impeller/geometry/path_component.cc - mirrors/engine - Git at Google

 // Copyright 2013 The Flutter Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.

 #include "path_component.h"

 #include <cmath>

 namespace impeller {

 /*
  *  Based on: https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Specific_cases
  */

 static inline Scalar LinearSolve(Scalar t, Scalar p0, Scalar p1) {
   return p0 + t * (p1 - p0);
 }

 static inline Scalar QuadraticSolve(Scalar t, Scalar p0, Scalar p1, Scalar p2) {
   return (1 - t) * (1 - t) * p0 +  //
          2 * (1 - t) * t * p1 +    //
          t * t * p2;
 }

 static inline Scalar QuadraticSolveDerivative(Scalar t,
                                               Scalar p0,
                                               Scalar p1,
                                               Scalar p2) {
   return 2 * (1 - t) * (p1 - p0) +  //
          2 * t * (p2 - p1);
 }

 static inline Scalar CubicSolve(Scalar t,
                                 Scalar p0,
                                 Scalar p1,
                                 Scalar p2,
                                 Scalar p3) {
   return (1 - t) * (1 - t) * (1 - t) * p0 +  //
          3 * (1 - t) * (1 - t) * t * p1 +    //
          3 * (1 - t) * t * t * p2 +          //
          t * t * t * p3;
 }

 static inline Scalar CubicSolveDerivative(Scalar t,
                                           Scalar p0,
                                           Scalar p1,
                                           Scalar p2,
                                           Scalar p3) {
   return -3 * p0 * (1 - t) * (1 - t) +  //
          p1 * (3 * (1 - t) * (1 - t) - 6 * (1 - t) * t) +
          p2 * (6 * (1 - t) * t - 3 * t * t) +  //
          3 * p3 * t * t;
 }

 Point LinearPathComponent::Solve(Scalar time) const {
   return {
       LinearSolve(time, p1.x, p2.x),  // x
       LinearSolve(time, p1.y, p2.y),  // y
   };
 }

 std::vector<Point> LinearPathComponent::CreatePolyline() const {
   return {p2};
 }

 std::vector<Point> LinearPathComponent::Extrema() const {
   return {p1, p2};
 }

 Point QuadraticPathComponent::Solve(Scalar time) const {
   return {
       QuadraticSolve(time, p1.x, cp.x, p2.x),  // x
       QuadraticSolve(time, p1.y, cp.y, p2.y),  // y
   };
 }

 Point QuadraticPathComponent::SolveDerivative(Scalar time) const {
   return {
       QuadraticSolveDerivative(time, p1.x, cp.x, p2.x),  // x
       QuadraticSolveDerivative(time, p1.y, cp.y, p2.y),  // y
   };
 }

 static Scalar ApproximateParabolaIntegral(Scalar x) {
   constexpr Scalar d = 0.67;
   return x / (1.0 - d + sqrt(sqrt(pow(d, 4) + 0.25 * x * x)));
 }

 std::vector<Point> QuadraticPathComponent::CreatePolyline(
     Scalar tolerance) const {
   std::vector<Point> points;
   FillPointsForPolyline(points, tolerance);
   return points;
 }

 void QuadraticPathComponent::FillPointsForPolyline(std::vector<Point>& points,
                                                    Scalar tolerance) const {
   auto sqrt_tolerance = sqrt(tolerance);

   auto d01 = cp - p1;
   auto d12 = p2 - cp;
   auto dd = d01 - d12;
   auto cross = (p2 - p1).Cross(dd);
   auto x0 = d01.Dot(dd) * 1 / cross;
   auto x2 = d12.Dot(dd) * 1 / cross;
   auto scale = abs(cross / (hypot(dd.x, dd.y) * (x2 - x0)));

   auto a0 = ApproximateParabolaIntegral(x0);
   auto a2 = ApproximateParabolaIntegral(x2);
   Scalar val = 0.f;
   if (std::isfinite(scale)) {
     auto da = abs(a2 - a0);
     auto sqrt_scale = sqrt(scale);
     if ((x0 < 0 && x2 < 0) || (x0 >= 0 && x2 >= 0)) {
       val = da * sqrt_scale;
     } else {
       // cusp case
       auto xmin = sqrt_tolerance / sqrt_scale;
       val = sqrt_tolerance * da / ApproximateParabolaIntegral(xmin);
     }
   }
   auto u0 = ApproximateParabolaIntegral(a0);
   auto u2 = ApproximateParabolaIntegral(a2);
   auto uscale = 1 / (u2 - u0);

