remove prototype pathops code

experimental/Intersection/QuadraticParameterization.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 "CurveIntersection.h"
-#include "QuadraticParameterization.h"
-#include "QuadraticUtilities.h"
-
-/* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1
- *
- * This paper proves that Syvester's method can compute the implicit form of
- * the quadratic from the parameterized form.
- *
- * Given x = a*t*t + b*t + c  (the parameterized form)
- *       y = d*t*t + e*t + f
- *
- * we want to find an equation of the implicit form:
- *
- * A*x*x + B*x*y + C*y*y + D*x + E*y + F = 0
- *
- * The implicit form can be expressed as a 4x4 determinant, as shown.
- *
- * The resultant obtained by Syvester's method is
- *
- * |   a   b   (c - x)     0     |
- * |   0   a      b     (c - x)  |
- * |   d   e   (f - y)     0     |
- * |   0   d      e     (f - y)  |
- *
- * which expands to
- *
- * d*d*x*x + -2*a*d*x*y + a*a*y*y
- *         + (-2*c*d*d + b*e*d - a*e*e + 2*a*f*d)*x
- *         + (-2*f*a*a + e*b*a - d*b*b + 2*d*c*a)*y
- *         +
- * |   a   b   c   0   |
- * |   0   a   b   c   | == 0.
- * |   d   e   f   0   |
- * |   0   d   e   f   |
- *
- * Expanding the constant determinant results in
- *
- *   | a b c |     | b c 0 |
- * a*| e f 0 | + d*| a b c | ==
- *   | d e f |     | d e f |
- *
- * a*(a*f*f + c*e*e - c*f*d - b*e*f) + d*(b*b*f + c*c*d - c*a*f - c*e*b)
- *
- */
-
-
-static bool straight_forward = true;
-
-QuadImplicitForm::QuadImplicitForm(const Quadratic& q) {
-    double a, b, c;
-    set_abc(&q[0].x, a, b, c);
-    double d, e, f;
-    set_abc(&q[0].y, d, e, f);
-    // compute the implicit coefficients
-    if (straight_forward) { // 42 muls, 13 adds
-        p[xx_coeff] = d * d;
-        p[xy_coeff] = -2 * a * d;
-        p[yy_coeff] = a * a;
-        p[x_coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d;
-        p[y_coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a;
-        p[c_coeff] = a*(a*f*f + c*e*e - c*f*d - b*e*f)
-                   + d*(b*b*f + c*c*d - c*a*f - c*e*b);
-    } else { // 26 muls, 11 adds
-        double aa = a * a;
-        double ad = a * d;
-        double dd = d * d;
-        p[xx_coeff] = dd;
-        p[xy_coeff] = -2 * ad;
-        p[yy_coeff] = aa;
-        double be = b * e;
-        double bde = be * d;
-        double cdd = c * dd;
-        double ee = e * e;
-        p[x_coeff] =  -2*cdd + bde - a*ee + 2*ad*f;
-        double aaf = aa * f;
-        double abe = a * be;
-        double ac = a * c;
-        double bb_2ac = b*b - 2*ac;
-        p[y_coeff] = -2*aaf + abe - d*bb_2ac;
-        p[c_coeff] = aaf*f + ac*ee + d*f*bb_2ac - abe*f + c*cdd - c*bde;
-    }
-}
-
- /* Given a pair of quadratics, determine their parametric coefficients.
-  * If the scaled coefficients are nearly equal, then the part of the quadratics
-  * may be coincident.
-  * FIXME: optimization -- since comparison short-circuits on no match,
-  * lazily compute the coefficients, comparing the easiest to compute first.
-  * xx and yy first; then xy; and so on.
-  */
-bool QuadImplicitForm::implicit_match(const QuadImplicitForm& p2) const {
-    int first = 0;
-    for (int index = 0; index < coeff_count; ++index) {
-        if (approximately_zero(p[index]) && approximately_zero(p2.p[index])) {
-            first += first == index;
-            continue;
-        }
-        if (first == index) {
-            continue;
-        }
-        if (!AlmostEqualUlps(p[index] * p2.p[first], p[first] * p2.p[index])) {
-            return false;
-        }
-    }
-    return true;
-}
-
-bool implicit_matches(const Quadratic& quad1, const Quadratic& quad2) {
-    QuadImplicitForm i1(quad1);  // a'xx , b'xy , c'yy , d'x , e'y , f
-    QuadImplicitForm i2(quad2);
-    return i1.implicit_match(i2);
-}
-
-static double tangent(const double* quadratic, double t) {
-    double a, b, c;
-    set_abc(quadratic, a, b, c);
-    return 2 * a * t + b;
-}
-
-void tangent(const Quadratic& quadratic, double t, _Point& result) {
-    result.x = tangent(&quadratic[0].x, t);
-    result.y = tangent(&quadratic[0].y, t);
-}
-
-
-
-// unit test to return and validate parametric coefficients
-#include "QuadraticParameterization_TestUtility.cpp"