The Tor Browser: gfx/skia/trunk/src/core/SkGeometry.cpp@129ffea94266 (annotated)

gfx/skia/trunk/src/core/SkGeometry.cpp@129ffea94266 (annotated)

gfx/skia/trunk/src/core/SkGeometry.cpp

Sat, 03 Jan 2015 20:18:00 +0100

author: Michael Schloh von Bennewitz <michael@schloh.com>
date: Sat, 03 Jan 2015 20:18:00 +0100
branch: TOR_BUG_3246
changeset 7: 129ffea94266
permissions: -rw-r--r--

Conditionally enable double key logic according to:
private browsing mode or privacy.thirdparty.isolate preference and
implement in GetCookieStringCommon and FindCookie where it counts...
With some reservations of how to convince FindCookie users to test
condition and pass a nullptr when disabling double key logic.

 /*
  * Copyright 2006 The Android Open Source Project
  *
  * Use of this source code is governed by a BSD-style license that can be
  * found in the LICENSE file.
  */
 #include "SkGeometry.h"
 #include "SkMatrix.h"
 bool SkXRayCrossesLine(const SkXRay& pt,
                        const SkPoint pts[2],
                        bool* ambiguous) {
     if (ambiguous) {
         *ambiguous = false;
     }
     // Determine quick discards.
     // Consider query line going exactly through point 0 to not
     // intersect, for symmetry with SkXRayCrossesMonotonicCubic.
     if (pt.fY == pts[0].fY) {
         if (ambiguous) {
             *ambiguous = true;
         }
         return false;
     }
     if (pt.fY < pts[0].fY && pt.fY < pts[1].fY)
         return false;
     if (pt.fY > pts[0].fY && pt.fY > pts[1].fY)
         return false;
     if (pt.fX > pts[0].fX && pt.fX > pts[1].fX)
         return false;
     // Determine degenerate cases
     if (SkScalarNearlyZero(pts[0].fY - pts[1].fY))
         return false;
     if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) {
         // We've already determined the query point lies within the
         // vertical range of the line segment.
         if (pt.fX <= pts[0].fX) {
             if (ambiguous) {
                 *ambiguous = (pt.fY == pts[1].fY);
             }
             return true;
         }
         return false;
     }
     // Ambiguity check
     if (pt.fY == pts[1].fY) {
         if (pt.fX <= pts[1].fX) {
             if (ambiguous) {
                 *ambiguous = true;
             }
             return true;
         }
         return false;
     }
     // Full line segment evaluation
     SkScalar delta_y = pts[1].fY - pts[0].fY;
     SkScalar delta_x = pts[1].fX - pts[0].fX;
     SkScalar slope = SkScalarDiv(delta_y, delta_x);
     SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX);
     // Solve for x coordinate at y = pt.fY
     SkScalar x = SkScalarDiv(pt.fY - b, slope);
     return pt.fX <= x;
 }
 /** If defined, this makes eval_quad and eval_cubic do more setup (sometimes
     involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul.
     May also introduce overflow of fixed when we compute our setup.
 */
 //    #define DIRECT_EVAL_OF_POLYNOMIALS
 ////////////////////////////////////////////////////////////////////////
 static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) {
     SkScalar ab = a - b;
     SkScalar bc = b - c;
     if (ab < 0) {
         bc = -bc;
     }
     return ab == 0 || bc < 0;
 }
 ////////////////////////////////////////////////////////////////////////
 static bool is_unit_interval(SkScalar x) {
     return x > 0 && x < SK_Scalar1;
 }
 static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio) {
     SkASSERT(ratio);
     if (numer < 0) {
         numer = -numer;
         denom = -denom;
     }
     if (denom == 0 || numer == 0 || numer >= denom) {
         return 0;
     }
     SkScalar r = SkScalarDiv(numer, denom);
     if (SkScalarIsNaN(r)) {
         return 0;
     }
     SkASSERT(r >= 0 && r < SK_Scalar1);
     if (r == 0) { // catch underflow if numer <<<< denom
         return 0;
     }
     *ratio = r;
     return 1;
 }
 /** From Numerical Recipes in C.
     Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
     x1 = Q / A
     x2 = C / Q
 */
 int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) {
     SkASSERT(roots);
     if (A == 0) {
         return valid_unit_divide(-C, B, roots);
     }
     SkScalar* r = roots;
     SkScalar R = B*B - 4*A*C;
     if (R < 0 || SkScalarIsNaN(R)) {  // complex roots
         return 0;
     }
     R = SkScalarSqrt(R);
     SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
     r += valid_unit_divide(Q, A, r);
     r += valid_unit_divide(C, Q, r);
     if (r - roots == 2) {
         if (roots[0] > roots[1])
             SkTSwap<SkScalar>(roots[0], roots[1]);
         else if (roots[0] == roots[1])  // nearly-equal?
