| /* |
| * 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" |
| #include "SkNx.h" |
| #include "SkPoint3.h" |
| #include "SkPointPriv.h" |
| |
| #include <utility> |
| |
| static SkVector to_vector(const Sk2s& x) { |
| SkVector vector; |
| x.store(&vector); |
| return vector; |
| } |
| |
| //////////////////////////////////////////////////////////////////////// |
| |
| 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 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 = numer / denom; |
| if (SkScalarIsNaN(r)) { |
| return 0; |
| } |
| SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r); |
| if (r == 0) { // catch underflow if numer <<<< denom |
| return 0; |
| } |
| *ratio = r; |
| return 1; |
| } |
| |
| // Just returns its argument, but makes it easy to set a break-point to know when |
| // SkFindUnitQuadRoots is going to return 0 (an error). |
| static int return_check_zero(int value) { |
| if (value == 0) { |
| return 0; |
| } |
| return value; |
| } |
| |
| /** 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 return_check_zero(valid_unit_divide(-C, B, roots)); |
| } |
| |
| SkScalar* r = roots; |
| |
| // use doubles so we don't overflow temporarily trying to compute R |
| double dr = (double)B * B - 4 * (double)A * C; |
| if (dr < 0) { |
| return return_check_zero(0); |
| } |
| dr = sqrt(dr); |
| SkScalar R = SkDoubleToScalar(dr); |
| if (!SkScalarIsFinite(R)) { |
| return return_check_zero(0); |
| } |
| |
| 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]) { |
| using std::swap; |
| swap(roots[0], roots[1]); |
| } else if (roots[0] == roots[1]) { // nearly-equal? |
| r -= 1; // skip the double root |
| } |
| } |
| return return_check_zero((int)(r - roots)); |
| } |
| |
| /////////////////////////////////////////////////////////////////////////////// |
| /////////////////////////////////////////////////////////////////////////////// |
| |
| void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) { |
| SkASSERT(src); |
| SkASSERT(t >= 0 && t <= SK_Scalar1); |
| |
| if (pt) { |
| *pt = SkEvalQuadAt(src, t); |
| } |
| if (tangent) { |
| *tangent = SkEvalQuadTangentAt(src, t); |
| } |
| } |
| |
| SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) { |
| return to_point(SkQuadCoeff(src).eval(t)); |
| } |
| |
| SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) { |
| // The derivative equation is 2(b - a +(a - 2b +c)t). This returns a |
| // zero tangent vector when t is 0 or 1, and the control point is equal |
| // to the end point. In this case, use the quad end points to compute the tangent. |
| if ((t == 0 && src[0] == src[1]) || (t == 1 && src[1] == src[2])) { |
| return src[2] - src[0]; |
| } |
| SkASSERT(src); |
| SkASSERT(t >= 0 && t <= SK_Scalar1); |
| |
| Sk2s P0 = from_point(src[0]); |
| Sk2s P1 = from_point(src[1]); |
| Sk2s P2 = from_point(src[2]); |
| |
| Sk2s B = P1 - P0; |
| Sk2s A = P2 - P1 - B; |
| Sk2s T = A * Sk2s(t) + B; |
| |
| return to_vector(T + T); |
| } |
| |
| static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) { |
| return v0 + (v1 - v0) * t; |
| } |
| |
| void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) { |
| SkASSERT(t > 0 && t < SK_Scalar1); |
| |
| Sk2s p0 = from_point(src[0]); |
| Sk2s p1 = from_point(src[1]); |
| Sk2s p2 = from_point(src[2]); |
| Sk2s tt(t); |
| |
| Sk2s p01 = interp(p0, p1, tt); |
| Sk2s p12 = interp(p1, p2, tt); |
| |
| dst[0] = to_point(p0); |
| dst[1] = to_point(p01); |
| dst[2] = to_point(interp(p01, p12, tt)); |
| dst[3] = to_point(p12); |
| dst[4] = to_point(p2); |
| } |
| |
