| /* libs/graphics/sgl/SkGeometry.cpp |
| ** |
| ** Copyright 2006, The Android Open Source Project |
| ** |
| ** Licensed under the Apache License, Version 2.0 (the "License"); |
| ** you may not use this file except in compliance with the License. |
| ** You may obtain a copy of the License at |
| ** |
| ** http://www.apache.org/licenses/LICENSE-2.0 |
| ** |
| ** Unless required by applicable law or agreed to in writing, software |
| ** distributed under the License is distributed on an "AS IS" BASIS, |
| ** WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| ** See the License for the specific language governing permissions and |
| ** limitations under the License. |
| */ |
| |
| #include "SkGeometry.h" |
| #include "Sk64.h" |
| #include "SkMatrix.h" |
| |
| /** 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); |
| 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) // 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 void flatten_double_quad_extrema(SkScalar coords[14]) |
| { |
| coords[2] = coords[6] = coords[4]; |
| } |
| |
| static void force_quad_monotonic_in_y(SkPoint pts[3]) |
| { |
| // zap pts[1].fY to the nearest value |
| SkScalar ab = SkScalarAbs(pts[0].fY - pts[1].fY); |
| SkScalar bc = SkScalarAbs(pts[1].fY - pts[2].fY); |
| pts[1].fY = ab < bc ? pts[0].fY : pts[2].fY; |
| } |
| |
| /* 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; |
| } |
| |
| // 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) |
| // |
| int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) |
| { |
| 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 |
| |
| if (t == 0) |
| { |
| memcpy(dst, src, 3 * sizeof(SkPoint)); |
| return 1; |
| } |
| else |
| { |
| SkChopQuadAt(src, dst, t); |
| return 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); |
| } |
| |
| 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; |
| |
| SkDEBUGCODE(int valid =) valid_unit_divide(tValues[i+1] - tValues[i], SK_Scalar1 - tValues[i], &t); |
| SkASSERT(valid); |
| |
| dst += 3; |
| memcpy(tmp, dst, 4 * sizeof(SkPoint)); |
| src = tmp; |
| } |
| } |
| } |
| } |
| |
| 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; |
| } |
| |
| /** 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 |
| |
| #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] |
| */ |
| 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; |
| |
| // now sort the roots |
| bubble_sort(tValues, (int)(roots - tValues)); |
| #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; |
| } |
| |
| //////////////////////////////////////////////////////////////////////////////// |
| |
| /* 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 return |
| the new quad's offCurve point. Should only return false if the computed pos |
| is the start of the curve (i.e. root == 0) |
| */ |
| static bool quad_pt2OffCurve(const SkPoint quad[3], SkScalar x, SkScalar y, SkPoint* offCurve) |
| { |
| 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); |
| *offCurve = tmp[1]; |
| 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 |
| { |
| *offCurve = quad[1]; |
| return true; |
| } |
| } |
| return false; |
| } |
| |
| static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = { |
| { SK_Scalar1, 0 }, |
| { SK_Scalar1, SK_ScalarTanPIOver8 }, |
| { SK_ScalarRoot2Over2, SK_ScalarRoot2Over2 }, |
| { SK_ScalarTanPIOver8, SK_Scalar1 }, |
| |
| { 0, SK_Scalar1 }, |
| { -SK_ScalarTanPIOver8, SK_Scalar1 }, |
| { -SK_ScalarRoot2Over2, SK_ScalarRoot2Over2 }, |
| { -SK_Scalar1, SK_ScalarTanPIOver8 }, |
| |
| { -SK_Scalar1, 0 }, |
| { -SK_Scalar1, -SK_ScalarTanPIOver8 }, |
| { -SK_ScalarRoot2Over2, -SK_ScalarRoot2Over2 }, |
| { -SK_ScalarTanPIOver8, -SK_Scalar1 }, |
| |
| { 0, -SK_Scalar1 }, |
| { SK_ScalarTanPIOver8, -SK_Scalar1 }, |
| { SK_ScalarRoot2Over2, -SK_ScalarRoot2Over2 }, |
| { 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 (quad_pt2OffCurve(arc, x, y, &quadPoints[wholeCount + 1])) |
| { |
| quadPoints[wholeCount + 2].set(x, y); |
| 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; |
| } |
| |
| |
| ///////////////////////////////////////////////////////////////////////////////////////// |
| ///////////////////////////////////////////////////////////////////////////////////////// |
| |
| #ifdef SK_DEBUG |
| |
| void SkGeometry::UnitTest() |
| { |
| #ifdef SK_SUPPORT_UNITTEST |
| SkPoint pts[3], dst[5]; |
| |
| pts[0].set(0, 0); |
| pts[1].set(100, 50); |
| pts[2].set(0, 100); |
| |
| int count = SkChopQuadAtMaxCurvature(pts, dst); |
| SkASSERT(count == 1 || count == 2); |
| #endif |
| } |
| |
| #endif |
| |
| |