             r -= 1; // skip the double root
     }
     return (int)(r - roots);
 }
 ///////////////////////////////////////////////////////////////////////////////
 ///////////////////////////////////////////////////////////////////////////////
 static SkScalar eval_quad(const SkScalar src[], SkScalar t) {
     SkASSERT(src);
     SkASSERT(t >= 0 && t <= SK_Scalar1);
 #ifdef DIRECT_EVAL_OF_POLYNOMIALS
     SkScalar    C = src[0];
     SkScalar    A = src[4] - 2 * src[2] + C;
     SkScalar    B = 2 * (src[2] - C);
     return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
 #else
     SkScalar    ab = SkScalarInterp(src[0], src[2], t);
     SkScalar    bc = SkScalarInterp(src[2], src[4], t);
     return SkScalarInterp(ab, bc, t);
 #endif
 }
 static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t) {
     SkScalar A = src[4] - 2 * src[2] + src[0];
     SkScalar B = src[2] - src[0];
     return 2 * SkScalarMulAdd(A, t, B);
 }
 static SkScalar eval_quad_derivative_at_half(const SkScalar src[]) {
     SkScalar A = src[4] - 2 * src[2] + src[0];
     SkScalar B = src[2] - src[0];
     return A + 2 * B;
 }
 void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt,
                   SkVector* tangent) {
     SkASSERT(src);
     SkASSERT(t >= 0 && t <= SK_Scalar1);
     if (pt) {
         pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t));
     }
     if (tangent) {
         tangent->set(eval_quad_derivative(&src[0].fX, t),
                      eval_quad_derivative(&src[0].fY, t));
     }
 }
 void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent) {
     SkASSERT(src);
     if (pt) {
         SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
         SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
         SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
         SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
         pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
     }
     if (tangent) {
         tangent->set(eval_quad_derivative_at_half(&src[0].fX),
                      eval_quad_derivative_at_half(&src[0].fY));
     }
 }
 static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t) {
     SkScalar    ab = SkScalarInterp(src[0], src[2], t);
     SkScalar    bc = SkScalarInterp(src[2], src[4], t);
     dst[0] = src[0];
     dst[2] = ab;
     dst[4] = SkScalarInterp(ab, bc, t);
     dst[6] = bc;
     dst[8] = src[4];
 }
 void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) {
     SkASSERT(t > 0 && t < SK_Scalar1);
     interp_quad_coords(&src[0].fX, &dst[0].fX, t);
     interp_quad_coords(&src[0].fY, &dst[0].fY, t);
 }
 void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) {
     SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
     SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
     SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
     SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
     dst[0] = src[0];
     dst[1].set(x01, y01);
     dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
     dst[3].set(x12, y12);
     dst[4] = src[2];
 }
 /** Quad'(t) = At + B, where
     A = 2(a - 2b + c)
     B = 2(b - a)
     Solve for t, only if it fits between 0 < t < 1
 */
 int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) {
     /*  At + B == 0
         t = -B / A
     */
     return valid_unit_divide(a - b, a - b - b + c, tValue);
 }
 static inline void flatten_double_quad_extrema(SkScalar coords[14]) {
     coords[2] = coords[6] = coords[4];
 }
 /*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
  stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
  */
 int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5]) {
     SkASSERT(src);
     SkASSERT(dst);
     SkScalar a = src[0].fY;
     SkScalar b = src[1].fY;
     SkScalar c = src[2].fY;
     if (is_not_monotonic(a, b, c)) {
         SkScalar    tValue;
         if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
             SkChopQuadAt(src, dst, tValue);
             flatten_double_quad_extrema(&dst[0].fY);
             return 1;
         }
         // if we get here, we need to force dst to be monotonic, even though
         // we couldn't compute a unit_divide value (probably underflow).
         b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
     }
     dst[0].set(src[0].fX, a);
     dst[1].set(src[1].fX, b);
     dst[2].set(src[2].fX, c);
     return 0;
 }
 /*  Returns 0 for 1 quad, and 1 for two quads, either way the answer is
     stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
  */
 int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5]) {
     SkASSERT(src);
     SkASSERT(dst);
     SkScalar a = src[0].fX;
     SkScalar b = src[1].fX;
     SkScalar c = src[2].fX;
     if (is_not_monotonic(a, b, c)) {
         SkScalar tValue;
         if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
             SkChopQuadAt(src, dst, tValue);
             flatten_double_quad_extrema(&dst[0].fX);
             return 1;
         }
         // if we get here, we need to force dst to be monotonic, even though
         // we couldn't compute a unit_divide value (probably underflow).