| void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) { |
| SkChopQuadAt(src, dst, 0.5f); |
| } |
| |
| /** 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 numer = -(Ax * Bx + Ay * By); |
| SkScalar denom = Bx * Bx + By * By; |
| if (denom < 0) { |
| numer = -numer; |
| denom = -denom; |
| } |
| if (numer <= 0) { |
| return 0; |
| } |
| if (numer >= denom) { // Also catches denom=0. |
| return 1; |
| } |
| SkScalar t = numer / denom; |
| SkASSERT((0 <= t && t < 1) || SkScalarIsNaN(t)); |
| return t; |
| } |
| |
| int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) { |
| SkScalar t = SkFindQuadMaxCurvature(src); |
| if (t == 0 || t == 1) { |
| memcpy(dst, src, 3 * sizeof(SkPoint)); |
| return 1; |
| } else { |
| SkChopQuadAt(src, dst, t); |
| return 2; |
| } |
| } |
| |
| void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) { |
| Sk2s scale(SkDoubleToScalar(2.0 / 3.0)); |
| Sk2s s0 = from_point(src[0]); |
| Sk2s s1 = from_point(src[1]); |
| Sk2s s2 = from_point(src[2]); |
| |
| dst[0] = to_point(s0); |
| dst[1] = to_point(s0 + (s1 - s0) * scale); |
| dst[2] = to_point(s2 + (s1 - s2) * scale); |
| dst[3] = to_point(s2); |
| } |
| |
| ////////////////////////////////////////////////////////////////////////////// |
| ///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS ///// |
| ////////////////////////////////////////////////////////////////////////////// |
| |
| static SkVector eval_cubic_derivative(const SkPoint src[4], SkScalar t) { |
| SkQuadCoeff coeff; |
| Sk2s P0 = from_point(src[0]); |
| Sk2s P1 = from_point(src[1]); |
| Sk2s P2 = from_point(src[2]); |
| Sk2s P3 = from_point(src[3]); |
| |
| coeff.fA = P3 + Sk2s(3) * (P1 - P2) - P0; |
| coeff.fB = times_2(P2 - times_2(P1) + P0); |
| coeff.fC = P1 - P0; |
| return to_vector(coeff.eval(t)); |
| } |
| |
| static SkVector eval_cubic_2ndDerivative(const SkPoint src[4], SkScalar t) { |
| Sk2s P0 = from_point(src[0]); |
| Sk2s P1 = from_point(src[1]); |
| Sk2s P2 = from_point(src[2]); |
| Sk2s P3 = from_point(src[3]); |
| Sk2s A = P3 + Sk2s(3) * (P1 - P2) - P0; |
| Sk2s B = P2 - times_2(P1) + P0; |
| |
| return to_vector(A * Sk2s(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 = to_point(SkCubicCoeff(src).eval(t)); |
| } |
| if (tangent) { |
| // The derivative equation returns a zero tangent vector when t is 0 or 1, and the |
| // adjacent control point is equal to the end point. In this case, use the |
| // next control point or the end points to compute the tangent. |
| if ((t == 0 && src[0] == src[1]) || (t == 1 && src[2] == src[3])) { |
| if (t == 0) { |
| *tangent = src[2] - src[0]; |
| } else { |
| *tangent = src[3] - src[1]; |
| } |
| if (!tangent->fX && !tangent->fY) { |
| *tangent = src[3] - src[0]; |
| } |
| } else { |
| *tangent = eval_cubic_derivative(src, t); |
| } |
| } |
| if (curvature) { |
| *curvature = eval_cubic_2ndDerivative(src, 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); |
| } |
| |
| void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) { |
| SkASSERT(t > 0 && t < SK_Scalar1); |
| |
| Sk2s p0 = from_point(src[0]); |
| Sk2s p1 = from_point(src[1]); |
| Sk2s p2 = from_point(src[2]); |
| Sk2s p3 = from_point(src[3]); |
| Sk2s tt(t); |
| |
| Sk2s ab = interp(p0, p1, tt); |
| Sk2s bc = interp(p1, p2, tt); |
| Sk2s cd = interp(p2, p3, tt); |
| Sk2s abc = interp(ab, bc, tt); |
| Sk2s bcd = interp(bc, cd, tt); |
| Sk2s abcd = interp(abc, bcd, tt); |
| |
| dst[0] = to_point(p0); |
| dst[1] = to_point(ab); |
| dst[2] = to_point(abc); |
| dst[3] = to_point(abcd); |
| dst[4] = to_point(bcd); |
