| |
| /* |
| * 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 "Sk64.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. |
| */ |
| #ifdef SK_SCALAR_IS_FIXED |
| #define DIRECT_EVAL_OF_POLYNOMIALS |
| #endif |
| |
| //////////////////////////////////////////////////////////////////////// |
| |
| #ifdef SK_SCALAR_IS_FIXED |
| static int is_not_monotonic(int a, int b, int c, int d) |
| { |
| return (((a - b) | (b - c) | (c - d)) & ((b - a) | (c - b) | (d - c))) >> 31; |
| } |
| |
| static int is_not_monotonic(int a, int b, int c) |
| { |
| return (((a - b) | (b - c)) & ((b - a) | (c - b))) >> 31; |
| } |
| #else |
| static int is_not_monotonic(float a, float b, float c) |
| { |
| float ab = a - b; |
| float bc = b - c; |
| if (ab < 0) |
| bc = -bc; |
| return ab == 0 || bc < 0; |
| } |
| #endif |
| |
| //////////////////////////////////////////////////////////////////////// |
| |
| 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; |
| |
| #ifdef SK_SCALAR_IS_FLOAT |
| float R = B*B - 4*A*C; |
| if (R < 0 || SkScalarIsNaN(R)) { // complex roots |
| return 0; |
| } |
| R = sk_float_sqrt(R); |
| #else |
| Sk64 RR, tmp; |
| |
| RR.setMul(B,B); |
| tmp.setMul(A,C); |
| tmp.shiftLeft(2); |
| RR.sub(tmp); |
| if (RR.isNeg()) |
| return 0; |
| SkFixed R = RR.getSqrt(); |
| #endif |
| |
| 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<SkScalar>(roots[0], roots[1]); |
| else if (roots[0] == roots[1]) // nearly-equal? |
| r -= 1; // skip the double root |
| } |
| return (int)(r - roots); |
| } |
| |
| #ifdef SK_SCALAR_IS_FIXED |
| /** Trim A/B/C down so that they are all <= 32bits |
| and then call SkFindUnitQuadRoots() |
| */ |
| static int Sk64FindFixedQuadRoots(const Sk64& A, const Sk64& B, const Sk64& C, SkFixed roots[2]) |
| { |
| int na = A.shiftToMake32(); |
| int nb = B.shiftToMake32(); |
| int nc = C.shiftToMake32(); |
| |
| int shift = SkMax32(na, SkMax32(nb, nc)); |
| SkASSERT(shift >= 0); |
| |
| return SkFindUnitQuadRoots(A.getShiftRight(shift), B.getShiftRight(shift), C.getShiftRight(shift), roots); |
| } |
| #endif |
| |
| ///////////////////////////////////////////////////////////////////////////////////// |
| ///////////////////////////////////////////////////////////////////////////////////// |
| |
| 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 |
| */ |
| #ifdef SK_SCALAR_IS_FIXED |
| return is_not_monotonic(a, b, c) && valid_unit_divide(a - b, a - b - b + c, tValue); |
| #else |
| return valid_unit_divide(a - b, a - b - b + c, tValue); |
| #endif |
| } |
| |
| 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); |
| |
| #if 0 |
| static bool once = true; |
| if (once) |
| { |
| once = false; |
| SkPoint s[3] = { 0, 26398, 0, 26331, 0, 20621428 }; |
| SkPoint d[6]; |
| |
| int n = SkChopQuadAtYExtrema(s, d); |
| SkDebugf("chop=%d, Y=[%x %x %x %x %x %x]\n", n, d[0].fY, d[1].fY, d[2].fY, d[3].fY, d[4].fY, d[5].fY); |
| } |
| #endif |
| |
| 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) |
| // |
| float 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 |
| |
| #ifdef SK_SCALAR_IS_FLOAT |
| (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t); |
| #else |
| // !!! should I use SkFloat here? seems like it |
| Sk64 numer, denom, tmp; |
| |
| numer.setMul(Ax, -Bx); |
| tmp.setMul(Ay, -By); |
| numer.add(tmp); |
| |
| if (numer.isPos()) // do nothing if numer <= 0 |
| { |
| denom.setMul(Bx, Bx); |
| tmp.setMul(By, By); |
| denom.add(tmp); |
| SkASSERT(!denom.isNeg()); |
| if (numer < denom) |
| { |
| t = numer.getFixedDiv(denom); |
| SkASSERT(t >= 0 && t <= SK_Fixed1); // assert that we're numerically stable (ha!) |
| if ((unsigned)t >= SK_Fixed1) // runtime check for numerical stability |
| t = 0; // ignore the chop |
| } |
| } |
| #endif |
| 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; |
| } |
| } |
| |
| #ifdef SK_SCALAR_IS_FLOAT |
| #define SK_ScalarTwoThirds (0.666666666f) |
| #else |
| #define SK_ScalarTwoThirds ((SkFixed)(43691)) |
| #endif |
| |
| 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 // 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]) |