         b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
     }
     dst[0].set(a, src[0].fY);
     dst[1].set(b, src[1].fY);
     dst[2].set(c, src[2].fY);
     return 0;
 }
 //  F(t)    = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
 //  F'(t)   = 2 (b - a) + 2 (a - 2b + c) t
 //  F''(t)  = 2 (a - 2b + c)
 //
 //  A = 2 (b - a)
 //  B = 2 (a - 2b + c)
 //
 //  Maximum curvature for a quadratic means solving
 //  Fx' Fx'' + Fy' Fy'' = 0
 //
 //  t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
 //
 SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) {
     SkScalar    Ax = src[1].fX - src[0].fX;
     SkScalar    Ay = src[1].fY - src[0].fY;
     SkScalar    Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
     SkScalar    By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
     SkScalar    t = 0;  // 0 means don't chop
     (void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
     return t;
 }
 int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) {
     SkScalar t = SkFindQuadMaxCurvature(src);
     if (t == 0) {
         memcpy(dst, src, 3 * sizeof(SkPoint));
         return 1;
     } else {
         SkChopQuadAt(src, dst, t);
         return 2;
     }
 }
 #define SK_ScalarTwoThirds  (0.666666666f)
 void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
     const SkScalar scale = SK_ScalarTwoThirds;
     dst[0] = src[0];
     dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale),
                src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale));
     dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale),
                src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale));
     dst[3] = src[2];
 }
 //////////////////////////////////////////////////////////////////////////////
 ///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
 //////////////////////////////////////////////////////////////////////////////
 static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4]) {
     coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0];
     coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]);
     coeff[2] = 3*(pt[2] - pt[0]);
     coeff[3] = pt[0];
 }
 void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4]) {
     SkASSERT(pts);
     if (cx) {
         get_cubic_coeff(&pts[0].fX, cx);
     }
     if (cy) {
         get_cubic_coeff(&pts[0].fY, cy);
     }
 }
 static SkScalar eval_cubic(const SkScalar src[], SkScalar t) {
     SkASSERT(src);
     SkASSERT(t >= 0 && t <= SK_Scalar1);
     if (t == 0) {
         return src[0];
     }
 #ifdef DIRECT_EVAL_OF_POLYNOMIALS
     SkScalar D = src[0];
     SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
     SkScalar B = 3*(src[4] - src[2] - src[2] + D);
     SkScalar C = 3*(src[2] - D);
     return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D);
 #else
     SkScalar    ab = SkScalarInterp(src[0], src[2], t);
     SkScalar    bc = SkScalarInterp(src[2], src[4], t);
     SkScalar    cd = SkScalarInterp(src[4], src[6], t);
     SkScalar    abc = SkScalarInterp(ab, bc, t);
     SkScalar    bcd = SkScalarInterp(bc, cd, t);
     return SkScalarInterp(abc, bcd, t);
 #endif
 }
 /** return At^2 + Bt + C
 */
 static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t) {
     SkASSERT(t >= 0 && t <= SK_Scalar1);
     return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
 }
 static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t) {
     SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
     SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
     SkScalar C = src[2] - src[0];
     return eval_quadratic(A, B, C, t);
 }
 static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t) {
     SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
     SkScalar B = src[4] - 2 * src[2] + src[0];
     return SkScalarMulAdd(A, t, B);
 }
 void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc,
                    SkVector* tangent, SkVector* curvature) {
     SkASSERT(src);
     SkASSERT(t >= 0 && t <= SK_Scalar1);
     if (loc) {
         loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
     }
     if (tangent) {
         tangent->set(eval_cubic_derivative(&src[0].fX, t),
                      eval_cubic_derivative(&src[0].fY, t));
     }
     if (curvature) {
         curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
                        eval_cubic_2ndDerivative(&src[0].fY, t));
     }
 }
 /** Cubic'(t) = At^2 + Bt + C, where
     A = 3(-a + 3(b - c) + d)
     B = 6(a - 2b + c)
     C = 3(b - a)
     Solve for t, keeping only those that fit betwee 0 < t < 1
 */
 int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d,
                        SkScalar tValues[2]) {
     // we divide A,B,C by 3 to simplify
     SkScalar A = d - a + 3*(b - c);
     SkScalar B = 2*(a - b - b + c);
     SkScalar C = b - a;
     return SkFindUnitQuadRoots(A, B, C, tValues);
 }
 static void interp_cubic_coords(const SkScalar* src, SkScalar* dst,
                                 SkScalar t) {
     SkScalar    ab = SkScalarInterp(src[0], src[2], t);
     SkScalar    bc = SkScalarInterp(src[2], src[4], t);
     SkScalar    cd = SkScalarInterp(src[4], src[6], t);
     SkScalar    abc = SkScalarInterp(ab, bc, t);
     SkScalar    bcd = SkScalarInterp(bc, cd, t);
     SkScalar    abcd = SkScalarInterp(abc, bcd, t);
     dst[0] = src[0];
     dst[2] = ab;
     dst[4] = abc;
     dst[6] = abcd;
     dst[8] = bcd;
     dst[10] = cd;
     dst[12] = src[6];
 }
 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) {
     SkASSERT(t > 0 && t < SK_Scalar1);
     interp_cubic_coords(&src[0].fX, &dst[0].fX, t);
     interp_cubic_coords(&src[0].fY, &dst[0].fY, t);
 }
 /*  http://code.google.com/p/skia/issues/detail?id=32
     This test code would fail when we didn't check the return result of
     valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
     that after the first chop, the parameters to valid_unit_divide are equal
     (thanks to finite float precision and rounding in the subtracts). Thus
     even though the 2nd tValue looks < 1.0, after we renormalize it, we end
     up with 1.0, hence the need to check and just return the last cubic as
     a degenerate clump of 4 points in the sampe place.