| dst[5] = to_point(cd); |
| dst[6] = to_point(p3); |
| } |
| |
| /* 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(0 < tValues[i] && tValues[i] < 1); |
| SkASSERT(0 < tValues[i+1] && tValues[i+1] < 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]) { |
| SkChopCubicAt(src, dst, 0.5f); |
| } |
| |
| 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; |
| } |
| |
| // Assumes the third component of points is 1. |
| // Calcs p0 . (p1 x p2) |
| static double calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { |
| const double xComp = (double) p0.fX * ((double) p1.fY - (double) p2.fY); |
| const double yComp = (double) p0.fY * ((double) p2.fX - (double) p1.fX); |
| const double wComp = (double) p1.fX * (double) p2.fY - (double) p1.fY * (double) p2.fX; |
| return (xComp + yComp + wComp); |
| } |
| |
| // Returns a positive power of 2 that, when multiplied by n, and excepting the two edge cases listed |
| // below, shifts the exponent of n to yield a magnitude somewhere inside [1..2). |
| // Returns 2^1023 if abs(n) < 2^-1022 (including 0). |
| // Returns NaN if n is Inf or NaN. |
| inline static double previous_inverse_pow2(double n) { |
| uint64_t bits; |
| memcpy(&bits, &n, sizeof(double)); |
| bits = ((1023llu*2 << 52) + ((1llu << 52) - 1)) - bits; // exp=-exp |
| bits &= (0x7ffllu) << 52; // mantissa=1.0, sign=0 |
| memcpy(&n, &bits, sizeof(double)); |
| return n; |
| } |
| |
| inline static void write_cubic_inflection_roots(double t0, double s0, double t1, double s1, |
| double* t, double* s) { |
| t[0] = t0; |
| s[0] = s0; |
| |
| // This copysign/abs business orients the implicit function so positive values are always on the |
| // "left" side of the curve. |
| t[1] = -copysign(t1, t1 * s1); |
| s[1] = -fabs(s1); |
| |
| // Ensure t[0]/s[0] <= t[1]/s[1] (s[1] is negative from above). |
| if (copysign(s[1], s[0]) * t[0] > -fabs(s[0]) * t[1]) { |
| using std::swap; |
| swap(t[0], t[1]); |
| swap(s[0], s[1]); |
| } |
| } |
| |
| SkCubicType SkClassifyCubic(const SkPoint P[4], double t[2], double s[2], double d[4]) { |
| // Find the cubic's inflection function, I = [T^3 -3T^2 3T -1] dot D. (D0 will always be 0 |
| // for integral cubics.) |
| // |
| // See "Resolution Independent Curve Rendering using Programmable Graphics Hardware", |
| // 4.2 Curve Categorization: |
| // |
| // https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf |
| double A1 = calc_dot_cross_cubic(P[0], P[3], P[2]); |
| double A2 = calc_dot_cross_cubic(P[1], P[0], P[3]); |
| double A3 = calc_dot_cross_cubic(P[2], P[1], P[0]); |
| |
| double D3 = 3 * A3; |
| double D2 = D3 - A2; |
| double D1 = D2 - A2 + A1; |
| |
| // Shift the exponents in D so the largest magnitude falls somewhere in 1..2. This protects us |
| // from overflow down the road while solving for roots and KLM functionals. |
| double Dmax = std::max(std::max(fabs(D1), fabs(D2)), fabs(D3)); |
| double norm = previous_inverse_pow2(Dmax); |
| D1 *= norm; |
| D2 *= norm; |
| D3 *= norm; |
| |
| if (d) { |
| d[3] = D3; |
| d[2] = D2; |
| d[1] = D1; |
| d[0] = 0; |
| } |
| |
| // Now use the inflection function to classify the cubic. |
| // |
| // See "Resolution Independent Curve Rendering using Programmable Graphics Hardware", |
| // 4.4 Integral Cubics: |
| // |
| // https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf |
| if (0 != D1) { |
| double discr = 3*D2*D2 - 4*D1*D3; |
| if (discr > 0) { // Serpentine. |
| if (t && s) { |
| double q = 3*D2 + copysign(sqrt(3*discr), D2); |
| write_cubic_inflection_roots(q, 6*D1, 2*D3, q, t, s); |
| } |