   auto line_count = std::max(1., ceil(0.5 * val / sqrt_tolerance));
   auto step = 1 / line_count;
   for (size_t i = 1; i < line_count; i += 1) {
     auto u = i * step;
     auto a = a0 + (a2 - a0) * u;
     auto t = (ApproximateParabolaIntegral(a) - u0) * uscale;
     points.emplace_back(Solve(t));
   }
   points.emplace_back(p2);
 }

 std::vector<Point> QuadraticPathComponent::Extrema() const {
   CubicPathComponent elevated(*this);
   return elevated.Extrema();
 }

 Point CubicPathComponent::Solve(Scalar time) const {
   return {
       CubicSolve(time, p1.x, cp1.x, cp2.x, p2.x),  // x
       CubicSolve(time, p1.y, cp1.y, cp2.y, p2.y),  // y
   };
 }

 Point CubicPathComponent::SolveDerivative(Scalar time) const {
   return {
       CubicSolveDerivative(time, p1.x, cp1.x, cp2.x, p2.x),  // x
       CubicSolveDerivative(time, p1.y, cp1.y, cp2.y, p2.y),  // y
   };
 }

 std::vector<Point> CubicPathComponent::CreatePolyline(Scalar tolerance) const {
   auto quads = ToQuadraticPathComponents(.1);
   std::vector<Point> points;
   for (const auto& quad : quads) {
     quad.FillPointsForPolyline(points, tolerance);
   }
   return points;
 }

 inline QuadraticPathComponent CubicPathComponent::Lower() const {
   return QuadraticPathComponent(3.0 * (cp1 - p1), 3.0 * (cp2 - cp1),
                                 3.0 * (p2 - cp2));
 }

 CubicPathComponent CubicPathComponent::Subsegment(Scalar t0, Scalar t1) const {
   auto p0 = Solve(t0);
   auto p3 = Solve(t1);
   auto d = Lower();
   auto scale = (t1 - t0) * (1.0 / 3.0);
   auto p1 = p0 + scale * d.Solve(t0);
   auto p2 = p3 - scale * d.Solve(t1);
   return CubicPathComponent(p0, p1, p2, p3);
 }

 std::vector<QuadraticPathComponent>
 CubicPathComponent::ToQuadraticPathComponents(Scalar accuracy) const {
   std::vector<QuadraticPathComponent> quads;
   // The maximum error, as a vector from the cubic to the best approximating
   // quadratic, is proportional to the third derivative, which is constant
   // across the segment. Thus, the error scales down as the third power of
   // the number of subdivisions. Our strategy then is to subdivide `t` evenly.
   //
   // This is an overestimate of the error because only the component
   // perpendicular to the first derivative is important. But the simplicity is
   // appealing.

   // This magic number is the square of 36 / sqrt(3).
   // See: http://caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html
   auto max_hypot2 = 432.0 * accuracy * accuracy;
   auto p1x2 = 3.0 * cp1 - p1;
   auto p2x2 = 3.0 * cp2 - p2;
   auto p = p2x2 - p1x2;
   auto err = p.Dot(p);
   auto quad_count = std::max(1., ceil(pow(err / max_hypot2, 1. / 6.0)));

   for (size_t i = 0; i < quad_count; i++) {
     auto t0 = i / quad_count;
     auto t1 = (i + 1) / quad_count;
     auto seg = Subsegment(t0, t1);
     auto p1x2 = 3.0 * seg.cp1 - seg.p1;
     auto p2x2 = 3.0 * seg.cp2 - seg.p2;
     quads.emplace_back(
         QuadraticPathComponent(seg.p1, ((p1x2 + p2x2) / 4.0), seg.p2));
   }
   return quads;
 }