     static void test_cubic() {
         SkPoint src[4] = {
             { 556.25000, 523.03003 },
             { 556.23999, 522.96002 },
             { 556.21997, 522.89001 },
             { 556.21997, 522.82001 }
         };
         SkPoint dst[10];
         SkScalar tval[] = { 0.33333334f, 0.99999994f };
         SkChopCubicAt(src, dst, tval, 2);
     }
  */
 void SkChopCubicAt(const SkPoint src[4], SkPoint dst[],
                    const SkScalar tValues[], int roots) {
 #ifdef SK_DEBUG
     {
         for (int i = 0; i < roots - 1; i++)
         {
             SkASSERT(is_unit_interval(tValues[i]));
             SkASSERT(is_unit_interval(tValues[i+1]));
             SkASSERT(tValues[i] < tValues[i+1]);
         }
     }
 #endif
     if (dst) {
         if (roots == 0) { // nothing to chop
             memcpy(dst, src, 4*sizeof(SkPoint));
         } else {
             SkScalar    t = tValues[0];
             SkPoint     tmp[4];
             for (int i = 0; i < roots; i++) {
                 SkChopCubicAt(src, dst, t);
                 if (i == roots - 1) {
                     break;
                 }
                 dst += 3;
                 // have src point to the remaining cubic (after the chop)
                 memcpy(tmp, dst, 4 * sizeof(SkPoint));
                 src = tmp;
                 // watch out in case the renormalized t isn't in range
                 if (!valid_unit_divide(tValues[i+1] - tValues[i],
                                        SK_Scalar1 - tValues[i], &t)) {
                     // if we can't, just create a degenerate cubic
                     dst[4] = dst[5] = dst[6] = src[3];
                     break;
                 }
             }
         }
     }
 }
 void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) {
     SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
     SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
     SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
     SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
     SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX);
     SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY);
     SkScalar x012 = SkScalarAve(x01, x12);
     SkScalar y012 = SkScalarAve(y01, y12);
     SkScalar x123 = SkScalarAve(x12, x23);
     SkScalar y123 = SkScalarAve(y12, y23);
     dst[0] = src[0];
     dst[1].set(x01, y01);
     dst[2].set(x012, y012);
     dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123));
     dst[4].set(x123, y123);
     dst[5].set(x23, y23);
     dst[6] = src[3];
 }
 static void flatten_double_cubic_extrema(SkScalar coords[14]) {
     coords[4] = coords[8] = coords[6];
 }
 /** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
     the resulting beziers are monotonic in Y. This is called by the scan
     converter.  Depending on what is returned, dst[] is treated as follows:
 dst[0..3] is the original cubic
 dst[0..3] and dst[3..6] are the two new cubics
 dst[0..3], dst[3..6], dst[6..9] are the three new cubics
     If dst == null, it is ignored and only the count is returned.
 */
 int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
     SkScalar    tValues[2];
     int         roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
                                            src[3].fY, tValues);
     SkChopCubicAt(src, dst, tValues, roots);
     if (dst && roots > 0) {
         // we do some cleanup to ensure our Y extrema are flat
         flatten_double_cubic_extrema(&dst[0].fY);
         if (roots == 2) {
             flatten_double_cubic_extrema(&dst[3].fY);
         }
     }
     return roots;
 }
 int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
     SkScalar    tValues[2];
     int         roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
                                            src[3].fX, tValues);
     SkChopCubicAt(src, dst, tValues, roots);
     if (dst && roots > 0) {
         // we do some cleanup to ensure our Y extrema are flat
         flatten_double_cubic_extrema(&dst[0].fX);
         if (roots == 2) {
             flatten_double_cubic_extrema(&dst[3].fX);
         }
     }
     return roots;
 }
 /** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
     Inflection means that curvature is zero.
     Curvature is [F' x F''] / [F'^3]
     So we solve F'x X F''y - F'y X F''y == 0
     After some canceling of the cubic term, we get
     A = b - a
     B = c - 2b + a
     C = d - 3c + 3b - a
     (BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
 */
 int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) {
     SkScalar    Ax = src[1].fX - src[0].fX;
     SkScalar    Ay = src[1].fY - src[0].fY;
     SkScalar    Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
     SkScalar    By = src[2].fY - 2 * src[1].fY + src[0].fY;
     SkScalar    Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
     SkScalar    Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
     return SkFindUnitQuadRoots(Bx*Cy - By*Cx,
                                Ax*Cy - Ay*Cx,
                                Ax*By - Ay*Bx,
                                tValues);
 }
 int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10]) {
     SkScalar    tValues[2];
     int         count = SkFindCubicInflections(src, tValues);
     if (dst) {
         if (count == 0) {
             memcpy(dst, src, 4 * sizeof(SkPoint));
         } else {
             SkChopCubicAt(src, dst, tValues, count);
         }
     }
     return count + 1;
 }
 template <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;
     SkScalar*   roots = tValues;
     SkScalar    r;
     if (R2MinusQ3 < 0) { // we have 3 real roots
         SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3));
         SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
         r = neg2RootQ * SkScalarCos(theta/3) - adiv3;
         if (is_unit_interval(r)) {
             *roots++ = r;
         }
         r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3;
         if (is_unit_interval(r)) {
             *roots++ = r;
         }
         r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3;
         if (is_unit_interval(r)) {
             *roots++ = r;
         }
         SkDEBUGCODE(test_collaps_duplicates();)
         // now sort the roots
         int count = (int)(roots - tValues);
         SkASSERT((unsigned)count <= 3);
         bubble_sort(tValues, count);
         count = collaps_duplicates(tValues, count);
