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-rw-r--r--src/2geom/bezier-clipping.cpp2584
1 files changed, 1292 insertions, 1292 deletions
diff --git a/src/2geom/bezier-clipping.cpp b/src/2geom/bezier-clipping.cpp
index 96a06376c..c8d99c430 100644
--- a/src/2geom/bezier-clipping.cpp
+++ b/src/2geom/bezier-clipping.cpp
@@ -1,1292 +1,1292 @@
-/*
- * Implement the Bezier clipping algorithm for finding
- * Bezier curve intersection points and collinear normals
- *
- * Authors:
- * Marco Cecchetti <mrcekets at gmail.com>
- *
- * Copyright 2008 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/choose.h>
-#include <2geom/point.h>
-#include <2geom/interval.h>
-#include <2geom/bezier.h>
-//#include <2geom/convex-cover.h>
-#include <2geom/numeric/matrix.h>
-
-#include <cassert>
-#include <vector>
-#include <algorithm>
-#include <utility>
-//#include <iomanip>
-
-
-
-
-#define VERBOSE 0
-#define CHECK 0
-
-
-namespace Geom {
-
-namespace detail { namespace bezier_clipping {
-
-////////////////////////////////////////////////////////////////////////////////
-// for debugging
-//
-
-inline
-void print(std::vector<Point> const& cp, const char* msg = "")
-{
- std::cerr << msg << std::endl;
- for (size_t i = 0; i < cp.size(); ++i)
- std::cerr << i << " : " << cp[i] << std::endl;
-}
-
-template< class charT >
-inline
-std::basic_ostream<charT> &
-operator<< (std::basic_ostream<charT> & os, const Interval & I)
-{
- os << "[" << I.min() << ", " << I.max() << "]";
- return os;
-}
-
-inline
-double angle (std::vector<Point> const& A)
-{
- size_t n = A.size() -1;
- double a = std::atan2(A[n][Y] - A[0][Y], A[n][X] - A[0][X]);
- return (180 * a / M_PI);
-}
-
-inline
-size_t get_precision(Interval const& I)
-{
- double d = I.extent();
- double e = 0.1, p = 10;
- int n = 0;
- while (n < 16 && d < e)
- {
- p *= 10;
- e = 1/p;
- ++n;
- }
- return n;
-}
-
-inline
-void range_assertion(int k, int m, int n, const char* msg)
-{
- if ( k < m || k > n)
- {
- std::cerr << "range assertion failed: \n"
- << msg << std::endl
- << "value: " << k
- << " range: " << m << ", " << n << std::endl;
- assert (k >= m && k <= n);
- }
-}
-
-
-////////////////////////////////////////////////////////////////////////////////
-// convex hull
-
-/*
- * return true in case the oriented polyline p0, p1, p2 is a right turn
- */
-inline
-bool is_a_right_turn (Point const& p0, Point const& p1, Point const& p2)
-{
- if (p1 == p2) return false;
- Point q1 = p1 - p0;
- Point q2 = p2 - p0;
- if (q1 == -q2) return false;
- return (cross (q1, q2) < 0);
-}
-
-/*
- * return true if p < q wrt the lexicographyc order induced by the coordinates
- */
-struct lex_less
-{
- bool operator() (Point const& p, Point const& q)
- {
- return ((p[X] < q[X]) || (p[X] == q[X] && p[Y] < q[Y]));
- }
-};
-
-/*
- * return true if p > q wrt the lexicographyc order induced by the coordinates
- */
-struct lex_greater
-{
- bool operator() (Point const& p, Point const& q)
- {
- return ((p[X] > q[X]) || (p[X] == q[X] && p[Y] > q[Y]));
- }
-};
-
-/*
- * Compute the convex hull of a set of points.
- * The implementation is based on the Andrew's scan algorithm
- * note: in the Bezier clipping for collinear normals it seems
- * to be more stable wrt the Graham's scan algorithm and in general
- * a bit quikier
- */
-void convex_hull (std::vector<Point> & P)
-{
- size_t n = P.size();
- if (n < 2) return;
- std::sort(P.begin(), P.end(), lex_less());
- if (n < 4) return;
- // upper hull
- size_t u = 2;
- for (size_t i = 2; i < n; ++i)
- {
- while (u > 1 && !is_a_right_turn(P[u-2], P[u-1], P[i]))
- {
- --u;
- }
- std::swap(P[u], P[i]);
- ++u;
- }
- std::sort(P.begin() + u, P.end(), lex_greater());
- std::rotate(P.begin(), P.begin() + 1, P.end());
- // lower hull
- size_t l = u;
- size_t k = u - 1;
- for (size_t i = l; i < n; ++i)
- {
- while (l > k && !is_a_right_turn(P[l-2], P[l-1], P[i]))
- {
- --l;
- }
- std::swap(P[l], P[i]);
- ++l;
- }
- P.resize(l);
-}
-
-
-////////////////////////////////////////////////////////////////////////////////
-// numerical routines
-
-/*
- * Compute the binomial coefficient (n, k)
- */
-inline
-double binomial(unsigned int n, unsigned int k)
-{
- return choose<double>(n, k);
-}
-
-/*
- * Compute the determinant of the 2x2 matrix with column the point P1, P2
- */
-inline
-double det(Point const& P1, Point const& P2)
-{
- return P1[X]*P2[Y] - P1[Y]*P2[X];
-}
-
-/*
- * Solve the linear system [P1,P2] * P = Q
- * in case there isn't exactly one solution the routine returns false
- */
-inline
-bool solve(Point & P, Point const& P1, Point const& P2, Point const& Q)
-{
- double d = det(P1, P2);
- if (d == 0) return false;
- d = 1 / d;
- P[X] = det(Q, P2) * d;
- P[Y] = det(P1, Q) * d;
- return true;
-}
-
