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+/* -*- Mode: C++; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 2 -*- */
+/* vim: set ts=8 sts=2 et sw=2 tw=80: */
+/* This Source Code Form is subject to the terms of the Mozilla Public
+ * License, v. 2.0. If a copy of the MPL was not distributed with this
+ * file, You can obtain one at http://mozilla.org/MPL/2.0/. */
+
+#include "BezierUtils.h"
+
+#include "PathHelpers.h"
+
+namespace mozilla {
+namespace gfx {
+
+Point GetBezierPoint(const Bezier& aBezier, Float t) {
+ Float s = 1.0f - t;
+
+ return Point(aBezier.mPoints[0].x * s * s * s +
+ 3.0f * aBezier.mPoints[1].x * t * s * s +
+ 3.0f * aBezier.mPoints[2].x * t * t * s +
+ aBezier.mPoints[3].x * t * t * t,
+ aBezier.mPoints[0].y * s * s * s +
+ 3.0f * aBezier.mPoints[1].y * t * s * s +
+ 3.0f * aBezier.mPoints[2].y * t * t * s +
+ aBezier.mPoints[3].y * t * t * t);
+}
+
+Point GetBezierDifferential(const Bezier& aBezier, Float t) {
+ // Return P'(t).
+
+ Float s = 1.0f - t;
+
+ return Point(
+ -3.0f * ((aBezier.mPoints[0].x - aBezier.mPoints[1].x) * s * s +
+ 2.0f * (aBezier.mPoints[1].x - aBezier.mPoints[2].x) * t * s +
+ (aBezier.mPoints[2].x - aBezier.mPoints[3].x) * t * t),
+ -3.0f * ((aBezier.mPoints[0].y - aBezier.mPoints[1].y) * s * s +
+ 2.0f * (aBezier.mPoints[1].y - aBezier.mPoints[2].y) * t * s +
+ (aBezier.mPoints[2].y - aBezier.mPoints[3].y) * t * t));
+}
+
+Point GetBezierDifferential2(const Bezier& aBezier, Float t) {
+ // Return P''(t).
+
+ Float s = 1.0f - t;
+
+ return Point(6.0f * ((aBezier.mPoints[0].x - aBezier.mPoints[1].x) * s -
+ (aBezier.mPoints[1].x - aBezier.mPoints[2].x) * (s - t) -
+ (aBezier.mPoints[2].x - aBezier.mPoints[3].x) * t),
+ 6.0f * ((aBezier.mPoints[0].y - aBezier.mPoints[1].y) * s -
+ (aBezier.mPoints[1].y - aBezier.mPoints[2].y) * (s - t) -
+ (aBezier.mPoints[2].y - aBezier.mPoints[3].y) * t));
+}
+
+Float GetBezierLength(const Bezier& aBezier, Float a, Float b) {
+ if (a < 0.5f && b > 0.5f) {
+ // To increase the accuracy, split into two parts.
+ return GetBezierLength(aBezier, a, 0.5f) +
+ GetBezierLength(aBezier, 0.5f, b);
+ }
+
+ // Calculate length of simple bezier curve with Simpson's rule.
+ // _
+ // / b
+ // length = | |P'(x)| dx
+ // _/ a
+ //
+ // b - a a + b
+ // = ----- [ |P'(a)| + 4 |P'(-----)| + |P'(b)| ]
+ // 6 2
+
+ Float fa = GetBezierDifferential(aBezier, a).Length();
+ Float fab = GetBezierDifferential(aBezier, (a + b) / 2.0f).Length();
+ Float fb = GetBezierDifferential(aBezier, b).Length();
+
+ return (b - a) / 6.0f * (fa + 4.0f * fab + fb);
+}
+
+static void SplitBezierA(Bezier* aSubBezier, const Bezier& aBezier, Float t) {
+ // Split bezier curve into [0,t] and [t,1] parts, and return [0,t] part.
+
+ Float s = 1.0f - t;
+
+ Point tmp1;
+ Point tmp2;
+
+ aSubBezier->mPoints[0] = aBezier.mPoints[0];
+
+ aSubBezier->mPoints[1] = aBezier.mPoints[0] * s + aBezier.mPoints[1] * t;
+ tmp1 = aBezier.mPoints[1] * s + aBezier.mPoints[2] * t;
+ tmp2 = aBezier.mPoints[2] * s + aBezier.mPoints[3] * t;
+
+ aSubBezier->mPoints[2] = aSubBezier->mPoints[1] * s + tmp1 * t;
+ tmp1 = tmp1 * s + tmp2 * t;
+
+ aSubBezier->mPoints[3] = aSubBezier->mPoints[2] * s + tmp1 * t;
+}
+
+static void SplitBezierB(Bezier* aSubBezier, const Bezier& aBezier, Float t) {
+ // Split bezier curve into [0,t] and [t,1] parts, and return [t,1] part.