         roots = tValues + count;    // so we compute the proper count below
     } else {              // we have 1 real root
         SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
         A = SkScalarCubeRoot(A);
         if (R > 0) {
             A = -A;
         }
         if (A != 0) {
             A += Q / A;
         }
         r = A - adiv3;
         if (is_unit_interval(r)) {
             *roots++ = r;
         }
     }
     return (int)(roots - tValues);
 }
 /*  Looking for F' dot F'' == 0
     A = b - a
     B = c - 2b + a
     C = d - 3c + 3b - a
     F' = 3Ct^2 + 6Bt + 3A
     F'' = 6Ct + 6B
     F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
 */
 static void formulate_F1DotF2(const SkScalar src[], SkScalar coeff[4]) {
     SkScalar    a = src[2] - src[0];
     SkScalar    b = src[4] - 2 * src[2] + src[0];
     SkScalar    c = src[6] + 3 * (src[2] - src[4]) - src[0];
     coeff[0] = c * c;
     coeff[1] = 3 * b * c;
     coeff[2] = 2 * b * b + c * a;
     coeff[3] = a * b;
 }
 /*  Looking for F' dot F'' == 0
     A = b - a
     B = c - 2b + a
     C = d - 3c + 3b - a
     F' = 3Ct^2 + 6Bt + 3A
     F'' = 6Ct + 6B
     F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
 */
 int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3]) {
     SkScalar coeffX[4], coeffY[4];
     int      i;
     formulate_F1DotF2(&src[0].fX, coeffX);
     formulate_F1DotF2(&src[0].fY, coeffY);
     for (i = 0; i < 4; i++) {
         coeffX[i] += coeffY[i];
     }
     SkScalar    t[3];
     int         count = solve_cubic_poly(coeffX, t);
     int         maxCount = 0;
     // now remove extrema where the curvature is zero (mins)
     // !!!! need a test for this !!!!
     for (i = 0; i < count; i++) {
         // if (not_min_curvature())
         if (t[i] > 0 && t[i] < SK_Scalar1) {
             tValues[maxCount++] = t[i];
         }
     }
     return maxCount;
 }
 int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
                               SkScalar tValues[3]) {
     SkScalar    t_storage[3];
     if (tValues == NULL) {
         tValues = t_storage;
     }
     int count = SkFindCubicMaxCurvature(src, tValues);
     if (dst) {
         if (count == 0) {
             memcpy(dst, src, 4 * sizeof(SkPoint));
         } else {
             SkChopCubicAt(src, dst, tValues, count);
         }
     }
     return count + 1;
 }
 bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4],
                                  bool* ambiguous) {
     if (ambiguous) {
         *ambiguous = false;
     }
     // Find the minimum and maximum y of the extrema, which are the
     // first and last points since this cubic is monotonic
     SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY);
     SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY);
     if (pt.fY == cubic[0].fY
         || pt.fY < min_y
         || pt.fY > max_y) {
         // The query line definitely does not cross the curve
         if (ambiguous) {
             *ambiguous = (pt.fY == cubic[0].fY);
         }
         return false;
     }
     bool pt_at_extremum = (pt.fY == cubic[3].fY);
     SkScalar min_x =
         SkMinScalar(
             SkMinScalar(
                 SkMinScalar(cubic[0].fX, cubic[1].fX),
                 cubic[2].fX),
             cubic[3].fX);
     if (pt.fX < min_x) {
         // The query line definitely crosses the curve
         if (ambiguous) {
             *ambiguous = pt_at_extremum;
         }
         return true;
     }
     SkScalar max_x =
         SkMaxScalar(
             SkMaxScalar(
                 SkMaxScalar(cubic[0].fX, cubic[1].fX),
                 cubic[2].fX),
             cubic[3].fX);
     if (pt.fX > max_x) {
         // The query line definitely does not cross the curve
         return false;
     }
     // Do a binary search to find the parameter value which makes y as
     // close as possible to the query point. See whether the query
     // line's origin is to the left of the associated x coordinate.
     // kMaxIter is chosen as the number of mantissa bits for a float,
     // since there's no way we are going to get more precision by
     // iterating more times than that.
     const int kMaxIter = 23;
     SkPoint eval;
     int iter = 0;
     SkScalar upper_t;
     SkScalar lower_t;
     // Need to invert direction of t parameter if cubic goes up
     // instead of down
     if (cubic[3].fY > cubic[0].fY) {
         upper_t = SK_Scalar1;
         lower_t = 0;
     } else {
         upper_t = 0;
         lower_t = SK_Scalar1;
     }
     do {
         SkScalar t = SkScalarAve(upper_t, lower_t);
         SkEvalCubicAt(cubic, t, &eval, NULL, NULL);
         if (pt.fY > eval.fY) {
             lower_t = t;
         } else {
             upper_t = t;
         }
     } while (++iter < kMaxIter
              && !SkScalarNearlyZero(eval.fY - pt.fY));
     if (pt.fX <= eval.fX) {
         if (ambiguous) {
             *ambiguous = pt_at_extremum;
         }
         return true;
     }
     return false;
 }
 int SkNumXRayCrossingsForCubic(const SkXRay& pt,
                                const SkPoint cubic[4],
                                bool* ambiguous) {
     int num_crossings = 0;
     SkPoint monotonic_cubics[10];
     int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics);
     if (ambiguous) {
         *ambiguous = false;
     }
     bool locally_ambiguous;
     if (SkXRayCrossesMonotonicCubic(pt,
                                     &monotonic_cubics[0],
                                     &locally_ambiguous))
         ++num_crossings;
     if (ambiguous) {
         *ambiguous |= locally_ambiguous;
     }
     if (num_monotonic_cubics > 0)
         if (SkXRayCrossesMonotonicCubic(pt,
                                         &monotonic_cubics[3],
                                         &locally_ambiguous))
             ++num_crossings;
     if (ambiguous) {
         *ambiguous |= locally_ambiguous;
     }
     if (num_monotonic_cubics > 1)
         if (SkXRayCrossesMonotonicCubic(pt,
                                         &monotonic_cubics[6],
                                         &locally_ambiguous))
             ++num_crossings;
     if (ambiguous) {
         *ambiguous |= locally_ambiguous;
     }
     return num_crossings;
 }
 ///////////////////////////////////////////////////////////////////////////////
 /*  Find t value for quadratic [a, b, c] = d.
