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+/** @file
+ * @brief Basic intersection routines
+ *//*
+ * Authors:
+ * Nathan Hurst <njh@njhurst.com>
+ * Marco Cecchetti <mrcekets at gmail.com>
+ * Jean-François Barraud <jf.barraud@gmail.com>
+ *
+ * Copyright 2008-2009 Authors
+ *
+ * This library is free software; you can redistribute it and/or
+ * modify it either under the terms of the GNU Lesser General Public
+ * License version 2.1 as published by the Free Software Foundation
+ * (the "LGPL") or, at your option, under the terms of the Mozilla
+ * Public License Version 1.1 (the "MPL"). If you do not alter this
+ * notice, a recipient may use your version of this file under either
+ * the MPL or the LGPL.
+ *
+ * You should have received a copy of the LGPL along with this library
+ * in the file COPYING-LGPL-2.1; if not, write to the Free Software
+ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * You should have received a copy of the MPL along with this library
+ * in the file COPYING-MPL-1.1
+ *
+ * The contents of this file are subject to the Mozilla Public License
+ * Version 1.1 (the "License"); you may not use this file except in
+ * compliance with the License. You may obtain a copy of the License at
+ * http://www.mozilla.org/MPL/
+ *
+ * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
+ * OF ANY KIND, either express or implied. See the LGPL or the MPL for
+ * the specific language governing rights and limitations.
+ *
+ */
+
+#include <2geom/basic-intersection.h>
+#include <2geom/sbasis-to-bezier.h>
+#include <2geom/exception.h>
+
+#ifdef HAVE_GSL
+#include <gsl/gsl_vector.h>
+#include <gsl/gsl_multiroots.h>
+#endif
+
+using std::vector;
+namespace Geom {
+
+//#ifdef USE_RECURSIVE_INTERSECTOR
+
+// void find_intersections(std::vector<std::pair<double, double> > &xs,
+// D2<SBasis> const & A,
+// D2<SBasis> const & B) {
+// vector<Point> BezA, BezB;
+// sbasis_to_bezier(BezA, A);
+// sbasis_to_bezier(BezB, B);
+
+// xs.clear();
+
+// find_intersections_bezier_recursive(xs, BezA, BezB);
+// }
+// void find_intersections(std::vector< std::pair<double, double> > & xs,
+// std::vector<Point> const& A,
+// std::vector<Point> const& B,
+// double precision){
+// find_intersections_bezier_recursive(xs, A, B, precision);
+// }
+
+//#else
+
+namespace detail{ namespace bezier_clipping {
+void portion(std::vector<Point> &B, Interval const &I);
+void derivative(std::vector<Point> &D, std::vector<Point> const &B);
+}; };
+
+void find_intersections(std::vector<std::pair<double, double> > &xs,
+ D2<Bezier> const & A,
+ D2<Bezier> const & B,
+ double precision)
+{
+ find_intersections_bezier_clipping(xs, bezier_points(A), bezier_points(B), precision);
+}
+
+void find_intersections(std::vector<std::pair<double, double> > &xs,
+ D2<SBasis> const & A,
+ D2<SBasis> const & B,
+ double precision)
+{
+ vector<Point> BezA, BezB;
+ sbasis_to_bezier(BezA, A);
+ sbasis_to_bezier(BezB, B);
+
+ find_intersections_bezier_clipping(xs, BezA, BezB, precision);
+}
+
+void find_intersections(std::vector< std::pair<double, double> > & xs,
+ std::vector<Point> const& A,
+ std::vector<Point> const& B,
+ double precision)
+{
+ find_intersections_bezier_clipping(xs, A, B, precision);
+}
+
+//#endif
+
+/*
+ * split the curve at the midpoint, returning an array with the two parts
+ * Temporary storage is minimized by using part of the storage for the result
+ * to hold an intermediate value until it is no longer needed.