-////////////////////////////////////////////////////////////////////////////////
-// interval routines
-
-/*
- * Map the sub-interval I in [0,1] into the interval J and assign it to J
- */
-inline
-void map_to(Interval & J, Interval const& I)
-{
- double length = J.extent();
- J[1] = I.max() * length + J[0];
- J[0] = I.min() * length + J[0];
-}
-
-/*
- * The interval [1,0] is used to represent the empty interval, this routine
- * is just an helper function for creating such an interval
- */
-inline
-Interval make_empty_interval()
-{
- Interval I(0);
- I[0] = 1;
- return I;
-}
-
-
-////////////////////////////////////////////////////////////////////////////////
-// bezier curve routines
-
-/*
- * Return true if all the Bezier curve control points are near,
- * false otherwise
- */
-inline
-bool is_constant(std::vector<Point> const& A, double precision = EPSILON)
-{
- for (unsigned int i = 1; i < A.size(); ++i)
- {
- if(!are_near(A[i], A[0], precision))
- return false;
- }
- return true;
-}
-
-/*
- * Compute the hodograph of the bezier curve B and return it in D
- */
-inline
-void derivative(std::vector<Point> & D, std::vector<Point> const& B)
-{
- D.clear();
- size_t sz = B.size();
- if (sz == 0) return;
- if (sz == 1)
- {
- D.resize(1, Point(0,0));
- return;
- }
- size_t n = sz-1;
- D.reserve(n);
- for (size_t i = 0; i < n; ++i)
- {
- D.push_back(n*(B[i+1] - B[i]));
- }
-}
-
-/*
- * Compute the hodograph of the Bezier curve B rotated of 90 degree
- * and return it in D; we have N(t) orthogonal to B(t) for any t
- */
-inline
-void normal(std::vector<Point> & N, std::vector<Point> const& B)
-{
- derivative(N,B);
- for (size_t i = 0; i < N.size(); ++i)
- {
- N[i] = rot90(N[i]);
- }
-}
-
-/*
- * Compute the portion of the Bezier curve "B" wrt the interval [0,t]
- */
-inline
-void left_portion(Coord t, std::vector<Point> & B)
-{
- size_t n = B.size();
- for (size_t i = 1; i < n; ++i)
- {
- for (size_t j = n-1; j > i-1 ; --j)
- {
- B[j] = lerp(t, B[j-1], B[j]);
- }
- }
-}
-
-/*
- * Compute the portion of the Bezier curve "B" wrt the interval [t,1]
- */
-inline
-void right_portion(Coord t, std::vector<Point> & B)
-{
- size_t n = B.size();
- for (size_t i = 1; i < n; ++i)
- {
- for (size_t j = 0; j < n-i; ++j)
- {
- B[j] = lerp(t, B[j], B[j+1]);
- }
- }
-}
-
-/*
- * Compute the portion of the Bezier curve "B" wrt the interval "I"
- */
-inline
-void portion (std::vector<Point> & B , Interval const& I)
-{
- if (I.min() == 0)
- {
- if (I.max() == 1) return;
- left_portion(I.max(), B);
- return;
- }
- right_portion(I.min(), B);
- if (I.max() == 1) return;
- double t = I.extent() / (1 - I.min());
- left_portion(t, B);
-}
-
-
-////////////////////////////////////////////////////////////////////////////////
-// tags
-
-struct intersection_point_tag;
-struct collinear_normal_tag;
-template <typename Tag>
-void clip(Interval & dom,
- std::vector<Point> const& A,
- std::vector<Point> const& B);
-template <typename Tag>
-void iterate(std::vector<Interval>& domsA,
- std::vector<Interval>& domsB,
- std::vector<Point> const& A,
- std::vector<Point> const& B,
- Interval const& domA,
- Interval const& domB,
- double precision );
-
-
-////////////////////////////////////////////////////////////////////////////////
-// intersection
-
-/*
- * Make up an orientation line using the control points c[i] and c[j]
- * the line is returned in the output parameter "l" in the form of a 3 element
- * vector : l[0] * x + l[1] * y + l[2] == 0; the line is normalized.
- */
-inline
-void orientation_line (std::vector<double> & l,
- std::vector<Point> const& c,
- size_t i, size_t j)
-{
- l[0] = c[j][Y] - c[i][Y];
- l[1] = c[i][X] - c[j][X];
- l[2] = cross(c[i], c[j]);
- double length = std::sqrt(l[0] * l[0] + l[1] * l[1]);
- assert (length != 0);
- l[0] /= length;
- l[1] /= length;
- l[2] /= length;
-}
-
-/*
- * Pick up an orientation line for the Bezier curve "c" and return it in
- * the output parameter "l"
- */
-inline
-void pick_orientation_line (std::vector<double> & l,
- std::vector<Point> const& c)
-{
- size_t i = c.size();
- while (--i > 0 && are_near(c[0], c[i]))
- {}
- if (i == 0)
- {
- // this should never happen because when a new curve portion is created
- // we check that it is not constant;
- // however this requires that the precision used in the is_constant
- // routine has to be the same used here in the are_near test
- assert(i != 0);
- }
- orientation_line(l, c, 0, i);
- //std::cerr << "i = " << i << std::endl;
-}
-
-/*
- * Make up an orientation line for constant bezier curve;
- * the orientation line is made up orthogonal to the other curve base line;
- * the line is returned in the output parameter "l" in the form of a 3 element
- * vector : l[0] * x + l[1] * y + l[2] == 0; the line is normalized.
