remove prototype pathops code

experimental/Intersection/QuadraticUtilities.cpp

-/*
- * 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 "QuadraticUtilities.h"
-#include "TriangleUtilities.h"
-
-// from http://blog.gludion.com/2009/08/distance-to-quadratic-bezier-curve.html
-double nearestT(const Quadratic& quad, const _Point& pt) {
-    _Vector pos = quad[0] - pt;
-    // search points P of bezier curve with PM.(dP / dt) = 0
-    // a calculus leads to a 3d degree equation :
-    _Vector A = quad[1] - quad[0];
-    _Vector B = quad[2] - quad[1];
-    B -= A;
-    double a = B.dot(B);
-    double b = 3 * A.dot(B);
-    double c = 2 * A.dot(A) + pos.dot(B);
-    double d = pos.dot(A);
-    double ts[3];
-    int roots = cubicRootsValidT(a, b, c, d, ts);
-    double d0 = pt.distanceSquared(quad[0]);
-    double d2 = pt.distanceSquared(quad[2]);
-    double distMin = SkTMin(d0, d2);
-    int bestIndex = -1;
-    for (int index = 0; index < roots; ++index) {
-        _Point onQuad;
-        xy_at_t(quad, ts[index], onQuad.x, onQuad.y);
-        double dist = pt.distanceSquared(onQuad);
-        if (distMin > dist) {
-            distMin = dist;
-            bestIndex = index;
-        }
-    }
-    if (bestIndex >= 0) {
-        return ts[bestIndex];
-    }
-    return d0 < d2 ? 0 : 1;
-}
-
-bool point_in_hull(const Quadratic& quad, const _Point& pt) {
-    return pointInTriangle((const Triangle&) quad, pt);
-}
-
-_Point top(const Quadratic& quad, double startT, double endT) {
-    Quadratic sub;
-    sub_divide(quad, startT, endT, sub);
-    _Point topPt = sub[0];
-    if (topPt.y > sub[2].y || (topPt.y == sub[2].y && topPt.x > sub[2].x)) {
-        topPt = sub[2];
-    }
-    if (!between(sub[0].y, sub[1].y, sub[2].y)) {
-        double extremeT;
-        if (findExtrema(sub[0].y, sub[1].y, sub[2].y, &extremeT)) {
-            extremeT = startT + (endT - startT) * extremeT;
-            _Point test;
-            xy_at_t(quad, extremeT, test.x, test.y);
-            if (topPt.y > test.y || (topPt.y == test.y && topPt.x > test.x)) {
-                topPt = test;
-            }
-        }
-    }
-    return topPt;
-}
-
-/*
-Numeric Solutions (5.6) suggests to solve the quadratic by computing
-       Q = -1/2(B + sgn(B)Sqrt(B^2 - 4 A C))
-and using the roots
-      t1 = Q / A
-      t2 = C / Q
-*/
-int add_valid_ts(double s[], int realRoots, double* t) {
-    int foundRoots = 0;
-    for (int index = 0; index < realRoots; ++index) {
-        double tValue = s[index];
-        if (approximately_zero_or_more(tValue) && approximately_one_or_less(tValue)) {
-            if (approximately_less_than_zero(tValue)) {
-                tValue = 0;
-            } else if (approximately_greater_than_one(tValue)) {
-                tValue = 1;
-            }
-            for (int idx2 = 0; idx2 < foundRoots; ++idx2) {
-                if (approximately_equal(t[idx2], tValue)) {
-                    goto nextRoot;
-                }
-            }
-            t[foundRoots++] = tValue;
-        }
-nextRoot:
-        ;
-    }
-    return foundRoots;
-}
-
-// note: caller expects multiple results to be sorted smaller first
-// note: http://en.wikipedia.org/wiki/Loss_of_significance has an interesting
-//  analysis of the quadratic equation, suggesting why the following looks at
-//  the sign of B -- and further suggesting that the greatest loss of precision
-//  is in b squared less two a c
-int quadraticRootsValidT(double A, double B, double C, double t[2]) {
-#if 0
-    B *= 2;
-    double square = B * B - 4 * A * C;
-    if (approximately_negative(square)) {
-        if (!approximately_positive(square)) {
-            return 0;
-        }
-        square = 0;
-    }
-    double squareRt = sqrt(square);
-    double Q = (B + (B < 0 ? -squareRt : squareRt)) / -2;
-    int foundRoots = 0;
-    double ratio = Q / A;
-    if (approximately_zero_or_more(ratio) && approximately_one_or_less(ratio)) {