| return SkCubicType::kSerpentine; |
| } else if (discr < 0) { // Loop. |
| if (t && s) { |
| double q = D2 + copysign(sqrt(-discr), D2); |
| write_cubic_inflection_roots(q, 2*D1, 2*(D2*D2 - D3*D1), D1*q, t, s); |
| } |
| return SkCubicType::kLoop; |
| } else { // Cusp. |
| if (t && s) { |
| write_cubic_inflection_roots(D2, 2*D1, D2, 2*D1, t, s); |
| } |
| return SkCubicType::kLocalCusp; |
| } |
| } else { |
| if (0 != D2) { // Cusp at T=infinity. |
| if (t && s) { |
| write_cubic_inflection_roots(D3, 3*D2, 1, 0, t, s); // T1=infinity. |
| } |
| return SkCubicType::kCuspAtInfinity; |
| } else { // Degenerate. |
| if (t && s) { |
| write_cubic_inflection_roots(1, 0, 1, 0, t, s); // T0=T1=infinity. |
| } |
| return 0 != D3 ? SkCubicType::kQuadratic : SkCubicType::kLineOrPoint; |
| } |
| } |
| } |
| |
| 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; |
| } |
| } |
| |
| /** |
| * 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; |
| |
| if (R2MinusQ3 < 0) { // we have 3 real roots |
| // the divide/root can, due to finite precisions, be slightly outside of -1...1 |
| SkScalar theta = SkScalarACos(SkScalarPin(R / SkScalarSqrt(Q3), -1, 1)); |
| SkScalar neg2RootQ = -2 * SkScalarSqrt(Q); |
| |
| tValues[0] = SkScalarPin(neg2RootQ * SkScalarCos(theta/3) - adiv3, 0, 1); |
| tValues[1] = SkScalarPin(neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3, 0, 1); |
| tValues[2] = SkScalarPin(neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3, 0, 1); |
| SkDEBUGCODE(test_collaps_duplicates();) |
| |
| // now sort the roots |
| bubble_sort(tValues, 3); |
| return collaps_duplicates(tValues, 3); |
| } 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; |
| } |
| tValues[0] = SkScalarPin(A - adiv3, 0, 1); |
| return 1; |
| } |
| } |
| |
| /* 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]; |
| } |
| |
| int numRoots = solve_cubic_poly(coeffX, tValues); |
| // now remove extrema where the curvature is zero (mins) |
| // !!!! need a test for this !!!! |
| return numRoots; |
| } |
| |
| int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], |
| SkScalar tValues[3]) { |
| SkScalar t_storage[3]; |
| |
| if (tValues == nullptr) { |
| tValues = t_storage; |
| } |
| |
| SkScalar roots[3]; |
| int rootCount = SkFindCubicMaxCurvature(src, roots); |
| |
| // Throw out values not inside 0..1. |
| int count = 0; |
| for (int i = 0; i < rootCount; ++i) { |
| if (0 < roots[i] && roots[i] < 1) { |
| tValues[count++] = roots[i]; |
| } |
| } |
| |
| if (dst) { |
| if (count == 0) { |
| memcpy(dst, src, 4 * sizeof(SkPoint)); |
| } else { |
| SkChopCubicAt(src, dst, tValues, count); |
| } |
| } |
| return count + 1; |
| } |
| |
| // Returns a constant proportional to the dimensions of the cubic. |
| // Constant found through experimentation -- maybe there's a better way.... |
| static SkScalar calc_cubic_precision(const SkPoint src[4]) { |
| return (SkPointPriv::DistanceToSqd(src[1], src[0]) + SkPointPriv::DistanceToSqd(src[2], src[1]) |
| + SkPointPriv::DistanceToSqd(src[3], src[2])) * 1e-8f; |
| } |
| |
| // Returns true if both points src[testIndex], src[testIndex+1] are in the same half plane defined |
| // by the line segment src[lineIndex], src[lineIndex+1]. |
| static bool on_same_side(const SkPoint src[4], int testIndex, int lineIndex) { |
| SkPoint origin = src[lineIndex]; |
| SkVector line = src[lineIndex + 1] - origin; |
| SkScalar crosses[2]; |
| for (int index = 0; index < 2; ++index) { |
| SkVector testLine = src[testIndex + index] - origin; |
| crosses[index] = line.cross(testLine); |
| } |