+ */
+// TODO replace with Bezier method
+void split(vector<Point> const &p, double t,
+ vector<Point> &left, vector<Point> &right) {
+ const unsigned sz = p.size();
+ //Geom::Point Vtemp[sz][sz];
+ vector<vector<Point> > Vtemp(sz);
+ for ( size_t i = 0; i < sz; ++i )
+ Vtemp[i].reserve(sz);
+
+ /* Copy control points */
+ std::copy(p.begin(), p.end(), Vtemp[0].begin());
+
+ /* Triangle computation */
+ for (unsigned i = 1; i < sz; i++) {
+ for (unsigned j = 0; j < sz - i; j++) {
+ Vtemp[i][j] = lerp(t, Vtemp[i-1][j], Vtemp[i-1][j+1]);
+ }
+ }
+
+ left.resize(sz);
+ right.resize(sz);
+ for (unsigned j = 0; j < sz; j++)
+ left[j] = Vtemp[j][0];
+ for (unsigned j = 0; j < sz; j++)
+ right[j] = Vtemp[sz-1-j][j];
+}
+
+
+
+void find_self_intersections(std::vector<std::pair<double, double> > &xs,
+ D2<Bezier> const &A,
+ double precision)
+{
+ std::vector<double> dr = derivative(A[X]).roots();
+ {
+ std::vector<double> dyr = derivative(A[Y]).roots();
+ dr.insert(dr.begin(), dyr.begin(), dyr.end());
+ }
+ dr.push_back(0);
+ dr.push_back(1);
+ // We want to be sure that we have no empty segments
+ std::sort(dr.begin(), dr.end());
+ std::vector<double>::iterator new_end = std::unique(dr.begin(), dr.end());
+ dr.resize( new_end - dr.begin() );
+
+ std::vector< D2<Bezier> > pieces;
+ for (unsigned i = 0; i < dr.size() - 1; ++i) {
+ pieces.push_back(portion(A, dr[i], dr[i+1]));
+ }
+ /*{
+ vector<Point> l, r, in = A;
+ for(unsigned i = 0; i < dr.size()-1; i++) {
+ split(in, (dr[i+1]-dr[i]) / (1 - dr[i]), l, r);
+ pieces.push_back(l);
+ in = r;
+ }
+ }*/
+
+ for(unsigned i = 0; i < dr.size()-1; i++) {
+ for(unsigned j = i+1; j < dr.size()-1; j++) {
+ std::vector<std::pair<double, double> > section;
+
+ find_intersections(section, pieces[i], pieces[j], precision);
+ for(auto & k : section) {
+ double l = k.first;
+ double r = k.second;
+// XXX: This condition will prune out false positives, but it might create some false negatives. Todo: Confirm it is correct.
+ if(j == i+1)
+ //if((l == 1) && (r == 0))
+ if( ( l > precision ) && (r < precision) )//FIXME: what precision should be used here???
+ continue;
+ xs.emplace_back((1-l)*dr[i] + l*dr[i+1],
+ (1-r)*dr[j] + r*dr[j+1]);
+ }
+ }
+ }
+
+ // Because i is in order, xs should be roughly already in order?
+ //sort(xs.begin(), xs.end());
+ //unique(xs.begin(), xs.end());
+}
+
+void find_self_intersections(std::vector<std::pair<double, double> > &xs,
+ D2<SBasis> const &A,
+ double precision)
+{
+ D2<Bezier> in;
+ sbasis_to_bezier(in, A);
+ find_self_intersections(xs, in, precision);
+}
+
+
+void subdivide(D2<Bezier> const &a,
+ D2<Bezier> const &b,
+ std::vector< std::pair<double, double> > const &xs,
+ std::vector< D2<Bezier> > &av,
+ std::vector< D2<Bezier> > &bv)
+{
+ if (xs.empty()) {
+ av.push_back(a);
+ bv.push_back(b);
+ return;
+ }
+
+ std::pair<double, double> prev = std::make_pair(0., 0.);
+ for (const auto & x : xs) {
+ av.push_back(portion(a, prev.first, x.first));
+ bv.push_back(portion(b, prev.second, x.second));
+ av.back()[X].at0() = bv.back()[X].at0() = lerp(0.5, av.back()[X].at0(), bv.back()[X].at0());
+ av.back()[X].at1() = bv.back()[X].at1() = lerp(0.5, av.back()[X].at1(), bv.back()[X].at1());
+ av.back()[Y].at0() = bv.back()[Y].at0() = lerp(0.5, av.back()[Y].at0(), bv.back()[Y].at0());