- */
-inline
-void orthogonal_orientation_line (std::vector<double> & l,
- std::vector<Point> const& c,
- Point const& p)
-{
- if (is_constant(c))
- {
- // this should never happen
- assert(!is_constant(c));
- }
- std::vector<Point> ol(2);
- ol[0] = p;
- ol[1] = (c.back() - c.front()).cw() + p;
- orientation_line(l, ol, 0, 1);
-}
-
-/*
- * Compute the signed distance of the point "P" from the normalized line l
- */
-inline
-double distance (Point const& P, std::vector<double> const& l)
-{
- return l[X] * P[X] + l[Y] * P[Y] + l[2];
-}
-
-/*
- * Compute the min and max distance of the control points of the Bezier
- * curve "c" from the normalized orientation line "l".
- * This bounds are returned through the output Interval parameter"bound".
- */
-inline
-void fat_line_bounds (Interval& bound,
- std::vector<Point> const& c,
- std::vector<double> const& l)
-{
- bound[0] = 0;
- bound[1] = 0;
- double d;
- for (size_t i = 0; i < c.size(); ++i)
- {
- d = distance(c[i], l);
- if (bound[0] > d) bound[0] = d;
- if (bound[1] < d) bound[1] = d;
- }
-}
-
-/*
- * return the x component of the intersection point between the line
- * passing through points p1, p2 and the line Y = "y"
- */
-inline
-double intersect (Point const& p1, Point const& p2, double y)
-{
- // we are sure that p2[Y] != p1[Y] because this routine is called
- // only when the lower or the upper bound is crossed
- double dy = (p2[Y] - p1[Y]);
- double s = (y - p1[Y]) / dy;
- return (p2[X]-p1[X])*s + p1[X];
-}
-
-/*
- * Clip the Bezier curve "B" wrt the fat line defined by the orientation
- * line "l" and the interval range "bound", the new parameter interval for
- * the clipped curve is returned through the output parameter "dom"
- */
-void clip_interval (Interval& dom,
- std::vector<Point> const& B,
- std::vector<double> const& l,
- Interval const& bound)
-{
- double n = B.size() - 1; // number of sub-intervals
- std::vector<Point> D; // distance curve control points
- D.reserve (B.size());
- double d;
- for (size_t i = 0; i < B.size(); ++i)
- {
- d = distance (B[i], l);
- D.push_back (Point(i/n, d));
- }
- //print(D);
-
- convex_hull(D);
- std::vector<Point> & p = D;
- //print(p);
-
- bool plower, phigher;
- bool clower, chigher;
- double t, tmin = 1, tmax = 0;
-// std::cerr << "bound : " << bound << std::endl;
-
- plower = (p[0][Y] < bound.min());
- phigher = (p[0][Y] > bound.max());
- if (!(plower || phigher)) // inside the fat line
- {
- if (tmin > p[0][X]) tmin = p[0][X];
- if (tmax < p[0][X]) tmax = p[0][X];
-// std::cerr << "0 : inside " << p[0]
-// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
- }
-
- for (size_t i = 1; i < p.size(); ++i)
- {
- clower = (p[i][Y] < bound.min());
- chigher = (p[i][Y] > bound.max());
- if (!(clower || chigher)) // inside the fat line
- {
- if (tmin > p[i][X]) tmin = p[i][X];
- if (tmax < p[i][X]) tmax = p[i][X];
-// std::cerr << i << " : inside " << p[i]
-// << " : tmin = " << tmin << ", tmax = " << tmax
-// << std::endl;
- }
- if (clower != plower) // cross the lower bound
- {
- t = intersect(p[i-1], p[i], bound.min());
- if (tmin > t) tmin = t;
- if (tmax < t) tmax = t;
- plower = clower;
-// std::cerr << i << " : lower " << p[i]
-// << " : tmin = " << tmin << ", tmax = " << tmax
-// << std::endl;
- }
- if (chigher != phigher) // cross the upper bound
- {
- t = intersect(p[i-1], p[i], bound.max());
- if (tmin > t) tmin = t;
- if (tmax < t) tmax = t;
- phigher = chigher;
-// std::cerr << i << " : higher " << p[i]
-// << " : tmin = " << tmin << ", tmax = " << tmax
-// << std::endl;
- }
- }
-
- // we have to test the closing segment for intersection
- size_t last = p.size() - 1;
- clower = (p[0][Y] < bound.min());
- chigher = (p[0][Y] > bound.max());
- if (clower != plower) // cross the lower bound
- {
- t = intersect(p[last], p[0], bound.min());
- if (tmin > t) tmin = t;
- if (tmax < t) tmax = t;
-// std::cerr << "0 : lower " << p[0]
-// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
- }
- if (chigher != phigher) // cross the upper bound
- {
- t = intersect(p[last], p[0], bound.max());
- if (tmin > t) tmin = t;
- if (tmax < t) tmax = t;
-// std::cerr << "0 : higher " << p[0]
-// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
- }
-
- dom[0] = tmin;
- dom[1] = tmax;
-}
-
-/*
- * Clip the Bezier curve "B" wrt the Bezier curve "A" for individuating
- * intersection points the new parameter interval for the clipped curve
- * is returned through the output parameter "dom"
- */
-template <>
-inline
-void clip<intersection_point_tag> (Interval & dom,
- std::vector<Point> const& A,
- std::vector<Point> const& B)
-{
- std::vector<double> bl(3);
- Interval bound;
- if (is_constant(A))
- {
- Point M = middle_point(A.front(), A.back());
- orthogonal_orientation_line(bl, B, M);
- }
- else
- {
- pick_orientation_line(bl, A);
- }
- fat_line_bounds(bound, A, bl);
- clip_interval(dom, B, bl, bound);
-}
-
-
-///////////////////////////////////////////////////////////////////////////////
-// collinear normal
-
-/*
- * Compute a closed focus for the Bezier curve B and return it in F
- * A focus is any curve through which all lines perpendicular to B(t) pass.