     Return 0 if there is no solution within [0, 1)
 */
 static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d) {
     // At^2 + Bt + C = d
     SkScalar A = a - 2 * b + c;
     SkScalar B = 2 * (b - a);
     SkScalar C = a - d;
     SkScalar    roots[2];
     int         count = SkFindUnitQuadRoots(A, B, C, roots);
     SkASSERT(count <= 1);
     return count == 1 ? roots[0] : 0;
 }
 /*  given a quad-curve and a point (x,y), chop the quad at that point and place
     the new off-curve point and endpoint into 'dest'.
     Should only return false if the computed pos is the start of the curve
     (i.e. root == 0)
 */
 static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y,
                                 SkPoint* dest) {
     const SkScalar* base;
     SkScalar        value;
     if (SkScalarAbs(x) < SkScalarAbs(y)) {
         base = &quad[0].fX;
         value = x;
     } else {
         base = &quad[0].fY;
         value = y;
     }
     // note: this returns 0 if it thinks value is out of range, meaning the
     // root might return something outside of [0, 1)
     SkScalar t = quad_solve(base[0], base[2], base[4], value);
     if (t > 0) {
         SkPoint tmp[5];
         SkChopQuadAt(quad, tmp, t);
         dest[0] = tmp[1];
         dest[1].set(x, y);
         return true;
     } else {
         /*  t == 0 means either the value triggered a root outside of [0, 1)
             For our purposes, we can ignore the <= 0 roots, but we want to
             catch the >= 1 roots (which given our caller, will basically mean
             a root of 1, give-or-take numerical instability). If we are in the
             >= 1 case, return the existing offCurve point.
             The test below checks to see if we are close to the "end" of the
             curve (near base[4]). Rather than specifying a tolerance, I just
             check to see if value is on to the right/left of the middle point
             (depending on the direction/sign of the end points).
         */
         if ((base[0] < base[4] && value > base[2]) ||
             (base[0] > base[4] && value < base[2]))   // should root have been 1
         {
             dest[0] = quad[1];
             dest[1].set(x, y);
             return true;
         }
     }
     return false;
 }
 static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
 // The mid point of the quadratic arc approximation is half way between the two
 // control points. The float epsilon adjustment moves the on curve point out by
 // two bits, distributing the convex test error between the round rect
 // approximation and the convex cross product sign equality test.
 #define SK_MID_RRECT_OFFSET \
     (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2
     { SK_Scalar1,            0                      },
     { SK_Scalar1,            SK_ScalarTanPIOver8    },
     { SK_MID_RRECT_OFFSET,   SK_MID_RRECT_OFFSET    },
     { SK_ScalarTanPIOver8,   SK_Scalar1             },
     { 0,                     SK_Scalar1             },
     { -SK_ScalarTanPIOver8,  SK_Scalar1             },
     { -SK_MID_RRECT_OFFSET,  SK_MID_RRECT_OFFSET    },
     { -SK_Scalar1,           SK_ScalarTanPIOver8    },
     { -SK_Scalar1,           0                      },
     { -SK_Scalar1,           -SK_ScalarTanPIOver8   },
     { -SK_MID_RRECT_OFFSET,  -SK_MID_RRECT_OFFSET   },
     { -SK_ScalarTanPIOver8,  -SK_Scalar1            },
     { 0,                     -SK_Scalar1            },
     { SK_ScalarTanPIOver8,   -SK_Scalar1            },
     { SK_MID_RRECT_OFFSET,   -SK_MID_RRECT_OFFSET   },
     { SK_Scalar1,            -SK_ScalarTanPIOver8   },
     { SK_Scalar1,            0                      }
 #undef SK_MID_RRECT_OFFSET
 };
 int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
                    SkRotationDirection dir, const SkMatrix* userMatrix,
                    SkPoint quadPoints[]) {
     // rotate by x,y so that uStart is (1.0)
     SkScalar x = SkPoint::DotProduct(uStart, uStop);
     SkScalar y = SkPoint::CrossProduct(uStart, uStop);
     SkScalar absX = SkScalarAbs(x);
     SkScalar absY = SkScalarAbs(y);
     int pointCount;
     // check for (effectively) coincident vectors
     // this can happen if our angle is nearly 0 or nearly 180 (y == 0)
     // ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
     if (absY <= SK_ScalarNearlyZero && x > 0 &&
         ((y >= 0 && kCW_SkRotationDirection == dir) ||
          (y <= 0 && kCCW_SkRotationDirection == dir))) {
         // just return the start-point
         quadPoints[0].set(SK_Scalar1, 0);
         pointCount = 1;
     } else {
         if (dir == kCCW_SkRotationDirection) {
             y = -y;
         }
         // what octant (quadratic curve) is [xy] in?
