/* * 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; }