 static inline bool NearEqual(Scalar a, Scalar b, Scalar epsilon) {
   return (a > (b - epsilon)) && (a < (b + epsilon));
 }

 static inline bool NearZero(Scalar a) {
   return NearEqual(a, 0.0, 1e-12);
 }

 static void CubicPathBoundingPopulateValues(std::vector<Scalar>& values,
                                             Scalar p1,
                                             Scalar p2,
                                             Scalar p3,
                                             Scalar p4) {
   const Scalar a = 3.0 * (-p1 + 3.0 * p2 - 3.0 * p3 + p4);
   const Scalar b = 6.0 * (p1 - 2.0 * p2 + p3);
   const Scalar c = 3.0 * (p2 - p1);

   /*
    *  Boundary conditions.
    */
   if (NearZero(a)) {
     if (NearZero(b)) {
       return;
     }

     Scalar t = -c / b;
     if (t >= 0.0 && t <= 1.0) {
       values.emplace_back(t);
     }
     return;
   }

   Scalar b2Minus4AC = (b * b) - (4.0 * a * c);

   if (b2Minus4AC < 0.0) {
     return;
   }

   Scalar rootB2Minus4AC = ::sqrt(b2Minus4AC);

   {
     Scalar t = (-b + rootB2Minus4AC) / (2.0 * a);
     if (t >= 0.0 && t <= 1.0) {
       values.emplace_back(t);
     }
   }

   {
     Scalar t = (-b - rootB2Minus4AC) / (2.0 * a);
     if (t >= 0.0 && t <= 1.0) {
       values.emplace_back(t);
     }
   }
 }

 std::vector<Point> CubicPathComponent::Extrema() const {
   /*
    *  As described in: https://pomax.github.io/bezierinfo/#extremities
    */
   std::vector<Scalar> values;

   CubicPathBoundingPopulateValues(values, p1.x, cp1.x, cp2.x, p2.x);
   CubicPathBoundingPopulateValues(values, p1.y, cp1.y, cp2.y, p2.y);

   std::vector<Point> points = {p1, p2};

   for (const auto& value : values) {
     points.emplace_back(Solve(value));
   }

   return points;
 }

 }  // namespace impeller
	// Copyright 2013 The Flutter Authors. All rights reserved.
	// Use of this source code is governed by a BSD-style license that can be
	// found in the LICENSE file.

	#include "path_component.h"

	#include <cmath>

	namespace impeller {

	/*
	* Based on: https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Specific_cases
	*/

	static inline Scalar LinearSolve(Scalar t, Scalar p0, Scalar p1) {
	return p0 + t * (p1 - p0);
	}

	static inline Scalar QuadraticSolve(Scalar t, Scalar p0, Scalar p1, Scalar p2) {
	return (1 - t) * (1 - t) * p0 + //
	2 * (1 - t) * t * p1 + //
	t * t * p2;
	}

	static inline Scalar QuadraticSolveDerivative(Scalar t,
	Scalar p0,
	Scalar p1,
	Scalar p2) {
	return 2 * (1 - t) * (p1 - p0) + //
	2 * t * (p2 - p1);
	}

	static inline Scalar CubicSolve(Scalar t,
	Scalar p0,
	Scalar p1,
	Scalar p2,
	Scalar p3) {
	return (1 - t) * (1 - t) * (1 - t) * p0 + //
	3 * (1 - t) * (1 - t) * t * p1 + //
	3 * (1 - t) * t * t * p2 + //
	t * t * t * p3;
	}

	static inline Scalar CubicSolveDerivative(Scalar t,
	Scalar p0,
	Scalar p1,
	Scalar p2,
	Scalar p3) {
	return -3 * p0 * (1 - t) * (1 - t) + //
	p1 * (3 * (1 - t) * (1 - t) - 6 * (1 - t) * t) +
	p2 * (6 * (1 - t) * t - 3 * t * t) + //
	3 * p3 * t * t;
	}

