diff -r 000000000000 -r 6474c204b198 gfx/skia/trunk/src/core/SkGeometry.cpp --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/gfx/skia/trunk/src/core/SkGeometry.cpp Wed Dec 31 06:09:35 2014 +0100 @@ -0,0 +1,1468 @@ +/* + * Copyright 2006 The Android Open Source Project + * + * Use of this source code is governed by a BSD-style license that can be + * found in the LICENSE file. + */ + +#include "SkGeometry.h" +#include "SkMatrix.h" + +bool SkXRayCrossesLine(const SkXRay& pt, + const SkPoint pts[2], + bool* ambiguous) { + if (ambiguous) { + *ambiguous = false; + } + // Determine quick discards. + // Consider query line going exactly through point 0 to not + // intersect, for symmetry with SkXRayCrossesMonotonicCubic. + if (pt.fY == pts[0].fY) { + if (ambiguous) { + *ambiguous = true; + } + return false; + } + if (pt.fY < pts[0].fY && pt.fY < pts[1].fY) + return false; + if (pt.fY > pts[0].fY && pt.fY > pts[1].fY) + return false; + if (pt.fX > pts[0].fX && pt.fX > pts[1].fX) + return false; + // Determine degenerate cases + if (SkScalarNearlyZero(pts[0].fY - pts[1].fY)) + return false; + if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) { + // We've already determined the query point lies within the + // vertical range of the line segment. + if (pt.fX <= pts[0].fX) { + if (ambiguous) { + *ambiguous = (pt.fY == pts[1].fY); + } + return true; + } + return false; + } + // Ambiguity check + if (pt.fY == pts[1].fY) { + if (pt.fX <= pts[1].fX) { + if (ambiguous) { + *ambiguous = true; + } + return true; + } + return false; + } + // Full line segment evaluation + SkScalar delta_y = pts[1].fY - pts[0].fY; + SkScalar delta_x = pts[1].fX - pts[0].fX; + SkScalar slope = SkScalarDiv(delta_y, delta_x); + SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX); + // Solve for x coordinate at y = pt.fY + SkScalar x = SkScalarDiv(pt.fY - b, slope); + return pt.fX <= x; +} + +/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes + involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul. + May also introduce overflow of fixed when we compute our setup. +*/ +// #define DIRECT_EVAL_OF_POLYNOMIALS + +//////////////////////////////////////////////////////////////////////// + +static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) { + SkScalar ab = a - b; + SkScalar bc = b - c; + if (ab < 0) { + bc = -bc; + } + return ab == 0 || bc < 0; +} + +//////////////////////////////////////////////////////////////////////// + +static bool is_unit_interval(SkScalar x) { + return x > 0 && x < SK_Scalar1; +} + +static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) { + SkASSERT(ratio); + + if (numer < 0) { + numer = -numer; + denom = -denom; + } + + if (denom == 0 || numer == 0 || numer >= denom) { + return 0; + } + + SkScalar r = SkScalarDiv(numer, denom); + if (SkScalarIsNaN(r)) { + return 0; + } + SkASSERT(r >= 0 && r < SK_Scalar1); + if (r == 0) { // catch underflow if numer <<<< denom + return 0; + } + *ratio = r; + return 1; +} + +/** From Numerical Recipes in C. + + Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C]) + x1 = Q / A + x2 = C / Q +*/ +int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) { + SkASSERT(roots); + + if (A == 0) { + return valid_unit_divide(-C, B, roots); + } + + SkScalar* r = roots; + + SkScalar R = B*B - 4*A*C; + if (R < 0 || SkScalarIsNaN(R)) { // complex roots + return 0; + } + R = SkScalarSqrt(R); + + SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2; + r += valid_unit_divide(Q, A, r); + r += valid_unit_divide(C, Q, r); + if (r - roots == 2) { + if (roots[0] > roots[1]) + SkTSwap(roots[0], roots[1]); + else if (roots[0] == roots[1]) // nearly-equal? + r -= 1; // skip the double root + } + return (int)(r - roots); +} + +/////////////////////////////////////////////////////////////////////////////// +/////////////////////////////////////////////////////////////////////////////// + +static SkScalar eval_quad(const SkScalar src[], SkScalar t) { + SkASSERT(src); + SkASSERT(t >= 0 && t <= SK_Scalar1); + +#ifdef DIRECT_EVAL_OF_POLYNOMIALS + SkScalar C = src[0]; + SkScalar A = src[4] - 2 * src[2] + C; + SkScalar B = 2 * (src[2] - C); + return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); +#else + SkScalar ab = SkScalarInterp(src[0], src[2], t); + SkScalar bc = SkScalarInterp(src[2], src[4], t); + return SkScalarInterp(ab, bc, t); +#endif +} + +static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) { + SkScalar A = src[4] - 2 * src[2] + src[0]; + SkScalar B = src[2] - src[0]; + + return 2 * SkScalarMulAdd(A, t, B); +} + +static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) { + SkScalar A = src[4] - 2 * src[2] + src[0]; + SkScalar B = src[2] - src[0]; + return A + 2 * B; +} + +void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, + SkVector* tangent) { + SkASSERT(src); + SkASSERT(t >= 0 && t <= SK_Scalar1); + + if (pt) { + pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t)); + } + if (tangent) { + tangent->set(eval_quad_derivative(&src[0].fX, t), + eval_quad_derivative(&src[0].fY, t)); + } +} + +void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) { + SkASSERT(src); + + if (pt) { + SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); + SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); + SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); + SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); + pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); + } + if (tangent) { + tangent->set(eval_quad_derivative_at_half(&src[0].fX), + eval_quad_derivative_at_half(&src[0].fY)); + } +} + +static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) { + SkScalar ab = SkScalarInterp(src[0], src[2], t); + SkScalar bc = SkScalarInterp(src[2], src[4], t); + + dst[0] = src[0]; + dst[2] = ab; + dst[4] = SkScalarInterp(ab, bc, t); + dst[6] = bc; + dst[8] = src[4]; +} + +void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { + SkASSERT(t > 0 && t < SK_Scalar1); + + interp_quad_coords(&src[0].fX, &dst[0].fX, t); + interp_quad_coords(&src[0].fY, &dst[0].fY, t); +} + +void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { + SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); + SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); + SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); + SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); + + dst[0] = src[0]; + dst[1].set(x01, y01); + dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12)); + dst[3].set(x12, y12); + dst[4] = src[2]; +} + +/** Quad'(t) = At + B, where + A = 2(a - 2b + c) + B = 2(b - a) + Solve for t, only if it fits between 0 < t < 1 +*/ +int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) { + /* At + B == 0 + t = -B / A + */ + return valid_unit_divide(a - b, a - b - b + c, tValue); +} + +static inline void flatten_double_quad_extrema(SkScalar coords[14]) { + coords[2] = coords[6] = coords[4]; +} + +/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is + stored in dst[]. Guarantees that the 1/2 quads will be monotonic. + */ +int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) { + SkASSERT(src); + SkASSERT(dst); + + SkScalar a = src[0].fY; + SkScalar b = src[1].fY; + SkScalar c = src[2].fY; + + if (is_not_monotonic(a, b, c)) { + SkScalar tValue; + if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { + SkChopQuadAt(src, dst, tValue); + flatten_double_quad_extrema(&dst[0].fY); + return 1; + } + // if we get here, we need to force dst to be monotonic, even though + // we couldn't compute a unit_divide value (probably underflow). + b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; + } + dst[0].set(src[0].fX, a); + dst[1].set(src[1].fX, b); + dst[2].set(src[2].fX, c); + return 0; +} + +/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is + stored in dst[]. Guarantees that the 1/2 quads will be monotonic. + */ +int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) { + SkASSERT(src); + SkASSERT(dst); + + SkScalar a = src[0].fX; + SkScalar b = src[1].fX; + SkScalar c = src[2].fX; + + if (is_not_monotonic(a, b, c)) { + SkScalar tValue; + if (valid_unit_divide(a - b, a - b - b + c, &tValue)) { + SkChopQuadAt(src, dst, tValue); + flatten_double_quad_extrema(&dst[0].fX); + return 1; + } + // if we get here, we need to force dst to be monotonic, even though + // we couldn't compute a unit_divide value (probably underflow). + b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c; + } + dst[0].set(a, src[0].fY); + dst[1].set(b, src[1].fY); + dst[2].set(c, src[2].fY); + return 0; +} + +// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2 +// F'(t) = 2 (b - a) + 2 (a - 2b + c) t +// F''(t) = 2 (a - 2b + c) +// +// A = 2 (b - a) +// B = 2 (a - 2b + c) +// +// Maximum curvature for a quadratic means solving +// Fx' Fx'' + Fy' Fy'' = 0 +// +// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2) +// +SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) { + SkScalar Ax = src[1].fX - src[0].fX; + SkScalar Ay = src[1].fY - src[0].fY; + SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX; + SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY; + SkScalar t = 0; // 0 means don't chop + + (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); + return t; +} + +int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { + SkScalar t = SkFindQuadMaxCurvature(src); + if (t == 0) { + memcpy(dst, src, 3 * sizeof(SkPoint)); + return 1; + } else { + SkChopQuadAt(src, dst, t); + return 2; + } +} + +#define SK_ScalarTwoThirds (0.666666666f) + +void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { + const SkScalar scale = SK_ScalarTwoThirds; + dst[0] = src[0]; + dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale), + src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale)); + dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale), + src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale)); + dst[3] = src[2]; +} + +////////////////////////////////////////////////////////////////////////////// +///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// +////////////////////////////////////////////////////////////////////////////// + +static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) { + coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0]; + coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]); + coeff[2] = 