-        if (approximately_less_than_zero(ratio)) {
-            ratio = 0;
-        } else if (approximately_greater_than_one(ratio)) {
-            ratio = 1;
-        }
-        t[0] = ratio;
-        ++foundRoots;
-    }
-    ratio = C / Q;
-    if (approximately_zero_or_more(ratio) && approximately_one_or_less(ratio)) {
-        if (approximately_less_than_zero(ratio)) {
-            ratio = 0;
-        } else if (approximately_greater_than_one(ratio)) {
-            ratio = 1;
-        }
-        if (foundRoots == 0 || !approximately_negative(ratio - t[0])) {
-            t[foundRoots++] = ratio;
-        } else if (!approximately_negative(t[0] - ratio)) {
-            t[foundRoots++] = t[0];
-            t[0] = ratio;
-        }
-    }
-#else
-    double s[2];
-    int realRoots = quadraticRootsReal(A, B, C, s);
-    int foundRoots = add_valid_ts(s, realRoots, t);
-#endif
-    return foundRoots;
-}
-
-// unlike quadratic roots, this does not discard real roots <= 0 or >= 1
-int quadraticRootsReal(const double A, const double B, const double C, double s[2]) {
-    const double p = B / (2 * A);
-    const double q = C / A;
-    if (approximately_zero(A) && (approximately_zero_inverse(p) || approximately_zero_inverse(q))) {
-        if (approximately_zero(B)) {
-            s[0] = 0;
-            return C == 0;
-        }
-        s[0] = -C / B;
-        return 1;
-    }
-    /* normal form: x^2 + px + q = 0 */
-    const double p2 = p * p;
-#if 0
-    double D = AlmostEqualUlps(p2, q) ? 0 : p2 - q;
-    if (D <= 0) {
-        if (D < 0) {
-            return 0;
-        }
-        s[0] = -p;
-        SkDebugf("[%d] %1.9g\n", 1, s[0]);
-        return 1;
-    }
-    double sqrt_D = sqrt(D);
-    s[0] = sqrt_D - p;
-    s[1] = -sqrt_D - p;
-    SkDebugf("[%d] %1.9g %1.9g\n", 2, s[0], s[1]);
-    return 2;
-#else
-    if (!AlmostEqualUlps(p2, q) && p2 < q) {
-        return 0;
-    }
-    double sqrt_D = 0;
-    if (p2 > q) {
-        sqrt_D = sqrt(p2 - q);
-    }
-    s[0] = sqrt_D - p;
-    s[1] = -sqrt_D - p;
-#if 0
-    if (AlmostEqualUlps(s[0], s[1])) {
-        SkDebugf("[%d] %1.9g\n", 1, s[0]);
-    } else {
-        SkDebugf("[%d] %1.9g %1.9g\n", 2, s[0], s[1]);
-    }
-#endif
-    return 1 + !AlmostEqualUlps(s[0], s[1]);
-#endif
-}
-
-void toCubic(const Quadratic& quad, Cubic& cubic) {
-    cubic[0] = quad[0];
-    cubic[2] = quad[1];
-    cubic[3] = quad[2];
-    cubic[1].x = (cubic[0].x + cubic[2].x * 2) / 3;
-    cubic[1].y = (cubic[0].y + cubic[2].y * 2) / 3;
-    cubic[2].x = (cubic[3].x + cubic[2].x * 2) / 3;
-    cubic[2].y = (cubic[3].y + cubic[2].y * 2) / 3;
-}
-
-static double derivativeAtT(const double* quad, double t) {
-    double a = t - 1;
-    double b = 1 - 2 * t;
-    double c = t;
-    return a * quad[0] + b * quad[2] + c * quad[4];
-}
-
-double dx_at_t(const Quadratic& quad, double t) {
-    return derivativeAtT(&quad[0].x, t);
-}
-
-double dy_at_t(const Quadratic& quad, double t) {
-    return derivativeAtT(&quad[0].y, t);
-}
-
-_Vector dxdy_at_t(const Quadratic& quad, double t) {
-    double a = t - 1;
-    double b = 1 - 2 * t;
-    double c = t;
-    _Vector result = { a * quad[0].x + b * quad[1].x + c * quad[2].x,
-            a * quad[0].y + b * quad[1].y + c * quad[2].y };
-    return result;
-}
-
-void xy_at_t(const Quadratic& quad, double t, double& x, double& y) {
-    double one_t = 1 - t;
-    double a = one_t * one_t;
-    double b = 2 * one_t * t;
-    double c = t * t;
-    if (&x) {
-        x = a * quad[0].x + b * quad[1].x + c * quad[2].x;
-    }
-    if (&y) {
-        y = a * quad[0].y + b * quad[1].y + c * quad[2].y;
-    }
-}
-
-_Point xy_at_t(const Quadratic& quad, double t) {
-    double one_t = 1 - t;
-    double a = one_t * one_t;
-    double b = 2 * one_t * t;
-    double c = t * t;
-    _Point result = { a * quad[0].x + b * quad[1].x + c * quad[2].x,
-            a * quad[0].y + b * quad[1].y + c * quad[2].y };
-    return result;
-}