| return crosses[0] * crosses[1] >= 0; |
| } |
| |
| // Return location (in t) of cubic cusp, if there is one. |
| // Note that classify cubic code does not reliably return all cusp'd cubics, so |
| // it is not called here. |
| SkScalar SkFindCubicCusp(const SkPoint src[4]) { |
| // When the adjacent control point matches the end point, it behaves as if |
| // the cubic has a cusp: there's a point of max curvature where the derivative |
| // goes to zero. Ideally, this would be where t is zero or one, but math |
| // error makes not so. It is not uncommon to create cubics this way; skip them. |
| if (src[0] == src[1]) { |
| return -1; |
| } |
| if (src[2] == src[3]) { |
| return -1; |
| } |
| // Cubics only have a cusp if the line segments formed by the control and end points cross. |
| // Detect crossing if line ends are on opposite sides of plane formed by the other line. |
| if (on_same_side(src, 0, 2) || on_same_side(src, 2, 0)) { |
| return -1; |
| } |
| // Cubics may have multiple points of maximum curvature, although at most only |
| // one is a cusp. |
| SkScalar maxCurvature[3]; |
| int roots = SkFindCubicMaxCurvature(src, maxCurvature); |
| for (int index = 0; index < roots; ++index) { |
| SkScalar testT = maxCurvature[index]; |
| if (0 >= testT || testT >= 1) { // no need to consider max curvature on the end |
| continue; |
| } |
| // A cusp is at the max curvature, and also has a derivative close to zero. |
| // Choose the 'close to zero' meaning by comparing the derivative length |
| // with the overall cubic size. |
| SkVector dPt = eval_cubic_derivative(src, testT); |
| SkScalar dPtMagnitude = SkPointPriv::LengthSqd(dPt); |
| SkScalar precision = calc_cubic_precision(src); |
| if (dPtMagnitude < precision) { |
| // All three max curvature t values may be close to the cusp; |
| // return the first one. |
| return testT; |
| } |
| } |
| return -1; |
| } |
| |
| #include "../pathops/SkPathOpsCubic.h" |
| |
| typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const; |
| |
| static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7], |
| InterceptProc method) { |
| SkDCubic cubic; |
| double roots[3]; |
| int count = (cubic.set(src).*method)(intercept, roots); |
| if (count > 0) { |
| SkDCubicPair pair = cubic.chopAt(roots[0]); |
| for (int i = 0; i < 7; ++i) { |
| dst[i] = pair.pts[i].asSkPoint(); |
| } |
| return true; |
| } |
| return false; |
| } |
| |
| bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) { |
| return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect); |
| } |
| |
| bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) { |
| return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect); |
| } |
| |
| /////////////////////////////////////////////////////////////////////////////// |
| // |
| // 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} |
| // |
| |
| // 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 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; |
| } |
| |
| // 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, SkPoint3 dst[3]) { |
| 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); |
| } |
| |
| static SkPoint project_down(const SkPoint3& src) { |
| return {src.fX / src.fZ, src.fY / src.fZ}; |
| } |
| |
| // return false if infinity or NaN is generated; caller must check |
| bool SkConic::chopAt(SkScalar t, SkConic dst[2]) const { |
| SkPoint3 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]; |
| dst[0].fPts[1] = project_down(tmp2[0]); |
| dst[0].fPts[2] = project_down(tmp2[1]); dst[1].fPts[0] = dst[0].fPts[2]; |