- */
-inline
-void make_focus (std::vector<Point> & F, std::vector<Point> const& B)
-{
- assert (B.size() > 2);
- size_t n = B.size() - 1;
- normal(F, B);
- Point c(1, 1);
-#if VERBOSE
- if (!solve(c, F[0], -F[n-1], B[n]-B[0]))
- {
- std::cerr << "make_focus: unable to make up a closed focus" << std::endl;
- }
-#else
- solve(c, F[0], -F[n-1], B[n]-B[0]);
-#endif
-// std::cerr << "c = " << c << std::endl;
-
-
- // B(t) + c(t) * N(t)
- double n_inv = 1 / (double)(n);
- Point c0ni;
- F.push_back(c[1] * F[n-1]);
- F[n] += B[n];
- for (size_t i = n-1; i > 0; --i)
- {
- F[i] *= -c[0];
- c0ni = F[i];
- F[i] += (c[1] * F[i-1]);
- F[i] *= (i * n_inv);
- F[i] -= c0ni;
- F[i] += B[i];
- }
- F[0] *= c[0];
- F[0] += B[0];
-}
-
-/*
- * Compute the projection on the plane (t, d) of the control points
- * (t, u, D(t,u)) where D(t,u) = <(B(t) - F(u)), B'(t)> with 0 <= t, u <= 1
- * B is a Bezier curve and F is a focus of another Bezier curve.
- * See Sederberg, Nishita, 1990 - Curve intersection using Bezier clipping.
- */
-void distance_control_points (std::vector<Point> & D,
- std::vector<Point> const& B,
- std::vector<Point> const& F)
-{
- assert (B.size() > 1);
- assert (F.size() > 0);
- const size_t n = B.size() - 1;
- const size_t m = F.size() - 1;
- const size_t r = 2 * n - 1;
- const double r_inv = 1 / (double)(r);
- D.clear();
- D.reserve (B.size() * F.size());
-
- std::vector<Point> dB;
- dB.reserve(n);
- for (size_t k = 0; k < n; ++k)
- {
- dB.push_back (B[k+1] - B[k]);
- }
- NL::Matrix dBB(n,B.size());
- for (size_t i = 0; i < n; ++i)
- for (size_t j = 0; j < B.size(); ++j)
- dBB(i,j) = dot (dB[i], B[j]);
- NL::Matrix dBF(n, F.size());
- for (size_t i = 0; i < n; ++i)
- for (size_t j = 0; j < F.size(); ++j)
- dBF(i,j) = dot (dB[i], F[j]);
-
- size_t k0, kn, l;
- double bc, bri;
- Point dij;
- std::vector<double> d(F.size());
- for (size_t i = 0; i <= r; ++i)
- {
- for (size_t j = 0; j <= m; ++j)
- {
- d[j] = 0;
- }
- k0 = std::max(i, n) - n;
- kn = std::min(i, n-1);
- bri = n / binomial(r,i);
- for (size_t k = k0; k <= kn; ++k)
- {
- //if (k > i || (i-k) > n) continue;
- l = i - k;
-#if CHECK
- assert (l <= n);
-#endif
- bc = bri * binomial(n,l) * binomial(n-1, k);
- for (size_t j = 0; j <= m; ++j)
- {
- //d[j] += bc * dot(dB[k], B[l] - F[j]);
- d[j] += bc * (dBB(k,l) - dBF(k,j));
- }
- }
- double dmin, dmax;
- dmin = dmax = d[m];
- for (size_t j = 0; j < m; ++j)
- {
- if (dmin > d[j]) dmin = d[j];
- if (dmax < d[j]) dmax = d[j];
- }
- dij[0] = i * r_inv;
- dij[1] = dmin;
- D.push_back (dij);
- dij[1] = dmax;
- D.push_back (dij);
- }
-}
-
-/*
- * Clip the Bezier curve "B" wrt the focus "F"; the new parameter interval for
- * the clipped curve is returned through the output parameter "dom"
- */
-void clip_interval (Interval& dom,
- std::vector<Point> const& B,
- std::vector<Point> const& F)
-{
- std::vector<Point> D; // distance curve control points
- distance_control_points(D, B, F);
- //print(D, "D");
-// ConvexHull chD(D);
-// std::vector<Point>& p = chD.boundary; // convex hull vertices
-
- convex_hull(D);
- std::vector<Point> & p = D;
- //print(p, "CH(D)");
-
- bool plower, clower;
- double t, tmin = 1, tmax = 0;
-
- plower = (p[0][Y] < 0);
- if (p[0][Y] == 0) // on the x axis
- {
- if (tmin > p[0][X]) tmin = p[0][X];
- if (tmax < p[0][X]) tmax = p[0][X];
-// std::cerr << "0 : on x axis " << p[0]
-// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
- }
-
- for (size_t i = 1; i < p.size(); ++i)
- {
- clower = (p[i][Y] < 0);
- if (p[i][Y] == 0) // on x axis
- {
- if (tmin > p[i][X]) tmin = p[i][X];
- if (tmax < p[i][X]) tmax = p[i][X];
-// std::cerr << i << " : on x axis " << p[i]
-// << " : tmin = " << tmin << ", tmax = " << tmax
-// << std::endl;
- }
- else if (clower != plower) // cross the x axis
- {
- t = intersect(p[i-1], p[i], 0);
- if (tmin > t) tmin = t;
- if (tmax < t) tmax = t;
- plower = clower;
-// std::cerr << i << " : lower " << p[i]
-// << " : tmin = " << tmin << ", tmax = " << tmax
-// << std::endl;
- }
- }
-
- // we have to test the closing segment for intersection
- size_t last = p.size() - 1;
- clower = (p[0][Y] < 0);
