remove prototype pathops code

experimental/Intersection/QuarticRoot.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    <math.h>
-#include "CubicUtilities.h"
-#include "QuadraticUtilities.h"
-#include "QuarticRoot.h"
-
-int reducedQuarticRoots(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));
-    sprintf(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 0 && SK_DEBUG
-    bool t4Or = 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);
-    bool t4And = 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 (t4Or != t4And) {
-        SkDebugf("%s t4 or and\n", __FUNCTION__);
-    }
-    bool t3Or = approximately_zero_when_compared_to(t3, t0)
-            || approximately_zero_when_compared_to(t3, t1)
-            || approximately_zero_when_compared_to(t3, t2);
-    bool t3And = approximately_zero_when_compared_to(t3, t0)
-            && approximately_zero_when_compared_to(t3, t1)
-            && approximately_zero_when_compared_to(t3, t2);
-    if (t3Or != t3And) {
-        SkDebugf("%s t3 or and\n", __FUNCTION__);
-    }
-    bool t0Or = approximately_zero_when_compared_to(t0, 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);
-    bool t0And = approximately_zero_when_compared_to(t0, 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);
-    if (t0Or != t0And) {
-        SkDebugf("%s t0 or and\n", __FUNCTION__);
-    }
-#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 quadraticRootsReal(t2, t1, t0, roots);
-        }
-        if (approximately_zero_when_compared_to(t4, t3)) {
-            return cubicRootsReal(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 = cubicRootsReal(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
-        int num = cubicRootsReal(t4, t4 + t3, -(t1 + t0), -t0, roots); // note that -C==A+B+D+E
-        for (int i = 0; i < num; ++i) {
-            if (approximately_equal(roots[i], 1)) {
-                return num;
-            }
-        }
-        roots[num++] = 1;
-        return num;
-    }
-    return -1;
-}
-
-int quarticRootsReal(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 = cubicRootsReal(1, 0, p, q, s);
-        s[num++] = 0;
-    } else {
-        /* solve the resolvent cubic ... */
-        double cubicRoots[3];
-        int roots = cubicRootsReal(1, -p / 2, -r, r * p / 2 - q * q / 8, cubicRoots);
-        int index;
-    #if 0 && SK_DEBUG // enable to verify that any cubic root is as good as any other
-        double tries[3][4];
-        int nums[3];
-        for (index = 0; index < roots; ++index) {
-            /* ... and take one real solution ... */
-            const double 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 {
-                SkDebugf("%s u=%1.9g <0\n", __FUNCTION__, u);
-                continue;
-            }
-            if (approximately_zero_squared(v)) {
-                v = 0;
-            } else if (v > 0) {
-                v = sqrt(v);
-            } else {
-                SkDebugf("%s v=%1.9g <0\n", __FUNCTION__, v);
-                continue;
-            }
-            nums[index] = quadraticRootsReal(1, q < 0 ? -v : v, z - u, tries[index]);
-            nums[index] += quadraticRootsReal(1, q < 0 ? v : -v, z + u, tries[index] + nums[index]);
-            /* resubstitute */
-            const double sub = a / 4;
-            for (int i = 0; i < nums[index]; ++i) {
-                tries[index][i] -= sub;
-            }
-        }
-        for (index = 0; index < roots; ++index) {
-            SkDebugf("%s", __FUNCTION__);
-            for (int idx2 = 0; idx2 < nums[index]; ++idx2) {
-                SkDebugf(" %1.9g", tries[index][idx2]);
-            }
-            SkDebugf("\n");
-        }
-    #endif
-        /* ... 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 = quadraticRootsReal(1, q < 0 ? -v : v, z - u, s);
-            num2 = quadraticRootsReal(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;
-}