| dst[1].fPts[1] = project_down(tmp2[2]); |
| 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; |
| SkASSERT(sizeof(dst[0]) == sizeof(SkScalar) * 7); |
| SkASSERT(0 == offsetof(SkConic, fPts[0].fX)); |
| return SkScalarsAreFinite(&dst[0].fPts[0].fX, 7 * 2); |
| } |
| |
| void SkConic::chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const { |
| if (0 == t1 || 1 == t2) { |
| if (0 == t1 && 1 == t2) { |
| *dst = *this; |
| return; |
| } else { |
| SkConic pair[2]; |
| if (this->chopAt(t1 ? t1 : t2, pair)) { |
| *dst = pair[SkToBool(t1)]; |
| return; |
| } |
| } |
| } |
| SkConicCoeff coeff(*this); |
| Sk2s tt1(t1); |
| Sk2s aXY = coeff.fNumer.eval(tt1); |
| Sk2s aZZ = coeff.fDenom.eval(tt1); |
| Sk2s midTT((t1 + t2) / 2); |
| Sk2s dXY = coeff.fNumer.eval(midTT); |
| Sk2s dZZ = coeff.fDenom.eval(midTT); |
| Sk2s tt2(t2); |
| Sk2s cXY = coeff.fNumer.eval(tt2); |
| Sk2s cZZ = coeff.fDenom.eval(tt2); |
| Sk2s bXY = times_2(dXY) - (aXY + cXY) * Sk2s(0.5f); |
| Sk2s bZZ = times_2(dZZ) - (aZZ + cZZ) * Sk2s(0.5f); |
| dst->fPts[0] = to_point(aXY / aZZ); |
| dst->fPts[1] = to_point(bXY / bZZ); |
| dst->fPts[2] = to_point(cXY / cZZ); |
| Sk2s ww = bZZ / (aZZ * cZZ).sqrt(); |
| dst->fW = ww[0]; |
| } |
| |
| SkPoint SkConic::evalAt(SkScalar t) const { |
| return to_point(SkConicCoeff(*this).eval(t)); |
| } |
| |
| SkVector SkConic::evalTangentAt(SkScalar t) const { |
| // The derivative equation returns a zero tangent vector when t is 0 or 1, |
| // and the control point is equal to the end point. |
| // In this case, use the conic endpoints to compute the tangent. |
| if ((t == 0 && fPts[0] == fPts[1]) || (t == 1 && fPts[1] == fPts[2])) { |
| return fPts[2] - fPts[0]; |
| } |
| Sk2s p0 = from_point(fPts[0]); |
| Sk2s p1 = from_point(fPts[1]); |
| Sk2s p2 = from_point(fPts[2]); |
| Sk2s ww(fW); |
| |
| Sk2s p20 = p2 - p0; |
| Sk2s p10 = p1 - p0; |
| |
| Sk2s C = ww * p10; |
| Sk2s A = ww * p20 - p20; |
| Sk2s B = p20 - C - C; |
| |
| return to_vector(SkQuadCoeff(A, B, C).eval(t)); |
| } |
| |
| void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const { |
| SkASSERT(t >= 0 && t <= SK_Scalar1); |
| |
| if (pt) { |
| *pt = this->evalAt(t); |
| } |
| if (tangent) { |
| *tangent = this->evalTangentAt(t); |
| } |
| } |
| |
| static SkScalar subdivide_w_value(SkScalar w) { |
| return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf); |
| } |
| |
| void SkConic::chop(SkConic * SK_RESTRICT dst) const { |
| Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW)); |
| SkScalar newW = subdivide_w_value(fW); |
| |
| Sk2s p0 = from_point(fPts[0]); |
| Sk2s p1 = from_point(fPts[1]); |
| Sk2s p2 = from_point(fPts[2]); |
| Sk2s ww(fW); |
| |
| Sk2s wp1 = ww * p1; |
| Sk2s m = (p0 + times_2(wp1) + p2) * scale * Sk2s(0.5f); |
| SkPoint mPt = to_point(m); |
| if (!mPt.isFinite()) { |
| double w_d = fW; |
| double w_2 = w_d * 2; |
| double scale_half = 1 / (1 + w_d) * 0.5; |
| mPt.fX = SkDoubleToScalar((fPts[0].fX + w_2 * fPts[1].fX + fPts[2].fX) * scale_half); |
| mPt.fY = SkDoubleToScalar((fPts[0].fY + w_2 * fPts[1].fY + fPts[2].fY) * scale_half); |
| } |
| dst[0].fPts[0] = fPts[0]; |
| dst[0].fPts[1] = to_point((p0 + wp1) * scale); |
| dst[0].fPts[2] = dst[1].fPts[0] = mPt; |
| dst[1].fPts[1] = to_point((wp1 + p2) * scale); |
| dst[1].fPts[2] = fPts[2]; |
| |
| dst[0].fW = dst[1].fW = newW; |
| } |
| |
| /* |
| * "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; |
| } |
| |
| // Limit the number of suggested quads to approximate a conic |
| #define kMaxConicToQuadPOW2 5 |
| |
| int SkConic::computeQuadPOW2(SkScalar tol) const { |