- if (clower != plower) // cross the x axis
- {
- t = intersect(p[last], p[0], 0);
- if (tmin > t) tmin = t;
- if (tmax < t) tmax = t;
-// std::cerr << "0 : lower " << p[0]
-// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
- }
- dom[0] = tmin;
- dom[1] = tmax;
-}
-
-/*
- * Clip the Bezier curve "B" wrt the Bezier curve "A" for individuating
- * points which have collinear normals; the new parameter interval
- * for the clipped curve is returned through the output parameter "dom"
- */
-template <>
-inline
-void clip<collinear_normal_tag> (Interval & dom,
- std::vector<Point> const& A,
- std::vector<Point> const& B)
-{
- std::vector<Point> F;
- make_focus(F, A);
- clip_interval(dom, B, F);
-}
-
-
-
-const double MAX_PRECISION = 1e-8;
-const double MIN_CLIPPED_SIZE_THRESHOLD = 0.8;
-const Interval UNIT_INTERVAL(0,1);
-const Interval EMPTY_INTERVAL = make_empty_interval();
-const Interval H1_INTERVAL(0, 0.5);
-const Interval H2_INTERVAL(0.5 + MAX_PRECISION, 1.0);
-
-/*
- * iterate
- *
- * input:
- * A, B: control point sets of two bezier curves
- * domA, domB: real parameter intervals of the two curves
- * precision: required computational precision of the returned parameter ranges
- * output:
- * domsA, domsB: sets of parameter intervals
- *
- * The parameter intervals are computed by using a Bezier clipping algorithm,
- * in case the clipping doesn't shrink the initial interval more than 20%,
- * a subdivision step is performed.
- * If during the computation both curves collapse to a single point
- * the routine exits indipendently by the precision reached in the computation
- * of the curve intervals.
- */
-template <>
-void iterate<intersection_point_tag> (std::vector<Interval>& domsA,
- std::vector<Interval>& domsB,
- std::vector<Point> const& A,
- std::vector<Point> const& B,
- Interval const& domA,
- Interval const& domB,
- double precision )
-{
- // in order to limit recursion
- static size_t counter = 0;
- if (domA.extent() == 1 && domB.extent() == 1) counter = 0;
- if (++counter > 100) return;
-#if VERBOSE
- std::cerr << std::fixed << std::setprecision(16);
- std::cerr << ">> curve subdision performed <<" << std::endl;
- std::cerr << "dom(A) : " << domA << std::endl;
- std::cerr << "dom(B) : " << domB << std::endl;
-// std::cerr << "angle(A) : " << angle(A) << std::endl;
-// std::cerr << "angle(B) : " << angle(B) << std::endl;
-#endif
-
- if (precision < MAX_PRECISION)
- precision = MAX_PRECISION;
-
- std::vector<Point> pA = A;
- std::vector<Point> pB = B;
- std::vector<Point>* C1 = &pA;
- std::vector<Point>* C2 = &pB;
-
- Interval dompA = domA;
- Interval dompB = domB;
- Interval* dom1 = &dompA;
- Interval* dom2 = &dompB;
-
- Interval dom;
-
- if ( is_constant(A) && is_constant(B) ){
- Point M1 = middle_point(C1->front(), C1->back());
- Point M2 = middle_point(C2->front(), C2->back());
- if (are_near(M1,M2)){
- domsA.push_back(domA);
- domsB.push_back(domB);
- }
- return;
- }
-
- size_t iter = 0;
- while (++iter < 100
- && (dompA.extent() >= precision || dompB.extent() >= precision))
- {
-#if VERBOSE
- std::cerr << "iter: " << iter << std::endl;
-#endif
- clip<intersection_point_tag>(dom, *C1, *C2);
-
- // [1,0] is utilized to represent an empty interval
- if (dom == EMPTY_INTERVAL)
- {
-#if VERBOSE
- std::cerr << "dom: empty" << std::endl;
-#endif
- return;
- }
-#if VERBOSE
- std::cerr << "dom : " << dom << std::endl;
-#endif
- // all other cases where dom[0] > dom[1] are invalid
- if (dom.min() > dom.max())
- {
- assert(dom.min() < dom.max());
- }
-
- map_to(*dom2, dom);
-
- portion(*C2, dom);
- if (is_constant(*C2) && is_constant(*C1))
- {
- Point M1 = middle_point(C1->front(), C1->back());
- Point M2 = middle_point(C2->front(), C2->back());
-#if VERBOSE
- std::cerr << "both curves are constant: \n"
- << "M1: " << M1 << "\n"
- << "M2: " << M2 << std::endl;
- print(*C2, "C2");
- print(*C1, "C1");
-#endif
- if (are_near(M1,M2))
- break; // append the new interval
- else
- return; // exit without appending any new interval
- }
-
-
- // if we have clipped less than 20% than we need to subdive the curve
