1 files changed, 424 insertions, 0 deletions
diff --git a/chromium/third_party/skia/experimental/Intersection/CubicUtilities.cpp b/chromium/third_party/skia/experimental/Intersection/CubicUtilities.cpp
new file mode 100644
index 00000000000..c9881c14809
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+++ b/chromium/third_party/skia/experimental/Intersection/CubicUtilities.cpp
@@ -0,0 +1,424 @@
+/*
+ * Copyright 2012 Google Inc.
+ *
+ * Use of this source code is governed by a BSD-style license that can be
+ * found in the LICENSE file.
+ */
+#include "CubicUtilities.h"
+#include "Extrema.h"
+#include "LineUtilities.h"
+#include "QuadraticUtilities.h"
+
+const int gPrecisionUnit = 256; // FIXME: arbitrary -- should try different values in test framework
+
+// FIXME: cache keep the bounds and/or precision with the caller?
+double calcPrecision(const Cubic& cubic) {
+    _Rect dRect;
+    dRect.setBounds(cubic); // OPTIMIZATION: just use setRawBounds ?
+    double width = dRect.right - dRect.left;
+    double height = dRect.bottom - dRect.top;
+    return (width > height ? width : height) / gPrecisionUnit;
+}
+
+#if SK_DEBUG
+double calcPrecision(const Cubic& cubic, double t, double scale) {
+    Cubic part;
+    sub_divide(cubic, SkTMax(0., t - scale), SkTMin(1., t + scale), part);
+    return calcPrecision(part);
+}
+#endif
+
+bool clockwise(const Cubic& c) {
+    double sum = (c[0].x - c[3].x) * (c[0].y + c[3].y);
+    for (int idx = 0; idx < 3; ++idx){
+        sum += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y);
+    }
+    return sum <= 0;
+}
+
+void coefficients(const double* cubic, double& A, double& B, double& C, double& D) {
+    A = cubic[6]; // d
+    B = cubic[4] * 3; // 3*c
+    C = cubic[2] * 3; // 3*b
+    D = cubic[0]; // a
+    A -= D - C + B;     // A =   -a + 3*b - 3*c + d
+    B += 3 * D - 2 * C; // B =  3*a - 6*b + 3*c
+    C -= 3 * D;         // C = -3*a + 3*b
+}
+
+bool controls_contained_by_ends(const Cubic& c) {
+    _Vector startTan = c[1] - c[0];
+    if (startTan.x == 0 && startTan.y == 0) {
+        startTan = c[2] - c[0];
+    }
+    _Vector endTan = c[2] - c[3];
+    if (endTan.x == 0 && endTan.y == 0) {
+        endTan = c[1] - c[3];
+    }
+    if (startTan.dot(endTan) >= 0) {
+        return false;
+    }
+    _Line startEdge = {c[0], c[0]};
+    startEdge[1].x -= startTan.y;
+    startEdge[1].y += startTan.x;
+    _Line endEdge = {c[3], c[3]};
+    endEdge[1].x -= endTan.y;
+    endEdge[1].y += endTan.x;
+    double leftStart1 = is_left(startEdge, c[1]);
+    if (leftStart1 * is_left(startEdge, c[2]) < 0) {
+        return false;
+    }
+    double leftEnd1 = is_left(endEdge, c[1]);
+    if (leftEnd1 * is_left(endEdge, c[2]) < 0) {
+        return false;
+    }
+    return leftStart1 * leftEnd1 >= 0;
+}
+
+bool ends_are_extrema_in_x_or_y(const Cubic& c) {
+    return (between(c[0].x, c[1].x, c[3].x) && between(c[0].x, c[2].x, c[3].x))
+            || (between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y));
+}
+
+bool monotonic_in_y(const Cubic& c) {
+    return between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y);
+}
+
+bool serpentine(const Cubic& c) {
+    if (!controls_contained_by_ends(c)) {
+        return false;
+    }
+    double wiggle = (c[0].x - c[2].x) * (c[0].y + c[2].y);
+    for (int idx = 0; idx < 2; ++idx){
+        wiggle += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y);
+    }
+    double waggle = (c[1].x - c[3].x) * (c[1].y + c[3].y);
+    for (int idx = 1; idx < 3; ++idx){
+        waggle += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y);
+    }
+    return wiggle * waggle < 0;
+}
+
+// cubic roots
+
+const double PI = 4 * atan(1);
+
+// from SkGeometry.cpp (and Numeric Solutions, 5.6)
+int cubicRootsValidT(double A, double B, double C, double D, double t[3]) {
+#if 0
+    if (approximately_zero(A)) {  // we're just a quadratic
+        return quadraticRootsValidT(B, C, D, t);
+    }
+    double a, b, c;
+    {
+        double invA = 1 / A;
+        a = B * invA;
+        b = C * invA;
+        c = D * invA;
+    }
+    double a2 = a * a;
+    double Q = (a2 - b * 3) / 9;
+    double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
+    double Q3 = Q * Q * Q;
+    double R2MinusQ3 = R * R - Q3;
+    double adiv3 = a / 3;
+    double* roots = t;
+    double r;
+
+    if (R2MinusQ3 < 0)   // we have 3 real roots
+    {
+        double theta = acos(R / sqrt(Q3));
+        double neg2RootQ = -2 * sqrt(Q);
+
+        r = neg2RootQ * cos(theta / 3) - adiv3;
+        if (is_unit_interval(r))
+            *roots++ = r;
+
+        r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3;
+        if (is_unit_interval(r))
+            *roots++ = r;
+