| if (tol < 0 || !SkScalarIsFinite(tol) || !SkPointPriv::AreFinite(fPts, 3)) { |
| return 0; |
| } |
| |
| AS_QUAD_ERROR_SETUP |
| |
| SkScalar error = SkScalarSqrt(x * x + y * y); |
| int pow2; |
| for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) { |
| if (error <= tol) { |
| break; |
| } |
| error *= 0.25f; |
| } |
| // float version -- using ceil gives the same results as the above. |
| if (false) { |
| SkScalar err = SkScalarSqrt(x * x + y * y); |
| if (err <= tol) { |
| return 0; |
| } |
| SkScalar tol2 = tol * tol; |
| if (tol2 == 0) { |
| return kMaxConicToQuadPOW2; |
| } |
| SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f; |
| int altPow2 = SkScalarCeilToInt(fpow2); |
| if (altPow2 != pow2) { |
| SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol); |
| } |
| pow2 = altPow2; |
| } |
| return pow2; |
| } |
| |
| // This was originally developed and tested for pathops: see SkOpTypes.h |
| // returns true if (a <= b <= c) || (a >= b >= c) |
| static bool between(SkScalar a, SkScalar b, SkScalar c) { |
| return (a - b) * (c - b) <= 0; |
| } |
| |
| 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); |
| const SkScalar startY = src.fPts[0].fY; |
| SkScalar endY = src.fPts[2].fY; |
| if (between(startY, src.fPts[1].fY, endY)) { |
| // If the input is monotonic and the output is not, the scan converter hangs. |
| // Ensure that the chopped conics maintain their y-order. |
| SkScalar midY = dst[0].fPts[2].fY; |
| if (!between(startY, midY, endY)) { |
| // If the computed midpoint is outside the ends, move it to the closer one. |
| SkScalar closerY = SkTAbs(midY - startY) < SkTAbs(midY - endY) ? startY : endY; |
| dst[0].fPts[2].fY = dst[1].fPts[0].fY = closerY; |
| } |
| if (!between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY)) { |
| // If the 1st control is not between the start and end, put it at the start. |
| // This also reduces the quad to a line. |
| dst[0].fPts[1].fY = startY; |
| } |
| if (!between(dst[1].fPts[0].fY, dst[1].fPts[1].fY, endY)) { |
| // If the 2nd control is not between the start and end, put it at the end. |
| // This also reduces the quad to a line. |
| dst[1].fPts[1].fY = endY; |
| } |
| // Verify that all five points are in order. |
| SkASSERT(between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY)); |
| SkASSERT(between(dst[0].fPts[1].fY, dst[0].fPts[2].fY, dst[1].fPts[1].fY)); |
| SkASSERT(between(dst[0].fPts[2].fY, dst[1].fPts[1].fY, endY)); |
| } |
| --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); |
| if (pow2 == kMaxConicToQuadPOW2) { // If an extreme weight generates many quads ... |
| SkConic dst[2]; |
| this->chop(dst); |
| // check to see if the first chop generates a pair of lines |
| if (SkPointPriv::EqualsWithinTolerance(dst[0].fPts[1], dst[0].fPts[2]) && |
| SkPointPriv::EqualsWithinTolerance(dst[1].fPts[0], dst[1].fPts[1])) { |
| pts[1] = pts[2] = pts[3] = dst[0].fPts[1]; // set ctrl == end to make lines |
| pts[4] = dst[1].fPts[2]; |
| pow2 = 1; |
| SkDEBUGCODE(endPts = &pts[5]); |
| goto commonFinitePtCheck; |
| } |
| } |
| SkDEBUGCODE(endPts = ) subdivide(*this, pts + 1, pow2); |
| commonFinitePtCheck: |
| const int quadCount = 1 << pow2; |
| const int ptCount = 2 * quadCount + 1; |
| SkASSERT(endPts - pts == ptCount); |
| if (!SkPointPriv::AreFinite(pts, ptCount)) { |
| // if we generated a non-finite, pin ourselves to the middle of the hull, |
| // as our first and last are already on the first/last pts of the hull. |
| for (int i = 1; i < ptCount - 1; ++i) { |