- // with the largest domain into two sub-curves
- if ( dom.extent() > MIN_CLIPPED_SIZE_THRESHOLD)
- {
-#if VERBOSE
- std::cerr << "clipped less than 20% : " << dom.extent() << std::endl;
- std::cerr << "angle(pA) : " << angle(pA) << std::endl;
- std::cerr << "angle(pB) : " << angle(pB) << std::endl;
-#endif
- std::vector<Point> pC1, pC2;
- Interval dompC1, dompC2;
- if (dompA.extent() > dompB.extent())
- {
- pC1 = pC2 = pA;
- portion(pC1, H1_INTERVAL);
- portion(pC2, H2_INTERVAL);
- dompC1 = dompC2 = dompA;
- map_to(dompC1, H1_INTERVAL);
- map_to(dompC2, H2_INTERVAL);
- iterate<intersection_point_tag>(domsA, domsB, pC1, pB,
- dompC1, dompB, precision);
- iterate<intersection_point_tag>(domsA, domsB, pC2, pB,
- dompC2, dompB, precision);
- }
- else
- {
- pC1 = pC2 = pB;
- portion(pC1, H1_INTERVAL);
- portion(pC2, H2_INTERVAL);
- dompC1 = dompC2 = dompB;
- map_to(dompC1, H1_INTERVAL);
- map_to(dompC2, H2_INTERVAL);
- iterate<intersection_point_tag>(domsB, domsA, pC1, pA,
- dompC1, dompA, precision);
- iterate<intersection_point_tag>(domsB, domsA, pC2, pA,
- dompC2, dompA, precision);
- }
- return;
- }
-
- std::swap(C1, C2);
- std::swap(dom1, dom2);
-#if VERBOSE
- std::cerr << "dom(pA) : " << dompA << std::endl;
- std::cerr << "dom(pB) : " << dompB << std::endl;
-#endif
- }
- domsA.push_back(dompA);
- domsB.push_back(dompB);
-}
-
-
-/*
- * iterate
- *
- * input:
- * A, B: control point sets of two bezier curves
- * domA, domB: real parameter intervals of the two curves
- * precision: required computational precision of the returned parameter ranges
- * output:
- * domsA, domsB: sets of parameter intervals
- *
- * The parameter intervals are computed by using a Bezier clipping algorithm,
- * in case the clipping doesn't shrink the initial interval more than 20%,
- * a subdivision step is performed.
- * If during the computation one of the two curve interval length becomes less
- * than MAX_PRECISION the routine exits indipendently by the precision reached
- * in the computation of the other curve interval.
- */
-template <>
-void iterate<collinear_normal_tag> (std::vector<Interval>& domsA,
- std::vector<Interval>& domsB,
- std::vector<Point> const& A,
- std::vector<Point> const& B,
- Interval const& domA,
- Interval const& domB,
- double precision)
-{
- // in order to limit recursion
- static size_t counter = 0;
- if (domA.extent() == 1 && domB.extent() == 1) counter = 0;
- if (++counter > 100) return;
-#if VERBOSE
- std::cerr << std::fixed << std::setprecision(16);
- std::cerr << ">> curve subdision performed <<" << std::endl;
- std::cerr << "dom(A) : " << domA << std::endl;
- std::cerr << "dom(B) : " << domB << std::endl;
-// std::cerr << "angle(A) : " << angle(A) << std::endl;
-// std::cerr << "angle(B) : " << angle(B) << std::endl;
-#endif
-
- if (precision < MAX_PRECISION)
- precision = MAX_PRECISION;
-
- std::vector<Point> pA = A;
- std::vector<Point> pB = B;
- std::vector<Point>* C1 = &pA;
- std::vector<Point>* C2 = &pB;
-
- Interval dompA = domA;
- Interval dompB = domB;
- Interval* dom1 = &dompA;
- Interval* dom2 = &dompB;
-
- Interval dom;
-
- size_t iter = 0;
- while (++iter < 100
- && (dompA.extent() >= precision || dompB.extent() >= precision))
- {
-#if VERBOSE
- std::cerr << "iter: " << iter << std::endl;
-#endif
- clip<collinear_normal_tag>(dom, *C1, *C2);
-
- // [1,0] is utilized to represent an empty interval
- if (dom == EMPTY_INTERVAL)
- {
-#if VERBOSE
- std::cerr << "dom: empty" << std::endl;
-#endif
- return;
- }
-#if VERBOSE
- std::cerr << "dom : " << dom << std::endl;
-#endif
- // all other cases where dom[0] > dom[1] are invalid
- if (dom.min() > dom.max())
- {
- assert(dom.min() < dom.max());
- }
-
- map_to(*dom2, dom);
-
- // it's better to stop before losing computational precision
- if (iter > 1 && (dom2->extent() <= MAX_PRECISION))
- {
-#if VERBOSE
- std::cerr << "beyond max precision limit" << std::endl;
-#endif
- break;
- }
-
- portion(*C2, dom);
- if (iter > 1 && is_constant(*C2))
- {
-#if VERBOSE
- std::cerr << "new curve portion pC1 is constant" << std::endl;
-#endif
- break;
- }
-
-