+
+ Float s = 1.0f - t;
+
+ Point tmp1;
+ Point tmp2;
+
+ aSubBezier->mPoints[3] = aBezier.mPoints[3];
+
+ aSubBezier->mPoints[2] = aBezier.mPoints[2] * s + aBezier.mPoints[3] * t;
+ tmp1 = aBezier.mPoints[1] * s + aBezier.mPoints[2] * t;
+ tmp2 = aBezier.mPoints[0] * s + aBezier.mPoints[1] * t;
+
+ aSubBezier->mPoints[1] = tmp1 * s + aSubBezier->mPoints[2] * t;
+ tmp1 = tmp2 * s + tmp1 * t;
+
+ aSubBezier->mPoints[0] = tmp1 * s + aSubBezier->mPoints[1] * t;
+}
+
+void GetSubBezier(Bezier* aSubBezier, const Bezier& aBezier, Float t1,
+ Float t2) {
+ Bezier tmp;
+ SplitBezierB(&tmp, aBezier, t1);
+
+ Float range = 1.0f - t1;
+ if (range == 0.0f) {
+ *aSubBezier = tmp;
+ } else {
+ SplitBezierA(aSubBezier, tmp, (t2 - t1) / range);
+ }
+}
+
+static Point BisectBezierNearestPoint(const Bezier& aBezier,
+ const Point& aTarget, Float* aT) {
+ // Find a nearest point on bezier curve with Binary search.
+ // Called from FindBezierNearestPoint.
+
+ Float lower = 0.0f;
+ Float upper = 1.0f;
+ Float t;
+
+ Point P, lastP;
+ const size_t MAX_LOOP = 32;
+ const Float DIST_MARGIN = 0.1f;
+ const Float DIST_MARGIN_SQUARE = DIST_MARGIN * DIST_MARGIN;
+ const Float DIFF = 0.0001f;
+ for (size_t i = 0; i < MAX_LOOP; i++) {
+ t = (upper + lower) / 2.0f;
+ P = GetBezierPoint(aBezier, t);
+
+ // Check if it converged.
+ if (i > 0 && (lastP - P).LengthSquare() < DIST_MARGIN_SQUARE) {
+ break;
+ }
+
+ Float distSquare = (P - aTarget).LengthSquare();
+ if ((GetBezierPoint(aBezier, t + DIFF) - aTarget).LengthSquare() <
+ distSquare) {
+ lower = t;
+ } else if ((GetBezierPoint(aBezier, t - DIFF) - aTarget).LengthSquare() <
+ distSquare) {
+ upper = t;
+ } else {
+ break;
+ }
+
+ lastP = P;
+ }
+
+ if (aT) {
+ *aT = t;
+ }
+
+ return P;
+}
+
+Point FindBezierNearestPoint(const Bezier& aBezier, const Point& aTarget,
+ Float aInitialT, Float* aT) {
+ // Find a nearest point on bezier curve with Newton's method.
+ // It converges within 4 iterations in most cases.
+ //
+ // f(t_n)
+ // t_{n+1} = t_n - ---------
+ // f'(t_n)
+ //
+ // d 2
+ // f(t) = ---- | P(t) - aTarget |
+ // dt
+
+ Float t = aInitialT;
+ Point P;
+ Point lastP = GetBezierPoint(aBezier, t);
+
+ const size_t MAX_LOOP = 4;
+ const Float DIST_MARGIN = 0.1f;
+ const Float DIST_MARGIN_SQUARE = DIST_MARGIN * DIST_MARGIN;
+ for (size_t i = 0; i <= MAX_LOOP; i++) {
+ Point dP = GetBezierDifferential(aBezier, t);
+ Point ddP = GetBezierDifferential2(aBezier, t);
+ Float f = 2.0f * (lastP.DotProduct(dP) - aTarget.DotProduct(dP));
+ Float df = 2.0f * (dP.DotProduct(dP) + lastP.DotProduct(ddP) -
+ aTarget.DotProduct(ddP));
+ t = t - f / df;
+ P = GetBezierPoint(aBezier, t);
+ if ((P - lastP).LengthSquare() < DIST_MARGIN_SQUARE) {
+ break;
+ }
+ lastP = P;
+
+ if (i == MAX_LOOP) {
+ // If aInitialT is too bad, it won't converge in a few iterations,
+ // fallback to binary search.