3*(pt[2] - pt[0]); + coeff[3] = pt[0]; +} + +void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) { + SkASSERT(pts); + + if (cx) { + get_cubic_coeff(&pts[0].fX, cx); + } + if (cy) { + get_cubic_coeff(&pts[0].fY, cy); + } +} + +static SkScalar eval_cubic(const SkScalar src[], SkScalar t) { + SkASSERT(src); + SkASSERT(t >= 0 && t <= SK_Scalar1); + + if (t == 0) { + return src[0]; + } + +#ifdef DIRECT_EVAL_OF_POLYNOMIALS + SkScalar D = src[0]; + SkScalar A = src[6] + 3*(src[2] - src[4]) - D; + SkScalar B = 3*(src[4] - src[2] - src[2] + D); + SkScalar C = 3*(src[2] - D); + + return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D); +#else + SkScalar ab = SkScalarInterp(src[0], src[2], t); + SkScalar bc = SkScalarInterp(src[2], src[4], t); + SkScalar cd = SkScalarInterp(src[4], src[6], t); + SkScalar abc = SkScalarInterp(ab, bc, t); + SkScalar bcd = SkScalarInterp(bc, cd, t); + return SkScalarInterp(abc, bcd, t); +#endif +} + +/** return At^2 + Bt + C +*/ +static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) { + SkASSERT(t >= 0 && t <= SK_Scalar1); + + return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); +} + +static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) { + SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; + SkScalar B = 2*(src[4] - 2 * src[2] + src[0]); + SkScalar C = src[2] - src[0]; + + return eval_quadratic(A, B, C, t); +} + +static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) { + SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0]; + SkScalar B = src[4] - 2 * src[2] + src[0]; + + return SkScalarMulAdd(A, t, B); +} + +void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, + SkVector* tangent, SkVector* curvature) { + SkASSERT(src); + SkASSERT(t >= 0 && t <= SK_Scalar1); + + if (loc) { + loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t)); + } + if (tangent) { + tangent->set(eval_cubic_derivative(&src[0].fX, t), + eval_cubic_derivative(&src[0].fY, t)); + } + if (curvature) { + curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t), + eval_cubic_2ndDerivative(&src[0].fY, t)); + } +} + +/** Cubic'(t) = At^2 + Bt + C, where + A = 3(-a + 3(b - c) + d) + B = 6(a - 2b + c) + C = 3(b - a) + Solve for t, keeping only those that fit betwee 0 < t < 1 +*/ +int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, + SkScalar tValues[2]) { + // we divide A,B,C by 3 to simplify + SkScalar A = d - a + 3*(b - c); + SkScalar B = 2*(a - b - b + c); + SkScalar C = b - a; + + return SkFindUnitQuadRoots(A, B, C, tValues); +} + +static void interp_cubic_coords(const SkScalar* src, SkScalar* dst, + SkScalar t) { + SkScalar ab = SkScalarInterp(src[0], src[2], t); + SkScalar bc = SkScalarInterp(src[2], src[4], t); + SkScalar cd = SkScalarInterp(src[4], src[6], t); + SkScalar abc = SkScalarInterp(ab, bc, t); + SkScalar bcd = SkScalarInterp(bc, cd, t); + SkScalar abcd = SkScalarInterp(abc, bcd, t); + + dst[0] = src[0]; + dst[2] = ab; + dst[4] = abc; + dst[6] = abcd; + dst[8] = bcd; + dst[10] = cd; + dst[12] = src[6]; +} + +void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { + SkASSERT(t > 0 && t < SK_Scalar1); + + interp_cubic_coords(&src[0].fX, &dst[0].fX, t); + interp_cubic_coords(&src[0].fY, &dst[0].fY, t); +} + +/* http://code.google.com/p/skia/issues/detail?id=32 + + This test code would fail when we didn't check the return result of + valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is + that after the first chop, the parameters to valid_unit_divide are equal + (thanks to finite float precision and rounding in the subtracts). Thus + even though the 2nd tValue looks < 1.0, after we renormalize it, we end + up with 1.0, hence the need to check and just return the last cubic as + a degenerate clump of 4 points in the sampe place. + + static void test_cubic() { + SkPoint src[4] = { + { 556.25000, 523.03003 }, + { 556.23999, 522.96002 }, + { 556.21997, 522.89001 }, + { 556.21997, 522.82001 } + }; + SkPoint dst[10]; + SkScalar tval[] = { 0.33333334f, 0.99999994f }; + SkChopCubicAt(src, dst, tval, 2); + } + */ + +void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], + const SkScalar tValues[], int roots) { +#ifdef SK_DEBUG + { + for (int i = 0; i < roots - 1; i++) + { + SkASSERT(is_unit_interval(tValues[i])); + SkASSERT(is_unit_interval(tValues[i+1])); + SkASSERT(tValues[i] < tValues[i+1]); + } + } +#endif + + if (dst) { + if (roots == 0) { // nothing to chop + memcpy(dst, src, 4*sizeof(SkPoint)); + } else { + SkScalar t = tValues[0]; + SkPoint tmp[4]; + + for (int i = 0; i < roots; i++) { + SkChopCubicAt(src, dst, t); + if (i == roots - 1) { + break; + } + + dst += 3; + // have src point to the