- // if we have clipped less than 20% than we need to subdive the curve
- // with the largest domain into two sub-curves
- if ( dom.extent() > MIN_CLIPPED_SIZE_THRESHOLD)
- {
-#if VERBOSE
- std::cerr << "clipped less than 20% : " << dom.extent() << std::endl;
- std::cerr << "angle(pA) : " << angle(pA) << std::endl;
- std::cerr << "angle(pB) : " << angle(pB) << std::endl;
-#endif
- std::vector<Point> pC1, pC2;
- Interval dompC1, dompC2;
- if (dompA.extent() > dompB.extent())
- {
- if ((dompA.extent() / 2) < MAX_PRECISION)
- {
- break;
- }
- pC1 = pC2 = pA;
- portion(pC1, H1_INTERVAL);
- if (false && is_constant(pC1))
- {
-#if VERBOSE
- std::cerr << "new curve portion pC1 is constant" << std::endl;
-#endif
- break;
- }
- portion(pC2, H2_INTERVAL);
- if (is_constant(pC2))
- {
-#if VERBOSE
- std::cerr << "new curve portion pC2 is constant" << std::endl;
-#endif
- break;
- }
- dompC1 = dompC2 = dompA;
- map_to(dompC1, H1_INTERVAL);
- map_to(dompC2, H2_INTERVAL);
- iterate<collinear_normal_tag>(domsA, domsB, pC1, pB,
- dompC1, dompB, precision);
- iterate<collinear_normal_tag>(domsA, domsB, pC2, pB,
- dompC2, dompB, precision);
- }
- else
- {
- if ((dompB.extent() / 2) < MAX_PRECISION)
- {
- break;
- }
- pC1 = pC2 = pB;
- portion(pC1, H1_INTERVAL);
- if (is_constant(pC1))
- {
-#if VERBOSE
- std::cerr << "new curve portion pC1 is constant" << std::endl;
-#endif
- break;
- }
- portion(pC2, H2_INTERVAL);
- if (is_constant(pC2))
- {
-#if VERBOSE
- std::cerr << "new curve portion pC2 is constant" << std::endl;
-#endif
- break;
- }
- dompC1 = dompC2 = dompB;
- map_to(dompC1, H1_INTERVAL);
- map_to(dompC2, H2_INTERVAL);
- iterate<collinear_normal_tag>(domsB, domsA, pC1, pA,
- dompC1, dompA, precision);
- iterate<collinear_normal_tag>(domsB, domsA, pC2, pA,
- dompC2, dompA, precision);
- }
- return;
- }
-
- std::swap(C1, C2);
- std::swap(dom1, dom2);
-#if VERBOSE
- std::cerr << "dom(pA) : " << dompA << std::endl;
- std::cerr << "dom(pB) : " << dompB << std::endl;
-#endif
- }
- domsA.push_back(dompA);
- domsB.push_back(dompB);
-}
-
-
-/*
- * get_solutions
- *
- * input: A, B - set of control points of two Bezier curve
- * input: precision - required precision of computation
- * input: clip - the routine used for clipping
- * output: xs - set of pairs of parameter values
- * at which the clipping algorithm converges
- *
- * This routine is based on the Bezier Clipping Algorithm,
- * see: Sederberg - Computer Aided Geometric Design
- */
-template <typename Tag>
-void get_solutions (std::vector< std::pair<double, double> >& xs,
- std::vector<Point> const& A,
- std::vector<Point> const& B,
- double precision)
-{
- std::pair<double, double> ci;
- std::vector<Interval> domsA, domsB;
- iterate<Tag> (domsA, domsB, A, B, UNIT_INTERVAL, UNIT_INTERVAL, precision);
- if (domsA.size() != domsB.size())
- {
- assert (domsA.size() == domsB.size());
- }
- xs.clear();
- xs.reserve(domsA.size());
- for (size_t i = 0; i < domsA.size(); ++i)
- {
-#if VERBOSE
- std::cerr << i << " : domA : " << domsA[i] << std::endl;
- std::cerr << "extent A: " << domsA[i].extent() << " ";
- std::cerr << "precision A: " << get_precision(domsA[i]) << std::endl;
- std::cerr << i << " : domB : " << domsB[i] << std::endl;
- std::cerr << "extent B: " << domsB[i].extent() << " ";
- std::cerr << "precision B: " << get_precision(domsB[i]) << std::endl;
-#endif
- ci.first = domsA[i].middle();
- ci.second = domsB[i].middle();
- xs.push_back(ci);
- }
-}
-
-} /* end namespace bezier_clipping */ } /* end namespace detail */
-
-
-/*
- * find_collinear_normal
- *
- * input: A, B - set of control points of two Bezier curve
- * input: precision - required precision of computation
- * output: xs - set of pairs of parameter values
- * at which there are collinear normals
- *
- * This routine is based on the Bezier Clipping Algorithm,
- * see: Sederberg, Nishita, 1990 - Curve intersection using Bezier clipping
- */
-void find_collinear_normal (std::vector< std::pair<double, double> >& xs,
- std::vector<Point> const& A,
- std::vector<Point> const& B,
- double precision)
-{
- using detail::bezier_clipping::get_solutions;