+ return BisectBezierNearestPoint(aBezier, aTarget, aT);
+ }
+ }
+
+ if (aT) {
+ *aT = t;
+ }
+
+ return P;
+}
+
+void GetBezierPointsForCorner(Bezier* aBezier, Corner aCorner,
+ const Point& aCornerPoint,
+ const Size& aCornerSize) {
+ // Calculate bezier control points for elliptic arc.
+
+ const Float signsList[4][2] = {
+ {+1.0f, +1.0f}, {-1.0f, +1.0f}, {-1.0f, -1.0f}, {+1.0f, -1.0f}};
+ const Float(&signs)[2] = signsList[aCorner];
+
+ aBezier->mPoints[0] = aCornerPoint;
+ aBezier->mPoints[0].x += signs[0] * aCornerSize.width;
+
+ aBezier->mPoints[1] = aBezier->mPoints[0];
+ aBezier->mPoints[1].x -= signs[0] * aCornerSize.width * kKappaFactor;
+
+ aBezier->mPoints[3] = aCornerPoint;
+ aBezier->mPoints[3].y += signs[1] * aCornerSize.height;
+
+ aBezier->mPoints[2] = aBezier->mPoints[3];
+ aBezier->mPoints[2].y -= signs[1] * aCornerSize.height * kKappaFactor;
+}
+
+Float GetQuarterEllipticArcLength(Float a, Float b) {
+ // Calculate the approximate length of a quarter elliptic arc formed by radii
+ // (a, b), by Ramanujan's approximation of the perimeter p of an ellipse.
+ // _ _
+ // | 2 |
+ // | 3 * (a - b) |
+ // p = PI | (a + b) + ------------------------------------------- |
+ // | 2 2 |
+ // |_ 10 * (a + b) + sqrt(a + 14 * a * b + b ) _|
+ //
+ // _ _
+ // | 2 |
+ // | 3 * (a - b) |
+ // = PI | (a + b) + -------------------------------------------------- |
+ // | 2 2 |
+ // |_ 10 * (a + b) + sqrt(4 * (a + b) - 3 * (a - b) ) _|
+ //
+ // _ _
+ // | 2 |
+ // | 3 * S |
+ // = PI | A + -------------------------------------- |
+ // | 2 2 |
+ // |_ 10 * A + sqrt(4 * A - 3 * S ) _|
+ //
+ // where A = a + b, S = a - b
+
+ Float A = a + b, S = a - b;
+ Float A2 = A * A, S2 = S * S;
+ Float p = M_PI * (A + 3.0f * S2 / (10.0f * A + sqrt(4.0f * A2 - 3.0f * S2)));
+ return p / 4.0f;
+}
+
+Float CalculateDistanceToEllipticArc(const Point& P, const Point& normal,
+ const Point& origin, Float width,
+ Float height) {
+ // Solve following equations with n and return smaller n.
+ //
+ // / (x, y) = P + n * normal
+ // |
+ // < _ _ 2 _ _ 2
+ // | | x - origin.x | | y - origin.y |
+ // | | ------------ | + | ------------ | = 1
+ // \ |_ width _| |_ height _|
+
+ Float a = (P.x - origin.x) / width;
+ Float b = normal.x / width;
+ Float c = (P.y - origin.y) / height;
+ Float d = normal.y / height;
+
+ Float A = b * b + d * d;
+ // In the quadratic formulat B would be 2*(a*b+c*d), however we factor the 2
+ // out Here which cancels out later.
+ Float B = a * b + c * d;
+ Float C = a * a + c * c - 1.0;
+
+ Float signB = 1.0;
+ if (B < 0.0) {
+ signB = -1.0;
+ }
+
+ // 2nd degree polynomials are typically computed using the formulae
+ // r1 = -(B - sqrt(delta)) / (2 * A)
+ // r2 = -(B + sqrt(delta)) / (2 * A)
+ // However B - sqrt(delta) can be an inportant source of precision loss for
+ // one of the roots when computing the difference between two similar and
+ // large numbers. To avoid that we pick the root with no precision loss in r1
+ // and compute r2 using the Citardauq formula.
+ // Factoring out 2 from B earlier let
+ Float S = B + signB * sqrt(B * B - A * C);
+ Float r1 = -S / A;
+ Float r2 = -C / S;
+
+#ifdef DEBUG
+ Float epsilon = (Float)0.001;
+ MOZ_ASSERT(r1 >= -epsilon);
+ MOZ_ASSERT(r2 >= -epsilon);
+#endif
+
+ return std::max((r1 < r2 ? r1 : r2), (Float)0.0);
+}
+
+} // namespace gfx
+} // namespace mozilla
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