| pts[i] = fPts[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)) { |
| if (!this->chopAt(t, dst)) { |
| // if chop can't return finite values, don't chop |
| return false; |
| } |
| // 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)) { |
| if (!this->chopAt(t, dst)) { |
| // if chop can't return finite values, don't chop |
| return false; |
| } |
| // 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); |
| } |
| |
| #if 0 // unimplemented |
| bool SkConic::findMaxCurvature(SkScalar* t) const { |
| // TODO: Implement me |
| return false; |
| } |
| #endif |
| |
| SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w, const SkMatrix& matrix) { |
| if (!matrix.hasPerspective()) { |
| return w; |
| } |
| |
| SkPoint3 src[3], dst[3]; |
| |
| ratquad_mapTo3D(pts, w, src); |
| |
| matrix.mapHomogeneousPoints(dst, src, 3); |
| |
| // w' = sqrt(w1*w1/w0*w2) |
| // use doubles temporarily, to handle small numer/denom |
| double w0 = dst[0].fZ; |
| double w1 = dst[1].fZ; |
| double w2 = dst[2].fZ; |
| return sk_double_to_float(sqrt(sk_ieee_double_divide(w1 * w1, w0 * w2))); |
| } |
| |
| int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir, |
| const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) { |
| // rotate by x,y so that uStart is (1.0) |
| SkScalar x = SkPoint::DotProduct(uStart, uStop); |
| SkScalar y = SkPoint::CrossProduct(uStart, uStop); |
| |
| SkScalar absY = SkScalarAbs(y); |
| |
| // 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))) { |
| return 0; |
| } |
| |
| if (dir == kCCW_SkRotationDirection) { |
| y = -y; |
| } |
| |
| // We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in? |
| // 0 == [0 .. 90) |
| // 1 == [90 ..180) |
| // 2 == [180..270) |
| // 3 == [270..360) |
| // |
| int quadrant = 0; |
| if (0 == y) { |
| quadrant = 2; // 180 |
| SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero); |
| } else if (0 == x) { |
| SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero); |
| quadrant = y > 0 ? 1 : 3; // 90 : 270 |
| } else { |
| if (y < 0) { |
| quadrant += 2; |
| } |
| if ((x < 0) != (y < 0)) { |
| quadrant += 1; |
| } |
| } |
| |
| const SkPoint quadrantPts[] = { |
| { 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 } |
| }; |
| const SkScalar quadrantWeight = SK_ScalarRoot2Over2; |
| |
| int conicCount = quadrant; |
| for (int i = 0; i < conicCount; ++i) { |
| dst[i].set(&quadrantPts[i * 2], quadrantWeight); |
| } |
| |
| // Now compute any remaing (sub-90-degree) arc for the last conic |
| const SkPoint finalP = { x, y }; |
| const SkPoint& lastQ = quadrantPts[quadrant * 2]; // will already be a unit-vector |
| const SkScalar dot = SkVector::DotProduct(lastQ, finalP); |
| SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero); |
| |
| if (dot < 1) { |
| SkVector offCurve = { lastQ.x() + x, lastQ.y() + y }; |
| // compute the bisector vector, and then rescale to be the off-curve point. |
| // we compute its length from cos(theta/2) = length / 1, using half-angle identity we get |
| // length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot. |
| // This is nice, since our computed weight is cos(theta/2) as well! |
| // |
| const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2); |
| offCurve.setLength(SkScalarInvert(cosThetaOver2)); |
| if (!SkPointPriv::EqualsWithinTolerance(lastQ, offCurve)) { |
| dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2); |
| conicCount += 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); |
| } |
| for (int i = 0; i < conicCount; ++i) { |
| matrix.mapPoints(dst[i].fPts, 3); |
| } |
| return conicCount; |
| } |