+ av.back()[Y].at1() = bv.back()[Y].at1() = lerp(0.5, av.back()[Y].at1(), bv.back()[Y].at1());
+ prev = x;
+ }
+ av.push_back(portion(a, prev.first, 1));
+ bv.push_back(portion(b, prev.second, 1));
+ av.back()[X].at0() = bv.back()[X].at0() = lerp(0.5, av.back()[X].at0(), bv.back()[X].at0());
+ av.back()[X].at1() = bv.back()[X].at1() = lerp(0.5, av.back()[X].at1(), bv.back()[X].at1());
+ av.back()[Y].at0() = bv.back()[Y].at0() = lerp(0.5, av.back()[Y].at0(), bv.back()[Y].at0());
+ av.back()[Y].at1() = bv.back()[Y].at1() = lerp(0.5, av.back()[Y].at1(), bv.back()[Y].at1());
+}
+
+#ifdef HAVE_GSL
+#include <gsl/gsl_multiroots.h>
+
+struct rparams
+{
+ D2<SBasis> const &A;
+ D2<SBasis> const &B;
+};
+
+static int
+intersect_polish_f (const gsl_vector * x, void *params,
+ gsl_vector * f)
+{
+ const double x0 = gsl_vector_get (x, 0);
+ const double x1 = gsl_vector_get (x, 1);
+
+ Geom::Point dx = ((struct rparams *) params)->A(x0) -
+ ((struct rparams *) params)->B(x1);
+
+ gsl_vector_set (f, 0, dx[0]);
+ gsl_vector_set (f, 1, dx[1]);
+
+ return GSL_SUCCESS;
+}
+#endif
+
+union dbl_64{
+ long long i64;
+ double d64;
+};
+
+static double EpsilonBy(double value, int eps)
+{
+ dbl_64 s;
+ s.d64 = value;
+ s.i64 += eps;
+ return s.d64;
+}
+
+
+static void intersect_polish_root (D2<SBasis> const &A, double &s,
+ D2<SBasis> const &B, double &t) {
+#ifdef HAVE_GSL
+ const gsl_multiroot_fsolver_type *T;
+ gsl_multiroot_fsolver *sol;
+
+ int status;
+ size_t iter = 0;
+#endif
+ std::vector<Point> as, bs;
+ as = A.valueAndDerivatives(s, 2);
+ bs = B.valueAndDerivatives(t, 2);
+ Point F = as[0] - bs[0];
+ double best = dot(F, F);
+
+ for(int i = 0; i < 4; i++) {
+
+ /**
+ we want to solve
+ J*(x1 - x0) = f(x0)
+
+ |dA(s)[0] -dB(t)[0]| (X1 - X0) = A(s) - B(t)
+ |dA(s)[1] -dB(t)[1]|
+ **/
+
+ // We're using the standard transformation matricies, which is numerically rather poor. Much better to solve the equation using elimination.
+
+ Affine jack(as[1][0], as[1][1],
+ -bs[1][0], -bs[1][1],
+ 0, 0);
+ Point soln = (F)*jack.inverse();
+ double ns = s - soln[0];
+ double nt = t - soln[1];
+
+ as = A.valueAndDerivatives(ns, 2);
+ bs = B.valueAndDerivatives(nt, 2);
+ F = as[0] - bs[0];
+ double trial = dot(F, F);
+ if (trial > best*0.1) {// we have standards, you know
+ // At this point we could do a line search
+ break;
+ }
+ best = trial;
+ s = ns;
+ t = nt;
+ }
+
+#ifdef HAVE_GSL
+ const size_t n = 2;
+ struct rparams p = {A, B};
+ gsl_multiroot_function f = {&intersect_polish_f, n, &p};
+
+ double x_init[2] = {s, t};
+ gsl_vector *x = gsl_vector_alloc (n);
+
+ gsl_vector_set (x, 0, x_init[0]);
+ gsl_vector_set (x, 1, x_init[1]);
+
+ T = gsl_multiroot_fsolver_hybrids;
+ sol = gsl_multiroot_fsolver_alloc (T, 2);
+ gsl_multiroot_fsolver_set (sol, &f, x);
+
+ do
+ {
+ iter++;
+ status = gsl_multiroot_fsolver_iterate (sol);
+
+ if (status) /* check if solver is stuck */
+ break;
+
+ status =
+ gsl_multiroot_test_residual (sol->f, 1e-12);
+ }
+ while (status == GSL_CONTINUE && iter < 1000);
+
+ s = gsl_vector_get (sol->x, 0);
+ t = gsl_vector_get (sol->x, 1);
+
+ gsl_multiroot_fsolver_free (sol);
+ gsl_vector_free (x);
+#endif
+
+ {
+ // This code does a neighbourhood search for minor improvements.