         int oct = 0;
         bool sameSign = true;
         if (0 == y) {
             oct = 4;        // 180
             SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
         } else if (0 == x) {
             SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
             oct = y > 0 ? 2 : 6; // 90 : 270
         } else {
             if (y < 0) {
                 oct += 4;
             }
             if ((x < 0) != (y < 0)) {
                 oct += 2;
                 sameSign = false;
             }
             if ((absX < absY) == sameSign) {
                 oct += 1;
             }
         }
         int wholeCount = oct << 1;
         memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));
         const SkPoint* arc = &gQuadCirclePts[wholeCount];
         if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1])) {
             wholeCount += 2;
         }
         pointCount = wholeCount + 1;
     }
     // now handle counter-clockwise and the initial unitStart rotation
     SkMatrix    matrix;
     matrix.setSinCos(uStart.fY, uStart.fX);
     if (dir == kCCW_SkRotationDirection) {
         matrix.preScale(SK_Scalar1, -SK_Scalar1);
     }
     if (userMatrix) {
         matrix.postConcat(*userMatrix);
     }
     matrix.mapPoints(quadPoints, pointCount);
     return pointCount;
 }
 ///////////////////////////////////////////////////////////////////////////////
 //
 // NURB representation for conics.  Helpful explanations at:
 //
 // http://citeseerx.ist.psu.edu/viewdoc/
 //   download?doi=10.1.1.44.5740&rep=rep1&type=ps
 // and
 // http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html
 //
 // F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
 //     ------------------------------------------
 //         ((1 - t)^2 + t^2 + 2 (1 - t) t w)
 //
 //   = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
 //     ------------------------------------------------
 //             {t^2 (2 - 2 w), t (-2 + 2 w), 1}
 //
 static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) {
     SkASSERT(src);
     SkASSERT(t >= 0 && t <= SK_Scalar1);
     SkScalar    src2w = SkScalarMul(src[2], w);
     SkScalar    C = src[0];
     SkScalar    A = src[4] - 2 * src2w + C;
     SkScalar    B = 2 * (src2w - C);
     SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
     B = 2 * (w - SK_Scalar1);
     C = SK_Scalar1;
     A = -B;
     SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
     return SkScalarDiv(numer, denom);
 }
 // F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
 //
 //  t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
 //  t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
 //  t^0 : -2 P0 w + 2 P1 w
 //
 //  We disregard magnitude, so we can freely ignore the denominator of F', and
 //  divide the numerator by 2
 //
 //    coeff[0] for t^2
 //    coeff[1] for t^1
 //    coeff[2] for t^0
 //
 static void conic_deriv_coeff(const SkScalar src[],
                               SkScalar w,
                               SkScalar coeff[3]) {
     const SkScalar P20 = src[4] - src[0];
     const SkScalar P10 = src[2] - src[0];
     const SkScalar wP10 = w * P10;
     coeff[0] = w * P20 - P20;
     coeff[1] = P20 - 2 * wP10;
     coeff[2] = wP10;
 }
 static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) {
     SkScalar coeff[3];
     conic_deriv_coeff(coord, w, coeff);
     return t * (t * coeff[0] + coeff[1]) + coeff[2];
 }
 static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
     SkScalar coeff[3];
     conic_deriv_coeff(src, w, coeff);
     SkScalar tValues[2];
     int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
     SkASSERT(0 == roots || 1 == roots);
     if (1 == roots) {
         *t = tValues[0];
         return true;
     }
     return false;
 }
 struct SkP3D {
     SkScalar fX, fY, fZ;
     void set(SkScalar x, SkScalar y, SkScalar z) {
         fX = x; fY = y; fZ = z;
     }
     void projectDown(SkPoint* dst) const {
         dst->set(fX / fZ, fY / fZ);
     }
 };
 // We only interpolate one dimension at a time (the first, at +0, +3, +6).