remaining cubic (after the chop) + memcpy(tmp, dst, 4 * sizeof(SkPoint)); + src = tmp; + + // watch out in case the renormalized t isn't in range + if (!valid_unit_divide(tValues[i+1] - tValues[i], + SK_Scalar1 - tValues[i], &t)) { + // if we can't, just create a degenerate cubic + dst[4] = dst[5] = dst[6] = src[3]; + break; + } + } + } + } +} + +void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) { + SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX); + SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY); + SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX); + SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY); + SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX); + SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY); + + SkScalar x012 = SkScalarAve(x01, x12); + SkScalar y012 = SkScalarAve(y01, y12); + SkScalar x123 = SkScalarAve(x12, x23); + SkScalar y123 = SkScalarAve(y12, y23); + + dst[0] = src[0]; + dst[1].set(x01, y01); + dst[2].set(x012, y012); + dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123)); + dst[4].set(x123, y123); + dst[5].set(x23, y23); + dst[6] = src[3]; +} + +static void flatten_double_cubic_extrema(SkScalar coords[14]) { + coords[4] = coords[8] = coords[6]; +} + +/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that + the resulting beziers are monotonic in Y. This is called by the scan + converter. Depending on what is returned, dst[] is treated as follows: + 0 dst[0..3] is the original cubic + 1 dst[0..3] and dst[3..6] are the two new cubics + 2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics + If dst == null, it is ignored and only the count is returned. +*/ +int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) { + SkScalar tValues[2]; + int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY, + src[3].fY, tValues); + + SkChopCubicAt(src, dst, tValues, roots); + if (dst && roots > 0) { + // we do some cleanup to ensure our Y extrema are flat + flatten_double_cubic_extrema(&dst[0].fY); + if (roots == 2) { + flatten_double_cubic_extrema(&dst[3].fY); + } + } + return roots; +} + +int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) { + SkScalar tValues[2]; + int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX, + src[3].fX, tValues); + + SkChopCubicAt(src, dst, tValues, roots); + if (dst && roots > 0) { + // we do some cleanup to ensure our Y extrema are flat + flatten_double_cubic_extrema(&dst[0].fX); + if (roots == 2) { + flatten_double_cubic_extrema(&dst[3].fX); + } + } + return roots; +} + +/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html + + Inflection means that curvature is zero. + Curvature is [F' x F''] / [F'^3] + So we solve F'x X F''y - F'y X F''y == 0 + After some canceling of the cubic term, we get + A = b - a + B = c - 2b + a + C = d - 3c + 3b - a + (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0 +*/ +int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) { + SkScalar Ax = src[1].fX - src[0].fX; + SkScalar Ay = src[1].fY - src[0].fY; + SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX; + SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY; + SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX; + SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY; + + return SkFindUnitQuadRoots(Bx*Cy - By*Cx, + Ax*Cy - Ay*Cx, + Ax*By - Ay*Bx, + tValues); +} + +int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) { + SkScalar tValues[2]; + int count = SkFindCubicInflections(src, tValues); + + if (dst) { + if (count == 0) { + memcpy(dst, src, 4 * sizeof(SkPoint)); + } else { + SkChopCubicAt(src, dst, tValues, count); + } + } + return count + 1; +} + +template void bubble_sort(T array[], int count) { + for (int i = count - 1; i > 0; --i) + for (int j = i; j > 0; --j) + if (array[j] < array[j-1]) + { + T tmp(array[j]); + array[j] = array[j-1]; + array[j-1] = tmp; + } +} + +/** + * Given an array and count, remove all pair-wise duplicates from the array, + * keeping the existing sorting, and return the new count + */ +static int collaps_duplicates(SkScalar array[], int count) { + for (int n = count; n > 1; --n) { + if (array[0] == array[1]) { + for (int i = 1; i < n; ++i) { + array[i - 1] = array[i]; + } + count -= 1; + } else { + array += 1; + } + } + return count; +} + +#ifdef SK_DEBUG + +#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array) + +static void test_collaps_duplicates() { + static bool gOnce; + if (gOnce) { return; } + gOnce = true; + const SkScalar src0[] = { 0 }; + const SkScalar src1[] = { 0, 0 }; + const SkScalar src2[] = { 0, 1 }; + const SkScalar src3[] = { 0, 0, 0 }; + const SkScalar src4[] = { 0, 0, 1 }; + const SkScalar src5[] = { 0, 1, 1 }; + const SkScalar src6[] = { 0, 1, 2 }; + const struct { + const SkScalar* fData; + int fCount; + int fCollapsedCount; + } data[] = { + { TEST_COLLAPS_ENTRY(src0), 1 }, + { TEST_COLLAPS_ENTRY(src1), 1 }, + { TEST_COLLAPS_ENTRY(src2), 2 }, + { TEST_COLLAPS_ENTRY(src3), 1 }, + { TEST_COLLAPS_ENTRY(src4), 