+        r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3;
+        if (is_unit_interval(r))
+            *roots++ = r;
+    }
+    else                // we have 1 real root
+    {
+        double A = fabs(R) + sqrt(R2MinusQ3);
+        A = cube_root(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 - t);
+#else
+    double s[3];
+    int realRoots = cubicRootsReal(A, B, C, D, s);
+    int foundRoots = add_valid_ts(s, realRoots, t);
+    return foundRoots;
+#endif
+}
+
+int cubicRootsReal(double A, double B, double C, double D, double s[3]) {
+#if SK_DEBUG
+    // create a string mathematica understands
+    // GDB set print repe 15 # if repeated digits is a bother
+    //     set print elements 400 # if line doesn't fit
+    char str[1024];
+    bzero(str, sizeof(str));
+    sprintf(str, "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]", A, B, C, D);
+    mathematica_ize(str, sizeof(str));
+#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA
+    SkDebugf("%s\n", str);
+#endif
+#endif
+    if (approximately_zero(A)
+            && approximately_zero_when_compared_to(A, B)
+            && approximately_zero_when_compared_to(A, C)
+            && approximately_zero_when_compared_to(A, D)) {  // we're just a quadratic
+        return quadraticRootsReal(B, C, D, s);
+    }
+    if (approximately_zero_when_compared_to(D, A)
+            && approximately_zero_when_compared_to(D, B)
+            && approximately_zero_when_compared_to(D, C)) { // 0 is one root
+        int num = quadraticRootsReal(A, B, C, s);
+        for (int i = 0; i < num; ++i) {
+            if (approximately_zero(s[i])) {
+                return num;
+            }
+        }
+        s[num++] = 0;
+        return num;
+    }
+    if (approximately_zero(A + B + C + D)) { // 1 is one root
+        int num = quadraticRootsReal(A, A + B, -D, s);
+        for (int i = 0; i < num; ++i) {
+            if (AlmostEqualUlps(s[i], 1)) {
+                return num;
+            }
+        }
+        s[num++] = 1;
+        return num;
+    }
+    double a, b, c;
+    {
+        double invA = 1 / A;
+        a = B * invA;
+        b = C * invA;
+        c = D * invA;
+    }
+    double a2 = a * a;
+    double Q = (a2 - b * 3) / 9;
+    double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
+    double R2 = R * R;
+    double Q3 = Q * Q * Q;
+    double R2MinusQ3 = R2 - Q3;
+    double adiv3 = a / 3;
+    double r;
+    double* roots = s;
+#if 0
+    if (approximately_zero_squared(R2MinusQ3) && AlmostEqualUlps(R2, Q3)) {
+        if (approximately_zero_squared(R)) {/* one triple solution */
+            *roots++ = -adiv3;
+        } else { /* one single and one double solution */
+
+            double u = cube_root(-R);
+            *roots++ = 2 * u - adiv3;
+            *roots++ = -u - adiv3;
+        }
+    }
+    else
+#endif
+    if (R2MinusQ3 < 0)   // we have 3 real roots
+    {
+        double theta = acos(R / sqrt(Q3));
+        double neg2RootQ = -2 * sqrt(Q);
+
+        r = neg2RootQ * cos(theta / 3) - adiv3;
+        *roots++ = r;
+
+        r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3;
+        if (!AlmostEqualUlps(s[0], r)) {
+            *roots++ = r;
+        }
+        r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3;
+        if (!AlmostEqualUlps(s[0], r) && (roots - s == 1 || !AlmostEqualUlps(s[1], r))) {
+            *roots++ = r;
+        }
+    }
+    else                // we have 1 real root
+    {
+        double sqrtR2MinusQ3 = sqrt(R2MinusQ3);
+        double A = fabs(R) + sqrtR2MinusQ3;
+        A = cube_root(A);
+        if (R > 0) {
+            A = -A;
+        }
+        if (A != 0) {
+            A += Q / A;
+        }
+        r = A - adiv3;
+        *roots++ = r;
+        if (AlmostEqualUlps(R2, Q3)) {
+            r = -A / 2 - adiv3;
+            if (!AlmostEqualUlps(s[0], r)) {
+                *roots++ = r;
+            }
+        }
+    }
+    return (int)(roots - s);
+}
+
+// from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf
+// c(t)  = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3
+// c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2
+//       = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2
+static double derivativeAtT(const double* cubic, double t) {
+    double one_t = 1 - t;
+    double a = cubic[0];
+    double b = cubic[2];
+    double c = cubic[4];
+    double d = cubic[6];
+    return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t);
+}
+
+double dx_at_t(const Cubic& cubic, double t) {
+    return derivativeAtT(&cubic[0].x, t);
+}
+
+double dy_at_t(const Cubic& cubic, double t) {
+    return derivativeAtT(&cubic[0].y, t);
+}
+
+// OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version of derivative at t?
