Add intersections for path ops

src/pathops/SkQuarticRoot.cpp

+// from http://tog.acm.org/resources/GraphicsGems/gems/Roots3And4.c
+/*
+ *  Roots3And4.c
+ *
+ *  Utility functions to find cubic and quartic roots,
+ *  coefficients are passed like this:
+ *
+ *      c[0] + c[1]*x + c[2]*x^2 + c[3]*x^3 + c[4]*x^4 = 0
+ *
+ *  The functions return the number of non-complex roots and
+ *  put the values into the s array.
+ *
+ *  Author:         Jochen Schwarze (schwarze@isa.de)
+ *
+ *  Jan 26, 1990    Version for Graphics Gems
+ *  Oct 11, 1990    Fixed sign problem for negative q's in SolveQuartic
+ *                  (reported by Mark Podlipec),
+ *                  Old-style function definitions,
+ *                  IsZero() as a macro
+ *  Nov 23, 1990    Some systems do not declare acos() and cbrt() in
+ *                  <math.h>, though the functions exist in the library.
+ *                  If large coefficients are used, EQN_EPS should be
+ *                  reduced considerably (e.g. to 1E-30), results will be
+ *                  correct but multiple roots might be reported more
+ *                  than once.
+ */
+
+#include "SkPathOpsCubic.h"
+#include "SkPathOpsQuad.h"
+#include "SkQuarticRoot.h"
+
+int SkReducedQuarticRoots(const double t4, const double t3, const double t2, const double t1,
+        const double t0, const bool oneHint, double roots[4]) {
+#ifdef 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));
+    snprintf(str, sizeof(str),
+            "Solve[%1.19g x^4 + %1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]",
+            t4, t3, t2, t1, t0);
+    mathematica_ize(str, sizeof(str));
+#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA
+    SkDebugf("%s\n", str);
+#endif
+#endif
+    if (approximately_zero_when_compared_to(t4, t0) // 0 is one root
+            && approximately_zero_when_compared_to(t4, t1)
+            && approximately_zero_when_compared_to(t4, t2)) {
+        if (approximately_zero_when_compared_to(t3, t0)
+            && approximately_zero_when_compared_to(t3, t1)
+            && approximately_zero_when_compared_to(t3, t2)) {
+            return SkDQuad::RootsReal(t2, t1, t0, roots);
+        }
+        if (approximately_zero_when_compared_to(t4, t3)) {
+            return SkDCubic::RootsReal(t3, t2, t1, t0, roots);
+        }
+    }
+    if ((approximately_zero_when_compared_to(t0, t1) || approximately_zero(t1))  // 0 is one root
+      //      && approximately_zero_when_compared_to(t0, t2)
+            && approximately_zero_when_compared_to(t0, t3)
+            && approximately_zero_when_compared_to(t0, t4)) {
+        int num = SkDCubic::RootsReal(t4, t3, t2, t1, roots);
+        for (int i = 0; i < num; ++i) {
+            if (approximately_zero(roots[i])) {
+                return num;
+            }
+        }
+        roots[num++] = 0;
+        return num;
+    }
+    if (oneHint) {
+        SkASSERT(approximately_zero(t4 + t3 + t2 + t1 + t0));  // 1 is one root
+        // note that -C == A + B + D + E
+        int num = SkDCubic::RootsReal(t4, t4 + t3, -(t1 + t0), -t0, roots);  
+        for (int i = 0; i < num; ++i) {
+            if (approximately_equal(roots[i], 1)) {
+                return num;
+            }
+        }
+        roots[num++] = 1;
+        return num;
+    }
+    return -1;
+}
+
+int SkQuarticRootsReal(int firstCubicRoot, const double A, const double B, const double C,
+        const double D, const double E, double s[4]) {
+    double  u, v;
+    /* normal form: x^4 + Ax^3 + Bx^2 + Cx + D = 0 */
+    const double invA = 1 / A;
+    const double a = B * invA;
+    const double b = C * invA;
+    const double c = D * invA;
+    const double d = E * invA;
+    /*  substitute x = y - a/4 to eliminate cubic term:
+    x^4 + px^2 + qx + r = 0 */
+    const double a2 = a * a;
+    const double p = -3 * a2 / 8 + b;
+    const double q = a2 * a / 8 - a * b / 2 + c;
+    const double r = -3 * a2 * a2 / 256 + a2 * b / 16 - a * c / 4 + d;
+    int num;
+    if (approximately_zero(r)) {
+    /* no absolute term: y(y^3 + py + q) = 0 */
+        num = SkDCubic::RootsReal(1, 0, p, q, s);
+        s[num++] = 0;
+    } else {
+        /* solve the resolvent cubic ... */
+        double cubicRoots[3];
+        int roots = SkDCubic::RootsReal(1, -p / 2, -r, r * p / 2 - q * q / 8, cubicRoots);
+        int index;
+        /* ... and take one real solution ... */
+        double z;
+        num = 0;
+        int num2 = 0;
+        for (index = firstCubicRoot; index < roots; ++index) {
+            z = cubicRoots[index];
+            /* ... to build two quadric equations */
+            u = z * z - r;
+            v = 2 * z - p;
+            if (approximately_zero_squared(u)) {
+                u = 0;
+            } else if (u > 0) {
+                u = sqrt(u);
+            } else {
+                continue;
+            }
+            if (approximately_zero_squared(v)) {
+                v = 0;
+            } else if (v > 0) {
+                v = sqrt(v);
+            } else {
+                continue;
+            }
+            num = SkDQuad::RootsReal(1, q < 0 ? -v : v, z - u, s);
+            num2 = SkDQuad::RootsReal(1, q < 0 ? v : -v, z + u, s + num);
+            if (!((num | num2) & 1)) {
+                break;  // prefer solutions without single quad roots
+            }
+        }
+        num += num2;
+        if (!num) {
+            return 0; // no valid cubic root
+        }
+    }
+    /* resubstitute */
+    const double sub = a / 4;
+    for (int i = 0; i < num; ++i) {
+        s[i] -= sub;
+    }
+    // eliminate duplicates
+    for (int i = 0; i < num - 1; ++i) {
+        for (int j = i + 1; j < num; ) {
+            if (AlmostEqualUlps(s[i], s[j])) {
+                if (j < --num) {
+                    s[j] = s[num];
+                }
+            } else {
+                ++j;
+            }
+        }
+    }
+    return num;
+}