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- /*
- * 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 "include/core/SkMatrix.h"
- #include "include/core/SkPoint3.h"
- #include "include/private/SkNx.h"
- #include "src/core/SkGeometry.h"
- #include "src/core/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 "src/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;
- }
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