	Point LinearPathComponent::Solve(Scalar time) const {
	return {
	LinearSolve(time, p1.x, p2.x), // x
	LinearSolve(time, p1.y, p2.y), // y
	};
	}

	std::vector<Point> LinearPathComponent::CreatePolyline() const {
	return {p2};
	}

	std::vector<Point> LinearPathComponent::Extrema() const {
	return {p1, p2};
	}

	Point QuadraticPathComponent::Solve(Scalar time) const {
	return {
	QuadraticSolve(time, p1.x, cp.x, p2.x), // x
	QuadraticSolve(time, p1.y, cp.y, p2.y), // y
	};
	}

	Point QuadraticPathComponent::SolveDerivative(Scalar time) const {
	return {
	QuadraticSolveDerivative(time, p1.x, cp.x, p2.x), // x
	QuadraticSolveDerivative(time, p1.y, cp.y, p2.y), // y
	};
	}

	static Scalar ApproximateParabolaIntegral(Scalar x) {
	constexpr Scalar d = 0.67;
	return x / (1.0 - d + sqrt(sqrt(pow(d, 4) + 0.25 * x * x)));
	}

	std::vector<Point> QuadraticPathComponent::CreatePolyline(
	Scalar tolerance) const {
	std::vector<Point> points;
	FillPointsForPolyline(points, tolerance);
	return points;
	}

	void QuadraticPathComponent::FillPointsForPolyline(std::vector<Point>& points,
	Scalar tolerance) const {
	auto sqrt_tolerance = sqrt(tolerance);

	auto d01 = cp - p1;
	auto d12 = p2 - cp;
	auto dd = d01 - d12;
	auto cross = (p2 - p1).Cross(dd);
	auto x0 = d01.Dot(dd) * 1 / cross;
	auto x2 = d12.Dot(dd) * 1 / cross;
	auto scale = abs(cross / (hypot(dd.x, dd.y) * (x2 - x0)));

	auto a0 = ApproximateParabolaIntegral(x0);
	auto a2 = ApproximateParabolaIntegral(x2);
	Scalar val = 0.f;
	if (std::isfinite(scale)) {
	auto da = abs(a2 - a0);
	auto sqrt_scale = sqrt(scale);
	if ((x0 < 0 && x2 < 0) \|\| (x0 >= 0 && x2 >= 0)) {
	val = da * sqrt_scale;
	} else {
	// cusp case
	auto xmin = sqrt_tolerance / sqrt_scale;
	val = sqrt_tolerance * da / ApproximateParabolaIntegral(xmin);
	}
	}
	auto u0 = ApproximateParabolaIntegral(a0);
	auto u2 = ApproximateParabolaIntegral(a2);
	auto uscale = 1 / (u2 - u0);

	auto line_count = std::max(1., ceil(0.5 * val / sqrt_tolerance));
	auto step = 1 / line_count;
	for (size_t i = 1; i < line_count; i += 1) {
	auto u = i * step;
	auto a = a0 + (a2 - a0) * u;
	auto t = (ApproximateParabolaIntegral(a) - u0) * uscale;
	points.emplace_back(Solve(t));
	}
	points.emplace_back(p2);
	}

	std::vector<Point> QuadraticPathComponent::Extrema() const {
	CubicPathComponent elevated(*this);
	return elevated.Extrema();
	}

	Point CubicPathComponent::Solve(Scalar time) const {
	return {
	CubicSolve(time, p1.x, cp1.x, cp2.x, p2.x), // x
	CubicSolve(time, p1.y, cp1.y, cp2.y, p2.y), // y
	};
	}

	Point CubicPathComponent::SolveDerivative(Scalar time) const {
	return {
	CubicSolveDerivative(time, p1.x, cp1.x, cp2.x, p2.x), // x
	CubicSolveDerivative(time, p1.y, cp1.y, cp2.y, p2.y), // y
	};
	}

	std::vector<Point> CubicPathComponent::CreatePolyline(Scalar tolerance) const {
	auto quads = ToQuadraticPathComponents(.1);
	std::vector<Point> points;
	for (const auto& quad : quads) {
	quad.FillPointsForPolyline(points, tolerance);
	}
	return points;
	}

	inline QuadraticPathComponent CubicPathComponent::Lower() const {
	return QuadraticPathComponent(3.0 * (cp1 - p1), 3.0 * (cp2 - cp1),
	3.0 * (p2 - cp2));
	}