| { |
| #ifdef SK_SCALAR_IS_FIXED |
| if (!is_not_monotonic(a, b, c, d)) |
| return 0; |
| #endif |
| |
| // 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; |
| int count; |
| |
| #ifdef SK_SCALAR_IS_FLOAT |
| count = SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues); |
| #else |
| Sk64 A, B, C, tmp; |
| |
| A.setMul(Bx, Cy); |
| tmp.setMul(By, Cx); |
| A.sub(tmp); |
| |
| B.setMul(Ax, Cy); |
| tmp.setMul(Ay, Cx); |
| B.sub(tmp); |
| |
| C.setMul(Ax, By); |
| tmp.setMul(Ay, Bx); |
| C.sub(tmp); |
| |
| count = Sk64FindFixedQuadRoots(A, B, C, tValues); |
| #endif |
| |
| return count; |
| } |
| |
| 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 <typename T> 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; |
| } |
| } |
| |
| #include "SkFP.h" |
| |
| // newton refinement |
| #if 0 |
| static SkScalar refine_cubic_root(const SkFP coeff[4], SkScalar root) |
| { |
| // x1 = x0 - f(t) / f'(t) |
| |
| SkFP T = SkScalarToFloat(root); |
| SkFP N, D; |
| |
| // f' = 3*coeff[0]*T^2 + 2*coeff[1]*T + coeff[2] |
| D = SkFPMul(SkFPMul(coeff[0], SkFPMul(T,T)), 3); |
| D = SkFPAdd(D, SkFPMulInt(SkFPMul(coeff[1], T), 2)); |
| D = SkFPAdd(D, coeff[2]); |
| |
| if (D == 0) |
| return root; |
| |
| // f = coeff[0]*T^3 + coeff[1]*T^2 + coeff[2]*T + coeff[3] |
| N = SkFPMul(SkFPMul(SkFPMul(T, T), T), coeff[0]); |
| N = SkFPAdd(N, SkFPMul(SkFPMul(T, T), coeff[1])); |
| N = SkFPAdd(N, SkFPMul(T, coeff[2])); |
| N = SkFPAdd(N, coeff[3]); |
| |
| if (N) |
| { |
| SkScalar delta = SkFPToScalar(SkFPDiv(N, D)); |
| |
| if (delta) |
| root -= delta; |
| } |
| return root; |
| } |
| #endif |
| |
| /** |
| * 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(float 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 float src0[] = { 0 }; |
| const float src1[] = { 0, 0 }; |
| const float src2[] = { 0, 1 }; |
| const float src3[] = { 0, 0, 0 }; |
| const float src4[] = { 0, 0, 1 }; |
| const float src5[] = { 0, 1, 1 }; |
| const float src6[] = { 0, 1, 2 }; |
| const struct { |
| const float* 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) { |
| float 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 |
| |
| #if defined _WIN32 && _MSC_VER >= 1300 && defined SK_SCALAR_IS_FIXED // disable warning : unreachable code if building fixed point for windows desktop |
| #pragma warning ( disable : 4702 ) |
| #endif |
| |
| /* 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_polynomial(const SkFP coeff[4], SkScalar tValues[3]) |
| { |
| #ifndef SK_SCALAR_IS_FLOAT |
| return 0; // this is not yet implemented for software float |
| #endif |
| |
| if (SkScalarNearlyZero(coeff[0])) // we're just a quadratic |
| { |
| return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); |
| } |
| |
| SkFP a, b, c, Q, R; |
| |
| { |
| SkASSERT(coeff[0] != 0); |
| |
| SkFP inva = SkFPInvert(coeff[0]); |
| a = SkFPMul(coeff[1], inva); |
| b = SkFPMul(coeff[2], inva); |
| c = SkFPMul(coeff[3], inva); |
| } |
| Q = SkFPDivInt(SkFPSub(SkFPMul(a,a), SkFPMulInt(b, 3)), 9); |
| // R = (2*a*a*a - 9*a*b + 27*c) / 54; |
| R = SkFPMulInt(SkFPMul(SkFPMul(a, a), a), 2); |
| R = SkFPSub(R, SkFPMulInt(SkFPMul(a, b), 9)); |
| R = SkFPAdd(R, SkFPMulInt(c, 27)); |
| R = SkFPDivInt(R, 54); |
| |
| SkFP Q3 = SkFPMul(SkFPMul(Q, Q), Q); |
| SkFP R2MinusQ3 = SkFPSub(SkFPMul(R,R), Q3); |
| SkFP adiv3 = SkFPDivInt(a, 3); |
| |
| SkScalar* roots = tValues; |
| SkScalar r; |
| |
| if (SkFPLT(R2MinusQ3, 0)) // we have 3 real roots |
| { |
| #ifdef SK_SCALAR_IS_FLOAT |
| float theta = sk_float_acos(R / sk_float_sqrt(Q3)); |
| float neg2RootQ = -2 * sk_float_sqrt(Q); |
| |
| r = neg2RootQ * sk_float_cos(theta/3) - adiv3; |
| if (is_unit_interval(r)) |
| *roots++ = r; |
| |
| r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3; |
| if (is_unit_interval(r)) |
| *roots++ = r; |
| |
| r = neg2RootQ * sk_float_cos((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 |
| #endif |
| } |
| else // we have 1 real root |
| { |
| SkFP A = SkFPAdd(SkFPAbs(R), SkFPSqrt(R2MinusQ3)); |
| A = SkFPCubeRoot(A); |
| if (SkFPGT(R, 0)) |
| A = SkFPNeg(A); |
