's formula for quads produces max error within tolerance.
TEST_CASE("wangs_formula_quad_within_tol", "[wangs_formula]")
{
// Wang's formula and the quad math starts to lose precision with very large
// coordinate values, so limit the magnitude a bit to prevent test failures
// due to loss of precision.
constexpr int maxExponent = 15;
for_random_beziers(
3,
[](const Vec2D pts[]) {
const int nsegs = static_cast<int>(
std::ceil(wangs_formula_quadratic_reference_impl(kPrecision, pts)));
const float tdelta = 1.f / nsegs;
for (int j = 0; j < nsegs; ++j)
{
const float tmin = j * tdelta, tmax = (j + 1) * tdelta;
// Get section of quad in [tmin,tmax]
const Vec2D* sectionPts;
Vec2D tmp0[5];
Vec2D tmp1[5];
if (tmin == 0)
{
if (tmax == 1)
{
sectionPts = pts;
}
else
{
chop_quad_at(pts, tmp0, tmax);
sectionPts = tmp0;
}
}
else
{
chop_quad_at(pts, tmp0, tmin);
if (tmax == 1)
{
sectionPts = tmp0 + 2;
}
else
{
chop_quad_at(tmp0 + 2, tmp1, (tmax - tmin) / (1 - tmin));
sectionPts = tmp1;
}
}
// For quads, max distance from baseline is always at t=0.5.
Vec2D p;
p = eval_quad_at(sectionPts, 0.5f);
// Get distance of p to baseline
const Vec2D n = {sectionPts[2].y - sectionPts[0].y,
sectionPts[0].x - sectionPts[2].x};
const float d = std::abs(Vec2D::dot(p - sectionPts[0], n)) / n.length();
// Check distance is within specified tolerance
REQUIRE(d <= (1.f / kPrecision) + 1e-2f);
}
},
maxExponent);
}
// Ensure the specialized version for rational quads reduces to regular Wang's
// formula when all weights are equal to one
TEST_CASE("wangs_formula_rational_quad_reduces", "[wangs_formula]")
{
constexpr static float kTessellationTolerance = 1 / 128.f;
for (int i = 0; i < 100; ++i)
{
for_random_beziers(3, [](const Vec2D pts[]) {
const float rational_nsegs = wangs_formula::conic(kPrecision, pts, 1.f);
const float integral_nsegs = wangs_formula_quadratic_reference_impl(kPrecision, pts);
REQUIRE(fuzzy_equal(rational_nsegs, integral_nsegs, kTessellationTolerance));
});
}
}
// Ensure the rational quad version (used for conics) produces max error within tolerance.
TEST_CASE("wangs_formula_conic_within_tol", "[wangs_formula]")
{
constexpr int maxExponent = 24;
srand(0);
// Single-precision functions in SkConic/SkGeometry lose too much accuracy with
// large-magnitude curves and large weights for this test to pass.
using Sk2d = simd::gvec<double, 2>;
const auto eval_conic = [](const Vec2D pts[3], double w, double t) -> Sk2d {
const auto eval = [](Sk2d A, Sk2d B, Sk2d C, double t) -> Sk2d {
return (A * t + B) * t + C;
};
const Sk2d p0 = {pts[0].x, pts[0].y};
const Sk2d p1 = {pts[1].x, pts[1].y};
const Sk2d p1w = p1 * w;
const Sk2d p2 = {pts[2].x, pts[2].y};
Sk2d numer = eval(p2 - p1w * 2.0 + p0, (p1w - p0) * 2.0, p0, t);
Sk2d denomC = {1, 1};
Sk2d denomB = {2 * (w - 1), 2 * (w - 1)};
Sk2d denomA = {-2 * (w - 1), -2 * (w - 1)};
Sk2d denom = eval(denomA, denomB, denomC, t);
return numer / denom;
};
const auto dot = [](const Sk2d& a, const Sk2d& b) -> double {
return a[0] * b[0] + a[1] * b[1];
};
const auto length = [](const Sk2d& p) -> double { return sqrt(p[0] * p[0] + p[1] * p[1]); };
for (int i = -10; i <= 10; ++i)
{
const float w = std::ldexp(1 + frand(), i);
for_random_beziers(
3,
[&](const Vec2D pts[]) {
const int nsegs = static_cast<int>(ceilf(wangs_formula::conic(kPrecision, pts, w)));
const float tdelta = 1.f / nsegs;
for (int j = 0; j < nsegs; ++j)
{
const float tmin = j * tdelta, tmax = (j + 1) * tdelta,
tmid = 0.5f * (tmin + tmax);
Sk2d p0, p1, p2;
p0 = eval_conic(pts, w, tmin);
p1 = eval_conic(pts, w, tmid);
p2 = eval_conic(pts, w, tmax);
// Get distance of p1 to baseline (p0, p2).
const Sk2d n = {p2[1] - p0[1], p0[0] - p2[0]};
assert(length(n) != 0);
const double d = std::abs(dot(p1 - p0, n)) / length(n);
// Check distance is within tolerance
REQUIRE(d <= (1.0 / kPrecision) + kEpsilon);
REQUIRE(d <= (1.0 / kPrecision) + kEpsilon);
}
},
maxExponent);
}
}
// Ensure the vectorized conic version equals the reference implementation
TEST_CASE("wangs_formula_conic_matches_reference", "[wangs_formula]")
{
srand(0);
for (int i = -10; i <= 10; ++i)
{
const float w = std::ldexp(1 + frand(), i);
for_random_beziers(3, [w](const Vec2D pts[]) {
const float ref_nsegs = wangs_formula_conic_reference_impl(kPrecision, pts, w);
const float nsegs = wangs_formula::conic(kPrecision, pts, w);
// Because the Gr version may implement the math differently for performance,
// allow different slack in the comparison based on the rough scale of the answer.
const float cmpThresh = ref_nsegs * (1.f / (1 << 20));
REQUIRE(fuzzy_equal(ref_nsegs, nsegs, cmpThresh));
});
}
}
// Ensure using transformations gives the same result as pre-transforming all points.
TEST_CASE("wangs_formula_conic_vectorXforms", "[wangs_formula]")
{
srand(0);
auto check_conic_with_transform = [&](const Vec2D* pts, float w, const Mat2D& m) {
Vec2D ptsXformed[3];
map_pts(m, ptsXformed, pts, 3);
float expected = wangs_formula::conic(kPrecision, ptsXformed, w);
float actual = wangs_formula::conic(kPrecision, pts, w, wangs_formula::VectorXform(m));
REQUIRE(actual == Approx(expected).margin(1e-4));
};
for (int i = -10; i <= 10; ++i)
{
const float w = std::ldexp(1 + frand(), i);
for_random_beziers(3, [&](const Vec2D pts[]) {
check_conic_with_transform(pts, w, Mat2D());
check_conic_with_transform(
pts,
w,
Mat2D::fromScale(frand_range(-10, 10), frand_range(-10, 10)));
// Random 2x2 matrix
Mat2D m;
m[0] = frand_range(-10, 10);
m[1] = frand_range(-10, 10);
m[2] = frand_range(-10, 10);
m[3] = frand_range(-10, 10);
check_conic_with_transform(pts, w, m);
});
}
}
} // namespace rive