	CubicPathComponent CubicPathComponent::Subsegment(Scalar t0, Scalar t1) const {
	auto p0 = Solve(t0);
	auto p3 = Solve(t1);
	auto d = Lower();
	auto scale = (t1 - t0) * (1.0 / 3.0);
	auto p1 = p0 + scale * d.Solve(t0);
	auto p2 = p3 - scale * d.Solve(t1);
	return CubicPathComponent(p0, p1, p2, p3);
	}

	std::vector<QuadraticPathComponent>
	CubicPathComponent::ToQuadraticPathComponents(Scalar accuracy) const {
	std::vector<QuadraticPathComponent> quads;
	// The maximum error, as a vector from the cubic to the best approximating
	// quadratic, is proportional to the third derivative, which is constant
	// across the segment. Thus, the error scales down as the third power of
	// the number of subdivisions. Our strategy then is to subdivide `t` evenly.
	//
	// This is an overestimate of the error because only the component
	// perpendicular to the first derivative is important. But the simplicity is
	// appealing.

	// This magic number is the square of 36 / sqrt(3).
	// See: http://caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html
	auto max_hypot2 = 432.0 * accuracy * accuracy;
	auto p1x2 = 3.0 * cp1 - p1;
	auto p2x2 = 3.0 * cp2 - p2;
	auto p = p2x2 - p1x2;
	auto err = p.Dot(p);
	auto quad_count = std::max(1., ceil(pow(err / max_hypot2, 1. / 6.0)));

	for (size_t i = 0; i < quad_count; i++) {
	auto t0 = i / quad_count;
	auto t1 = (i + 1) / quad_count;
	auto seg = Subsegment(t0, t1);
	auto p1x2 = 3.0 * seg.cp1 - seg.p1;
	auto p2x2 = 3.0 * seg.cp2 - seg.p2;
	quads.emplace_back(
	QuadraticPathComponent(seg.p1, ((p1x2 + p2x2) / 4.0), seg.p2));
	}
	return quads;
	}

	static inline bool NearEqual(Scalar a, Scalar b, Scalar epsilon) {
	return (a > (b - epsilon)) && (a < (b + epsilon));
	}

	static inline bool NearZero(Scalar a) {
	return NearEqual(a, 0.0, 1e-12);
	}

	static void CubicPathBoundingPopulateValues(std::vector<Scalar>& values,
	Scalar p1,
	Scalar p2,
	Scalar p3,
	Scalar p4) {
	const Scalar a = 3.0 * (-p1 + 3.0 * p2 - 3.0 * p3 + p4);
	const Scalar b = 6.0 * (p1 - 2.0 * p2 + p3);
	const Scalar c = 3.0 * (p2 - p1);

	/*
	* Boundary conditions.
	*/
	if (NearZero(a)) {
	if (NearZero(b)) {
	return;
	}

	Scalar t = -c / b;
	if (t >= 0.0 && t <= 1.0) {
	values.emplace_back(t);
	}
	return;
	}

	Scalar b2Minus4AC = (b * b) - (4.0 * a * c);

	if (b2Minus4AC < 0.0) {
	return;
	}

	Scalar rootB2Minus4AC = ::sqrt(b2Minus4AC);

	{
	Scalar t = (-b + rootB2Minus4AC) / (2.0 * a);
	if (t >= 0.0 && t <= 1.0) {
	values.emplace_back(t);
	}
	}

	{
	Scalar t = (-b - rootB2Minus4AC) / (2.0 * a);
	if (t >= 0.0 && t <= 1.0) {
	values.emplace_back(t);
	}
	}
	}

	std::vector<Point> CubicPathComponent::Extrema() const {
	/*
	* As described in: https://pomax.github.io/bezierinfo/#extremities
	*/
	std::vector<Scalar> values;

	CubicPathBoundingPopulateValues(values, p1.x, cp1.x, cp2.x, p2.x);
	CubicPathBoundingPopulateValues(values, p1.y, cp1.y, cp2.y, p2.y);

	std::vector<Point> points = {p1, p2};

	for (const auto& value : values) {
	points.emplace_back(Solve(value));
	}

	return points;
	}

	} // namespace impeller