| |
| if (A != 0) |
| A = SkFPAdd(A, SkFPDiv(Q, A)); |
| r = SkFPToScalar(SkFPSub(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[], SkFP 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]; |
| |
| SkFP A = SkScalarToFP(a); |
| SkFP B = SkScalarToFP(b); |
| SkFP C = SkScalarToFP(c); |
| |
| coeff[0] = SkFPMul(C, C); |
| coeff[1] = SkFPMulInt(SkFPMul(B, C), 3); |
| coeff[2] = SkFPMulInt(SkFPMul(B, B), 2); |
| coeff[2] = SkFPAdd(coeff[2], SkFPMul(C, A)); |
| coeff[3] = SkFPMul(A, B); |
| } |
| |
| // EXPERIMENTAL: can set this to zero to accept all t-values 0 < t < 1 |
| //#define kMinTValueForChopping (SK_Scalar1 / 256) |
| #define kMinTValueForChopping 0 |
| |
| /* 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]) |
| { |
| SkFP 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] = SkFPAdd(coeffX[i],coeffY[i]); |
| |
| SkScalar t[3]; |
| int count = solve_cubic_polynomial(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] > kMinTValueForChopping && t[i] < SK_Scalar1 - kMinTValueForChopping) |
| 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 = SkFloatToScalar(0); |
| } else { |
| upper_t = SkFloatToScalar(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'. The new end point is used |
| (rather than (x,y)) to compensate for numerical inaccuracies. |
| 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] = tmp[2]; |
| 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; |
| } |
| |
| #ifdef SK_SCALAR_IS_FLOAT |
| |
| // Due to floating point issues (i.e., 1.0f - SK_ScalarRoot2Over2 != |
| // SK_ScalarRoot2Over2 - SK_ScalarTanPIOver8) a cruder root2over2 |
| // approximation is required to make the quad circle points convex. The |
| // root of the problem is that with the root2over2 value in SkScalar.h |
| // the arcs really are ever so slightly concave. Some alternative fixes |
| // to this problem (besides just arbitrarily pushing out the mid-point as |
| // is done here) are: |
| // Adjust all the points (not just the middle) to both better approximate |
| // the curve and remain convex |
| // Switch over to using cubics rather then quads |
| // Use a different method to create the mid-point (e.g., compute |
| // the two side points, average them, then move it out as needed |
| #define SK_ScalarRoot2Over2_QuadCircle 0.7072f |
| |
| #else |
| #define SK_ScalarRoot2Over2_QuadCircle SK_FixedRoot2Over2 |
| #endif |
| |
| |
| static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { |
| { SK_Scalar1, 0 }, |
| { SK_Scalar1, SK_ScalarTanPIOver8 }, |
| { SK_ScalarRoot2Over2_QuadCircle, SK_ScalarRoot2Over2_QuadCircle }, |
| { SK_ScalarTanPIOver8, SK_Scalar1 }, |
| |
| { 0, SK_Scalar1 }, |
| { -SK_ScalarTanPIOver8, SK_Scalar1 }, |
| { -SK_ScalarRoot2Over2_QuadCircle, SK_ScalarRoot2Over2_QuadCircle }, |
| { -SK_Scalar1, SK_ScalarTanPIOver8 }, |
| |
| { -SK_Scalar1, 0 }, |
| { -SK_Scalar1, -SK_ScalarTanPIOver8 }, |
| { -SK_ScalarRoot2Over2_QuadCircle, -SK_ScalarRoot2Over2_QuadCircle }, |
| { -SK_ScalarTanPIOver8, -SK_Scalar1 }, |
| |
| { 0, -SK_Scalar1 }, |
| { SK_ScalarTanPIOver8, -SK_Scalar1 }, |
| { SK_ScalarRoot2Over2_QuadCircle, -SK_ScalarRoot2Over2_QuadCircle }, |
| { SK_Scalar1, -SK_ScalarTanPIOver8 }, |
| |
| { SK_Scalar1, 0 } |
| }; |
| |
| 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); |
| if (y > 0) |
| oct = 2; // 90 |
| else |
| oct = 6; // 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; |
| } |
| |
| /////////////////////////////////////////////////////////////////////////////// |
| |
| // 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} |
| // |
| |
| // Take the parametric specification for the conic (either X or Y) and return |
| // in coeff[] the coefficients for the simple quadratic polynomial |
| // coeff[0] for t^2 |
| // coeff[1] for t |
| // coeff[2] for constant term |
| // |
| 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 just return the middle 3 points, since the first and last are dups of src |
| // |
| static void p3d_interp(const SkScalar src[3], SkScalar dst[3], 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); |
| } |