2 }, + { TEST_COLLAPS_ENTRY(src5), 2 }, + { TEST_COLLAPS_ENTRY(src6), 3 }, + }; + for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) { + SkScalar dst[3]; + memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0])); + int count = collaps_duplicates(dst, data[i].fCount); + SkASSERT(data[i].fCollapsedCount == count); + for (int j = 1; j < count; ++j) { + SkASSERT(dst[j-1] < dst[j]); + } + } +} +#endif + +static SkScalar SkScalarCubeRoot(SkScalar x) { + return SkScalarPow(x, 0.3333333f); +} + +/* Solve coeff(t) == 0, returning the number of roots that + lie withing 0 < t < 1. + coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] + + Eliminates repeated roots (so that all tValues are distinct, and are always + in increasing order. +*/ +static int solve_cubic_poly(const SkScalar coeff[4], SkScalar tValues[3]) { + if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic + return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); + } + + SkScalar a, b, c, Q, R; + + { + SkASSERT(coeff[0] != 0); + + SkScalar inva = SkScalarInvert(coeff[0]); + a = coeff[1] * inva; + b = coeff[2] * inva; + c = coeff[3] * inva; + } + Q = (a*a - b*3) / 9; + R = (2*a*a*a - 9*a*b + 27*c) / 54; + + SkScalar Q3 = Q * Q * Q; + SkScalar R2MinusQ3 = R * R - Q3; + SkScalar adiv3 = a / 3; + + SkScalar* roots = tValues; + SkScalar r; + + if (R2MinusQ3 < 0) { // we have 3 real roots + SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3)); + SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); + + r = neg2RootQ * SkScalarCos(theta/3) - adiv3; + if (is_unit_interval(r)) { + *roots++ = r; + } + r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3; + if (is_unit_interval(r)) { + *roots++ = r; + } + r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3; + if (is_unit_interval(r)) { + *roots++ = r; + } + SkDEBUGCODE(test_collaps_duplicates();) + + // now sort the roots + int count = (int)(roots - tValues); + SkASSERT((unsigned)count <= 3); + bubble_sort(tValues, count); + count = collaps_duplicates(tValues, count); + roots = tValues + count; // so we compute the proper count below + } else { // we have 1 real root + SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3); + A = SkScalarCubeRoot(A); + if (R > 0) { + A = -A; + } + if (A != 0) { + A += Q / A; + } + r = A - adiv3; + if (is_unit_interval(r)) { + *roots++ = r; + } + } + + return (int)(roots - tValues); +} + +/* Looking for F' dot F'' == 0 + + A = b - a + B = c - 2b + a + C = d - 3c + 3b - a + + F' = 3Ct^2 + 6Bt + 3A + F'' = 6Ct + 6B + + F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB +*/ +static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) { + SkScalar a = src[2] - src[0]; + SkScalar b = src[4] - 2 * src[2] + src[0]; + SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0]; + + coeff[0] = c * c; + coeff[1] = 3 * b * c; + coeff[2] = 2 * b * b + c * a; + coeff[3] = a * b; +} + +/* Looking for F' dot F'' == 0 + + A = b - a + B = c - 2b + a + C = d - 3c + 3b - a + + F' = 3Ct^2 + 6Bt + 3A + F'' = 6Ct + 6B + + F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB +*/ +int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) { + SkScalar coeffX[4], coeffY[4]; + int i; + + formulate_F1DotF2(&src[0].fX, coeffX); + formulate_F1DotF2(&src[0].fY, coeffY); + + for (i = 0; i < 4; i++) { + coeffX[i] += coeffY[i]; + } + + SkScalar t[3]; + int count = solve_cubic_poly(coeffX, t); + int maxCount = 0; + + // now remove extrema where the curvature is zero (mins) + // !!!! need a test for this !!!! + for (i = 0; i < count; i++) { + // if (not_min_curvature()) + if (t[i] > 0 && t[i] < SK_Scalar1) { + tValues[maxCount++] = t[i]; + } + } + return maxCount; +} + +int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], + SkScalar tValues[3]) { + SkScalar t_storage[3]; + + if (tValues == NULL) { + tValues = t_storage; + } + + int count = SkFindCubicMaxCurvature(src, tValues); + + if (dst) { + if (count == 0) { + memcpy(dst, src, 4 * sizeof(SkPoint)); + } else { + SkChopCubicAt(src, dst, tValues, count); + } + } + return count + 1; +} + +bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4], + bool* ambiguous) { + if (ambiguous) { + *ambiguous = false; + } + + // Find the minimum and maximum y of the extrema, which are the + // first and last points since this cubic is monotonic + SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY); + SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY); + + if (pt.fY == cubic[0].fY + || pt.fY < min_y + || pt.fY > max_y) { + // The query line definitely does not cross the curve + if (ambiguous) { + *ambiguous = (pt.fY == cubic[0].fY); + } + return false; + } + + bool pt_at_extremum = (pt.fY == cubic[3].fY); + + SkScalar min_x = + SkMinScalar( + SkMinScalar( + SkMinScalar(cubic[0].fX, cubic[1].fX), + cubic[2].fX), + cubic[3].fX); + if (pt.fX < min_x) { + // The query line definitely crosses the curve + if (ambiguous) { + *ambiguous = pt_at_extremum; + } + return true; + } + + SkScalar max_x = + SkMaxScalar( + SkMaxScalar( + SkMaxScalar(cubic[0].fX, cubic[1].fX), + cubic[2].fX), + cubic[3].fX); + if (pt.fX > max_x) { + // The query line definitely does not cross the curve + return false; + } + + // Do a binary search to find the parameter value which makes y as + // close as possible to the query point. See whether the query + // line's origin is to the left of the associated x coordinate. + + // kMaxIter is chosen as the number of mantissa bits for a float, + // since there's no way we are going to get more precision by + // iterating more times than that. + const int kMaxIter = 23; + SkPoint eval; + int iter = 0; + SkScalar upper_t; + SkScalar lower_t; + // Need to invert direction of t parameter if cubic goes up + // instead of down + if (cubic[3].fY > cubic[0].fY) { + upper_t = SK_Scalar1; + lower_t = 0; + } else { + upper_t = 0; + lower_t = SK_Scalar1; + } + do { + SkScalar t = SkScalarAve(upper_t, lower_t); + SkEvalCubicAt(cubic, t, &eval, NULL, NULL); + if (pt.fY > eval.fY) { + lower_t = t; + } else { + upper_t = t; + } + } while (++iter < kMaxIter + && !SkScalarNearlyZero(eval.fY - pt.fY)); + if (pt.fX <= eval.fX) { + if (ambiguous) { + *ambiguous = pt_at_extremum; + } + return true; + } + return false; +} + +int SkNumXRayCrossingsForCubic(const SkXRay& pt, + const SkPoint cubic[4], + bool* ambiguous) { + int num_crossings = 0; + SkPoint monotonic_cubics[10]; + int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics); + if (ambiguous) { + *ambiguous = false; + } + bool locally_ambiguous; + if (SkXRayCrossesMonotonicCubic(pt, + &monotonic_cubics[0], + &locally_ambiguous)) + ++num_crossings; + if (ambiguous) { + *ambiguous |= locally_ambiguous; + } + if (num_monotonic_cubics > 0) + if (SkXRayCrossesMonotonicCubic(pt, + &monotonic_cubics[3], + &locally_ambiguous)) + ++num_crossings; + if (ambiguous) { + *ambiguous |= locally_ambiguous; + } + if (num_monotonic_cubics > 1) + if (SkXRayCrossesMonotonicCubic(pt, + &monotonic_cubics[6], + &locally_ambiguous)) + ++num_crossings; + if (ambiguous) { + *ambiguous |= locally_ambiguous; + } + return num_crossings; +} + +/////////////////////////////////////////////////////////////////////////////// + +/* Find t value for quadratic [a, b, c] = d. + Return 0 if there is no solution within [0, 1) +*/ +static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) { + // At^2 + Bt + C = d + SkScalar A = a - 2 * b + c; + SkScalar B = 2 * (b - a); + SkScalar C = a - d; + + SkScalar roots[2]; + int count = SkFindUnitQuadRoots(A, B, C, roots); + + SkASSERT(count <= 1); + return count == 1 ? roots[0] : 0; +} + +/* given a quad-curve and a point (x,y), chop the quad at that point and place + the new off-curve point and endpoint into 'dest'. + Should only return false if the computed pos is the start of the curve + (i.e. root == 0) +*/ +static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, + SkPoint* dest) { + const SkScalar* base; + SkScalar value; + + if (SkScalarAbs(x) < SkScalarAbs(y)) { + base = &quad[0].fX; + value = x; + } else { + base = &quad[0].fY; + value = y; + } + + // note: this returns 0 if it thinks value is out of range, meaning the + // root might return something outside of [0, 1) + SkScalar t = quad_solve(base[0], base[2], base[4], value); + + if (t > 0) { + SkPoint tmp[5]; + SkChopQuadAt(quad, tmp, t); + dest[0] = tmp[1]; + dest[1].set(x, y); + return true; + } else { + /* t == 0 means either the value triggered a root outside of [0, 1) + For our purposes, we can ignore the <= 0 roots, but we want to + catch the >= 1 roots (which given our caller, will basically mean + a root of 1, give-or-take numerical instability). If we are in the + >= 1 case, return the existing offCurve point. + + The test below checks to see if we are close to the "end" of the + curve (near base[4]). Rather than specifying a tolerance, I just + check to see if value is on to the right/left of the middle point + (depending on the direction/sign of the end points). + */ + if ((base[0] < base[4] && value > base[2]) || + (base[0] > base[4] && value < base[2])) // should root have been 1 + { + dest[0] = quad[1]; + dest[1].set(x, y); + return true; + } + } + return false; +} + +static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { +// The mid point of the quadratic arc approximation is half way between the two +// control points. The float epsilon adjustment moves the on curve point out by +// two bits, distributing the convex test error between the round rect +// approximation and the convex cross product sign equality test. +#define SK_MID_RRECT_OFFSET \ + (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2 + { SK_Scalar1, 0 }, + { SK_Scalar1, SK_ScalarTanPIOver8 }, + { SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, + { SK_ScalarTanPIOver8, SK_Scalar1 }, + + { 0, SK_Scalar1 }, + { -SK_ScalarTanPIOver8, SK_Scalar1 }, + { -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET }, + { -SK_Scalar1, SK_ScalarTanPIOver8 }, + + { -SK_Scalar1, 0 }, + { -SK_Scalar1, -SK_ScalarTanPIOver8 }, + { -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, + { -SK_ScalarTanPIOver8, -SK_Scalar1 }, + + { 0, -SK_Scalar1 }, + { SK_ScalarTanPIOver8, -SK_Scalar1 }, + { SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET }, + { SK_Scalar1, -SK_ScalarTanPIOver8 }, + + { SK_Scalar1, 0 } +#undef SK_MID_RRECT_OFFSET +}; + +int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop, + SkRotationDirection dir, const SkMatrix* userMatrix, + SkPoint quadPoints[]) { + // rotate by x,y so that uStart is (1.0) + SkScalar x = SkPoint::DotProduct(uStart, uStop); + SkScalar y = SkPoint::CrossProduct(uStart, uStop); + + SkScalar absX = SkScalarAbs(x); + SkScalar absY = SkScalarAbs(y); + + int pointCount; + + // check for (effectively) coincident vectors + // this can happen if our angle is nearly 0 or nearly 180 (y == 0) + // ... we use the dot-prod to distinguish between 0 and 180 (x > 0) + if (absY <= SK_ScalarNearlyZero && x > 0 && + ((y >= 0 && kCW_SkRotationDirection == dir) || + (y <= 0 && kCCW_SkRotationDirection == dir))) { + + // just return the start-point + quadPoints[0].set(SK_Scalar1, 0); + pointCount = 1; + } else { + if (dir == kCCW_SkRotationDirection) { + y = -y; + } + // what octant (quadratic curve) is [xy] in? + int oct = 0; + bool sameSign = true; + + if (0 == y) { + oct = 4; // 180 + SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); + } else if (0 == x) { + SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); + oct = y > 0 ? 2 : 6; // 90 : 270 + } else { + if (y < 0) { + oct += 4; + } + if ((x < 0) != (y < 0)) { + oct += 2; + sameSign = false; + } + if ((absX < absY) == sameSign) { + oct += 1; + } + } + + int wholeCount = oct << 1; + memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint)); + + const SkPoint* arc = &gQuadCirclePts[wholeCount]; + if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) { + wholeCount += 2; + } + pointCount = wholeCount + 1; + } + + // now handle counter-clockwise and the initial unitStart rotation + SkMatrix matrix; + matrix.setSinCos(uStart.fY, uStart.fX); + if (dir == kCCW_SkRotationDirection) { + matrix.preScale(SK_Scalar1, -SK_Scalar1); + } + if (userMatrix) { + matrix.postConcat(*userMatrix); + } + matrix.mapPoints(quadPoints, pointCount); + return pointCount; +} + + +/////////////////////////////////////////////////////////////////////////////// +// +// NURB representation for conics. Helpful explanations at: +// +// http://citeseerx.ist.psu.edu/viewdoc/ +// download?doi=10.1.1.44.5740&rep=rep1&type=ps +// and +// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html +// +// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w) +// ------------------------------------------ +// ((1 - t)^2 + t^2 + 2 (1 - t) t w) +// +// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0} +// ------------------------------------------------ +// {t^2 (2 - 2 w), t (-2 + 2 w), 1} +// + +static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) { + SkASSERT(src); + SkASSERT(t >= 0 && t <= SK_Scalar1); + + SkScalar src2w = SkScalarMul(src[2], w); + SkScalar C = src[0]; + SkScalar A = src[4] - 2 * src2w + C; + SkScalar B = 2 * (src2w - C); + SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); + + B = 2 * (w - SK_Scalar1); + C = SK_Scalar1; + A = -B; + SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C); + + return SkScalarDiv(numer, denom); +} + +// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w) +// +// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w) +// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w) +// t^0 : -2 P0 w + 2 P1 w +// +// We disregard magnitude, so we can freely ignore the denominator of F', and +// divide the numerator by 2 +// +// coeff[0] for t^2 +// coeff[1] for t^1 +// coeff[2] for t^0 +// +static void conic_deriv_coeff(const SkScalar src[], + SkScalar w, + SkScalar coeff[3]) { + const SkScalar P20 = src[4] - src[0]; + const SkScalar P10 = src[2] - src[0]; + const SkScalar wP10 = w * P10; + coeff[0] = w * P20 - P20; + coeff[1] = P20 - 2 * wP10; + coeff[2] = wP10; +} + +static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) { + SkScalar coeff[3]; + conic_deriv_coeff(coord, w, coeff); + return t * (t * coeff[0] + coeff[1]) + coeff[2]; +} + +static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) { + SkScalar coeff[3]; + conic_deriv_coeff(src, w, coeff); + + SkScalar tValues[2]; + int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues); + SkASSERT(0 == roots || 1 == roots); + + if (1 == roots) { + *t = tValues[0]; + return true; + } + return false; +} + +struct SkP3D { + SkScalar fX, fY, fZ; + + void set(SkScalar x, SkScalar y, SkScalar z) { + fX = x; fY = y; fZ = z; + } + + void projectDown(SkPoint* dst) const { + dst->set(fX / fZ, fY / fZ); + } +}; + +// We only interpolate one dimension at a time (the first, at +0, +3, +6). +static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) { + SkScalar ab = SkScalarInterp(src[0], src[3], t); + SkScalar bc = SkScalarInterp(src[3], src[6], t); + dst[0] = ab; + dst[3] = SkScalarInterp(ab, bc, t); + dst[6] = bc; +} + +static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) { + dst[0].set(src[0].fX * 1, src[0].fY * 1, 1); + dst[1].set(src[1].fX * w, src[1].fY * w, w); + dst[2].set(src[2].fX * 1, src[2].fY * 1, 1); +} + +void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { + SkASSERT(t >= 0 && t <= SK_Scalar1); + + if (pt) { + pt->set(conic_eval_pos(&fPts[0].fX, fW, t), + conic_eval_pos(&fPts[0].fY, fW, t)); + } + if (tangent) { + tangent->set(conic_eval_tan(&fPts[0].fX, fW, t), + conic_eval_tan(&fPts[0].fY, fW, t)); + } +} + +void SkConic::chopAt(SkScalar t, SkConic dst[2]) const { + SkP3D tmp[3], tmp2[3]; + + ratquad_mapTo3D(fPts, fW, tmp); + + p3d_interp(&tmp[0].fX, &tmp2[0].fX, t); + p3d_interp(&tmp[0].fY, &tmp2[0].fY, t); + p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t); + + dst[0].fPts[0] = fPts[0]; + tmp2[0].projectDown(&dst[0].fPts[1]); + tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2]; + tmp2[2].projectDown(&dst[1].fPts[1]); + dst[1].fPts[2] = fPts[2]; + + // to put in "standard form", where w0 and w2 are both 1, we compute the + // new w1 as sqrt(w1*w1/w0*w2) + // or + // w1 /= sqrt(w0*w2) + // + // However, in our case, we know that for dst[0]: + // w0 == 1, and for dst[1], w2 == 1 + // + SkScalar root = SkScalarSqrt(tmp2[1].fZ); + dst[0].fW = tmp2[0].fZ / root; + dst[1].fW = tmp2[2].fZ / root; +} + +static SkScalar subdivide_w_value(SkScalar w) { + return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); +} + +void SkConic::chop(SkConic dst[2]) const { + SkScalar scale = SkScalarInvert(SK_Scalar1 + fW); + SkScalar p1x = fW * fPts[1].fX; + SkScalar p1y = fW * fPts[1].fY; + SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf; + SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf; + + dst[0].fPts[0] = fPts[0]; + dst[0].fPts[1].set((fPts[0].fX + p1x) * scale, + (fPts[0].fY + p1y) * scale); + dst[0].fPts[2].set(mx, my); + + dst[1].fPts[0].set(mx, my); + dst[1].fPts[1].set((p1x + fPts[2].fX) * scale, + (p1y + fPts[2].fY) * scale); + dst[1].fPts[2] = fPts[2]; + + dst[0].fW = dst[1].fW = subdivide_w_value(fW); +} + +/* + * "High order approximation of conic sections by quadratic splines" + * by Michael Floater, 1993 + */ +#define AS_QUAD_ERROR_SETUP \ + SkScalar a = fW - 1; \ + SkScalar k = a / (4 * (2 + a)); \ + SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \ + SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY); + +void SkConic::computeAsQuadError(SkVector* err) const { + AS_QUAD_ERROR_SETUP + err->set(x, y); +} + +bool SkConic::asQuadTol(SkScalar tol) const { + AS_QUAD_ERROR_SETUP + return (x * x + y * y) <= tol * tol; +} + +int SkConic::computeQuadPOW2(SkScalar tol) const { + AS_QUAD_ERROR_SETUP + SkScalar error = SkScalarSqrt(x * x + y * y) - tol; + + if (error <= 0) { + return 0; + } + uint32_t ierr = (uint32_t)error; + return (34 - SkCLZ(ierr)) >> 1; +} + +static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) { + SkASSERT(level >= 0); + + if (0 == level) { + memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint)); + return pts + 2; + } else { + SkConic dst[2]; + src.chop(dst); + --level; + pts = subdivide(dst[0], pts, level); + return subdivide(dst[1], pts, level); + } +} + +int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const { + SkASSERT(pow2 >= 0); + *pts = fPts[0]; + SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2); + SkASSERT(endPts - pts == (2 * (1 << pow2) + 1)); + return 1 << pow2; +} + +bool SkConic::findXExtrema(SkScalar* t) const { + return conic_find_extrema(&fPts[0].fX, fW, t); +} + +bool SkConic::findYExtrema(SkScalar* t) const { + return conic_find_extrema(&fPts[0].fY, fW, t); +} + +bool SkConic::chopAtXExtrema(SkConic dst[2]) const { + SkScalar t; + if (this->findXExtrema(&t)) { + this->chopAt(t, dst); + // now clean-up the middle, since we know t was meant to be at + // an X-extrema + SkScalar value = dst[0].fPts[2].fX; + dst[0].fPts[1].fX = value; + dst[1].fPts[0].fX = value; + dst[1].fPts[1].fX = value; + return true; + } + return false; +} + +bool SkConic::chopAtYExtrema(SkConic dst[2]) const { + SkScalar t; + if (this->findYExtrema(&t)) { + this->chopAt(t, dst); + // now clean-up the middle, since we know t was meant to be at + // an Y-extrema + SkScalar value = dst[0].fPts[2].fY; + dst[0].fPts[1].fY = value; + dst[1].fPts[0].fY = value; + dst[1].fPts[1].fY = value; + return true; + } + return false; +} + +void SkConic::computeTightBounds(SkRect* bounds) const { + SkPoint pts[4]; + pts[0] = fPts[0]; + pts[1] = fPts[2]; + int count = 2; + + SkScalar t; + if (this->findXExtrema(&t)) { + this->evalAt(t, &pts[count++]); + } + if (this->findYExtrema(&t)) { + this->evalAt(t, &pts[count++]); + } + bounds->set(pts, count); +} + +void SkConic::computeFastBounds(SkRect* bounds) const { + bounds->set(fPts, 3); +} + +bool SkConic::findMaxCurvature(SkScalar* t) const { + // TODO: Implement me + return false; +}