 static void p3d_interp(const SkScalar src[7], SkScalar dst[7], SkScalar t) {
     SkScalar ab = SkScalarInterp(src[0], src[3], t);
     SkScalar bc = SkScalarInterp(src[3], src[6], t);
     dst[0] = ab;
     dst[3] = SkScalarInterp(ab, bc, t);
     dst[6] = bc;
 }
 static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
     dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
     dst[1].set(src[1].fX * w, src[1].fY * w, w);
     dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
 }
 void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
     SkASSERT(t >= 0 && t <= SK_Scalar1);
     if (pt) {
         pt->set(conic_eval_pos(&fPts[0].fX, fW, t),
                 conic_eval_pos(&fPts[0].fY, fW, t));
     }
     if (tangent) {
         tangent->set(conic_eval_tan(&fPts[0].fX, fW, t),
                      conic_eval_tan(&fPts[0].fY, fW, t));
     }
 }
 void SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
     SkP3D tmp[3], tmp2[3];
     ratquad_mapTo3D(fPts, fW, tmp);
     p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
     p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
     p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
     dst[0].fPts[0] = fPts[0];
     tmp2[0].projectDown(&dst[0].fPts[1]);
     tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
     tmp2[2].projectDown(&dst[1].fPts[1]);
     dst[1].fPts[2] = fPts[2];
     // to put in "standard form", where w0 and w2 are both 1, we compute the
     // new w1 as sqrt(w1*w1/w0*w2)
     // or
     // w1 /= sqrt(w0*w2)
     //
     // However, in our case, we know that for dst[0]:
     //     w0 == 1, and for dst[1], w2 == 1
     //
     SkScalar root = SkScalarSqrt(tmp2[1].fZ);
     dst[0].fW = tmp2[0].fZ / root;
     dst[1].fW = tmp2[2].fZ / root;
 }
 static SkScalar subdivide_w_value(SkScalar w) {
     return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
 }
 void SkConic::chop(SkConic dst[2]) const {
     SkScalar scale = SkScalarInvert(SK_Scalar1 + fW);
     SkScalar p1x = fW * fPts[1].fX;
     SkScalar p1y = fW * fPts[1].fY;
     SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf;
     SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf;
     dst[0].fPts[0] = fPts[0];
     dst[0].fPts[1].set((fPts[0].fX + p1x) * scale,
                        (fPts[0].fY + p1y) * scale);
     dst[0].fPts[2].set(mx, my);
     dst[1].fPts[0].set(mx, my);
     dst[1].fPts[1].set((p1x + fPts[2].fX) * scale,
                        (p1y + fPts[2].fY) * scale);
     dst[1].fPts[2] = fPts[2];
     dst[0].fW = dst[1].fW = subdivide_w_value(fW);
 }
 /*
  *  "High order approximation of conic sections by quadratic splines"
  *      by Michael Floater, 1993
  */
 #define AS_QUAD_ERROR_SETUP                                         \
     SkScalar a = fW - 1;                                            \
     SkScalar k = a / (4 * (2 + a));                                 \
     SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX);    \
     SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
 void SkConic::computeAsQuadError(SkVector* err) const {
     AS_QUAD_ERROR_SETUP
     err->set(x, y);
 }
 bool SkConic::asQuadTol(SkScalar tol) const {
     AS_QUAD_ERROR_SETUP
     return (x * x + y * y) <= tol * tol;
 }
 int SkConic::computeQuadPOW2(SkScalar tol) const {
     AS_QUAD_ERROR_SETUP
     SkScalar error = SkScalarSqrt(x * x + y * y) - tol;
     if (error <= 0) {
         return 0;
     }
     uint32_t ierr = (uint32_t)error;
     return (34 - SkCLZ(ierr)) >> 1;
 }
 static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
     SkASSERT(level >= 0);
     if (0 == level) {
         memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
         return pts + 2;
     } else {
         SkConic dst[2];
         src.chop(dst);
         --level;
         pts = subdivide(dst[0], pts, level);
         return subdivide(dst[1], pts, level);
     }
 }
 int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
     SkASSERT(pow2 >= 0);
     *pts = fPts[0];
     SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2);
     SkASSERT(endPts - pts == (2 * (1 << pow2) + 1));
     return 1 << pow2;
 }
 bool SkConic::findXExtrema(SkScalar* t) const {
     return conic_find_extrema(&fPts[0].fX, fW, t);
 }
 bool SkConic::findYExtrema(SkScalar* t) const {
     return conic_find_extrema(&fPts[0].fY, fW, t);
 }
 bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
     SkScalar t;
     if (this->findXExtrema(&t)) {
         this->chopAt(t, dst);
         // now clean-up the middle, since we know t was meant to be at
         // an X-extrema
         SkScalar value = dst[0].fPts[2].fX;
         dst[0].fPts[1].fX = value;
         dst[1].fPts[0].fX = value;
         dst[1].fPts[1].fX = value;
         return true;
     }
     return false;
 }
 bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
     SkScalar t;
     if (this->findYExtrema(&t)) {
         this->chopAt(t, dst);
         // now clean-up the middle, since we know t was meant to be at
         // an Y-extrema
         SkScalar value = dst[0].fPts[2].fY;
         dst[0].fPts[1].fY = value;
         dst[1].fPts[0].fY = value;
         dst[1].fPts[1].fY = value;
         return true;
     }
     return false;
 }
 void SkConic::computeTightBounds(SkRect* bounds) const {
     SkPoint pts[4];
     pts[0] = fPts[0];
     pts[1] = fPts[2];
     int count = 2;
     SkScalar t;
     if (this->findXExtrema(&t)) {
         this->evalAt(t, &pts[count++]);
     }
     if (this->findYExtrema(&t)) {
         this->evalAt(t, &pts[count++]);
     }
     bounds->set(pts, count);
 }
 void SkConic::computeFastBounds(SkRect* bounds) const {
     bounds->set(fPts, 3);
 }
 bool SkConic::findMaxCurvature(SkScalar* t) const {
     // TODO: Implement me
     return false;
 }

The Tor Browser / annotate

gfx/skia/trunk/src/core/SkGeometry.cpp@129ffea94266 (annotated)

gfx/skia/trunk/src/core/SkGeometry.cpp