+_Vector dxdy_at_t(const Cubic& cubic, double t) {
+    _Vector result = { derivativeAtT(&cubic[0].x, t), derivativeAtT(&cubic[0].y, t) };
+    return result;
+}
+
+// OPTIMIZE? share code with formulate_F1DotF2
+int find_cubic_inflections(const Cubic& src, double tValues[])
+{
+    double Ax = src[1].x - src[0].x;
+    double Ay = src[1].y - src[0].y;
+    double Bx = src[2].x - 2 * src[1].x + src[0].x;
+    double By = src[2].y - 2 * src[1].y + src[0].y;
+    double Cx = src[3].x + 3 * (src[1].x - src[2].x) - src[0].x;
+    double Cy = src[3].y + 3 * (src[1].y - src[2].y) - src[0].y;
+    return quadraticRootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By - Ay * Bx, tValues);
+}
+
+static void formulate_F1DotF2(const double src[], double coeff[4])
+{
+    double a = src[2] - src[0];
+    double b = src[4] - 2 * src[2] + src[0];
+    double 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;
+}
+
+/*  from SkGeometry.cpp
+    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 find_cubic_max_curvature(const Cubic& src, double tValues[])
+{
+    double coeffX[4], coeffY[4];
+    int i;
+    formulate_F1DotF2(&src[0].x, coeffX);
+    formulate_F1DotF2(&src[0].y, coeffY);
+    for (i = 0; i < 4; i++) {
+        coeffX[i] = coeffX[i] + coeffY[i];
+    }
+    return cubicRootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues);
+}
+
+
+bool rotate(const Cubic& cubic, int zero, int index, Cubic& rotPath) {
+    double dy = cubic[index].y - cubic[zero].y;
+    double dx = cubic[index].x - cubic[zero].x;
+    if (approximately_zero(dy)) {
+        if (approximately_zero(dx)) {
+            return false;
+        }
+        memcpy(rotPath, cubic, sizeof(Cubic));
+        return true;
+    }
+    for (int index = 0; index < 4; ++index) {
+        rotPath[index].x = cubic[index].x * dx + cubic[index].y * dy;
+        rotPath[index].y = cubic[index].y * dx - cubic[index].x * dy;
+    }
+    return true;
+}
+
+#if 0 // unused for now
+double secondDerivativeAtT(const double* cubic, double t) {
+    double a = cubic[0];
+    double b = cubic[2];
+    double c = cubic[4];
+    double d = cubic[6];
+    return (c - 2 * b + a) * (1 - t) + (d - 2 * c + b) * t;
+}
+#endif
+
+_Point top(const Cubic& cubic, double startT, double endT) {
+    Cubic sub;
+    sub_divide(cubic, startT, endT, sub);
+    _Point topPt = sub[0];
+    if (topPt.y > sub[3].y || (topPt.y == sub[3].y && topPt.x > sub[3].x)) {
+        topPt = sub[3];
+    }
+    double extremeTs[2];
+    if (!monotonic_in_y(sub)) {
+        int roots = findExtrema(sub[0].y, sub[1].y, sub[2].y, sub[3].y, extremeTs);
+        for (int index = 0; index < roots; ++index) {
+            _Point mid;
+            double t = startT + (endT - startT) * extremeTs[index];
+            xy_at_t(cubic, t, mid.x, mid.y);
+            if (topPt.y > mid.y || (topPt.y == mid.y && topPt.x > mid.x)) {
+                topPt = mid;
+            }
+        }
+    }
+    return topPt;
+}
+
+// OPTIMIZE: avoid computing the unused half
+void xy_at_t(const Cubic& cubic, double t, double& x, double& y) {
+    _Point xy = xy_at_t(cubic, t);
+    if (&x) {
+        x = xy.x;
+    }
+    if (&y) {
+        y = xy.y;
+    }
+}
+
+_Point xy_at_t(const Cubic& cubic, double t) {
+    double one_t = 1 - t;
+    double one_t2 = one_t * one_t;
+    double a = one_t2 * one_t;
+    double b = 3 * one_t2 * t;
+    double t2 = t * t;
+    double c = 3 * one_t * t2;
+    double d = t2 * t;
+    _Point result = {a * cubic[0].x + b * cubic[1].x + c * cubic[2].x + d * cubic[3].x,
+            a * cubic[0].y + b * cubic[1].y + c * cubic[2].y + d * cubic[3].y};
+    return result;
+}