+ double best_v = L1(A(s) - B(t));
+ //std::cout << "------\n" << best_v << std::endl;
+ Point best(s,t);
+ while (true) {
+ Point trial = best;
+ double trial_v = best_v;
+ for(int nsi = -1; nsi < 2; nsi++) {
+ for(int nti = -1; nti < 2; nti++) {
+ Point n(EpsilonBy(best[0], nsi),
+ EpsilonBy(best[1], nti));
+ double c = L1(A(n[0]) - B(n[1]));
+ //std::cout << c << "; ";
+ if (c < trial_v) {
+ trial = n;
+ trial_v = c;
+ }
+ }
+ }
+ if(trial == best) {
+ //std::cout << "\n" << s << " -> " << s - best[0] << std::endl;
+ //std::cout << t << " -> " << t - best[1] << std::endl;
+ //std::cout << best_v << std::endl;
+ s = best[0];
+ t = best[1];
+ return;
+ } else {
+ best = trial;
+ best_v = trial_v;
+ }
+ }
+ }
+}
+
+
+void polish_intersections(std::vector<std::pair<double, double> > &xs,
+ D2<SBasis> const &A, D2<SBasis> const &B)
+{
+ for(auto & x : xs)
+ intersect_polish_root(A, x.first,
+ B, x.second);
+}
+
+/**
+ * Compute the Hausdorf distance from A to B only.
+ */
+double hausdorfl(D2<SBasis>& A, D2<SBasis> const& B,
+ double m_precision,
+ double *a_t, double* b_t) {
+ std::vector< std::pair<double, double> > xs;
+ std::vector<Point> Az, Bz;
+ sbasis_to_bezier (Az, A);
+ sbasis_to_bezier (Bz, B);
+ find_collinear_normal(xs, Az, Bz, m_precision);
+ double h_dist = 0, h_a_t = 0, h_b_t = 0;
+ double dist = 0;
+ Point Ax = A.at0();
+ double t = Geom::nearest_time(Ax, B);
+ dist = Geom::distance(Ax, B(t));
+ if (dist > h_dist) {
+ h_a_t = 0;
+ h_b_t = t;
+ h_dist = dist;
+ }
+ Ax = A.at1();
+ t = Geom::nearest_time(Ax, B);
+ dist = Geom::distance(Ax, B(t));
+ if (dist > h_dist) {
+ h_a_t = 1;
+ h_b_t = t;
+ h_dist = dist;
+ }
+ for (auto & x : xs)
+ {
+ Point At = A(x.first);
+ Point Bu = B(x.second);
+ double distAtBu = Geom::distance(At, Bu);
+ t = Geom::nearest_time(At, B);
+ dist = Geom::distance(At, B(t));
+ //FIXME: we might miss it due to floating point precision...
+ if (dist >= distAtBu-.1 && distAtBu > h_dist) {
+ h_a_t = x.first;
+ h_b_t = x.second;
+ h_dist = distAtBu;
+ }
+
+ }
+ if(a_t) *a_t = h_a_t;
+ if(b_t) *b_t = h_b_t;
+
+ return h_dist;
+}
+
+/**
+ * Compute the symmetric Hausdorf distance.
+ */
+double hausdorf(D2<SBasis>& A, D2<SBasis> const& B,
+ double m_precision,
+ double *a_t, double* b_t) {
+ double h_dist = hausdorfl(A, B, m_precision, a_t, b_t);
+
+ double dist = 0;
+ Point Bx = B.at0();
+ double t = Geom::nearest_time(Bx, A);
+ dist = Geom::distance(Bx, A(t));
+ if (dist > h_dist) {
+ if(a_t) *a_t = t;
+ if(b_t) *b_t = 0;
+ h_dist = dist;
+ }
+ Bx = B.at1();
+ t = Geom::nearest_time(Bx, A);
+ dist = Geom::distance(Bx, A(t));
+ if (dist > h_dist) {
+ if(a_t) *a_t = t;
+ if(b_t) *b_t = 1;
+ h_dist = dist;
+ }
+
+ return h_dist;
+}
+
+bool non_collinear_segments_intersect(const Point &A, const Point &B, const Point &C, const Point &D)
+{
+ return cross(D - C, A - C) * cross(D - C, B - C) < 0 && //
+ cross(B - A, C - A) * cross(B - A, D - A) < 0;
+}
+};
+
+/*
+ Local Variables:
+ mode:c++
+ c-file-style:"stroustrup"
+ c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
+ indent-tabs-mode:nil
+ fill-column:99
+ End:
+*/
+// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 :
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