- using detail::bezier_clipping::collinear_normal_tag;
- get_solutions<collinear_normal_tag>(xs, A, B, precision);
-}
-
-
-/*
- * find_intersections_bezier_clipping
- *
- * input: A, B - set of control points of two Bezier curve
- * input: precision - required precision of computation
- * output: xs - set of pairs of parameter values
- * at which crossing happens
- *
- * This routine is based on the Bezier Clipping Algorithm,
- * see: Sederberg, Nishita, 1990 - Curve intersection using Bezier clipping
- */
-void find_intersections_bezier_clipping (std::vector< std::pair<double, double> >& xs,
- std::vector<Point> const& A,
- std::vector<Point> const& B,
- double precision)
-{
- using detail::bezier_clipping::get_solutions;
- using detail::bezier_clipping::intersection_point_tag;
- get_solutions<intersection_point_tag>(xs, A, B, precision);
-}
-
-} // end namespace Geom
-
-
-
-
-/*
- 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:encoding=utf-8:textwidth=99 :
+/*
+ * Implement the Bezier clipping algorithm for finding
+ * Bezier curve intersection points and collinear normals
+ *
+ * Authors:
+ * Marco Cecchetti <mrcekets at gmail.com>
+ *
+ * Copyright 2008 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/choose.h>
+#include <2geom/point.h>
+#include <2geom/interval.h>
+#include <2geom/bezier.h>
+//#include <2geom/convex-cover.h>
+#include <2geom/numeric/matrix.h>
+
+#include <cassert>
+#include <vector>
+#include <algorithm>
+#include <utility>
+//#include <iomanip>
+
+
+
+
+#define VERBOSE 0
+#define CHECK 0
+
+
+namespace Geom {
+
+namespace detail { namespace bezier_clipping {
+
+////////////////////////////////////////////////////////////////////////////////
+// for debugging
+//
+
+inline
+void print(std::vector<Point> const& cp, const char* msg = "")
+{
+ std::cerr << msg << std::endl;
+ for (size_t i = 0; i < cp.size(); ++i)
+ std::cerr << i << " : " << cp[i] << std::endl;
+}
+
+template< class charT >
+inline
+std::basic_ostream<charT> &
+operator<< (std::basic_ostream<charT> & os, const Interval & I)
+{
+ os << "[" << I.min() << ", " << I.max() << "]";
+ return os;
+}
+
+inline
+double angle (std::vector<Point> const& A)
+{
+ size_t n = A.size() -1;
+ double a = std::atan2(A[n][Y] - A[0][Y], A[n][X] - A[0][X]);
+ return (180 * a / M_PI);
+}
+
+inline
+size_t get_precision(Interval const& I)
+{
+ double d = I.extent();
+ double e = 0.1, p = 10;
+ int n = 0;
+ while (n < 16 && d < e)
+ {
+ p *= 10;
+ e = 1/p;
+ ++n;
+ }
+ return n;
+}
+
+inline
+void range_assertion(int k, int m, int n, const char* msg)
+{
+ if ( k < m || k > n)
+ {
+ std::cerr << "range assertion failed: \n"
+ << msg << std::endl
+ << "value: " << k
+ << " range: " << m << ", " << n << std::endl;
+ assert (k >= m && k <= n);
+ }
+}
+
+
+////////////////////////////////////////////////////////////////////////////////
+// convex hull
+
+/*
+ * return true in case the oriented polyline p0, p1, p2 is a right turn
+ */
+inline
+bool is_a_right_turn (Point const& p0, Point const& p1, Point const& p2)
+{
+ if (p1 == p2) return false;
+ Point q1 = p1 - p0;
+ Point q2 = p2 - p0;
+ if (q1 == -q2) return false;
+ return (cross (q1, q2) < 0);
+}
+
+/*
+ * return true if p < q wrt the lexicographyc order induced by the coordinates
+ */
+struct lex_less
+{
+ bool operator() (Point const& p, Point const& q)
+ {
+ return ((p[X] < q[X]) || (p[X] == q[X] && p[Y] < q[Y]));
+ }
+};
+
+/*
+ * return true if p > q wrt the lexicographyc order induced by the coordinates
+ */
+struct lex_greater
+{
+ bool operator() (Point const& p, Point const& q)
+ {
+ return ((p[X] > q[X]) || (p[X] == q[X] && p[Y] > q[Y]));
+ }
+};
+
+/*
+ * Compute the convex hull of a set of points.
+ * The implementation is based on the Andrew's scan algorithm
+ * note: in the Bezier clipping for collinear normals it seems
+ * to be more stable wrt the Graham's scan algorithm and in general
+ * a bit quikier
+ */
+void convex_hull (std::vector<Point> & P)
+{
+ size_t n = P.size();
+ if (n < 2) return;
+ std::sort(P.begin(), P.end(), lex_less());
+ if (n < 4) return;
+ // upper hull
+ size_t u = 2;
+ for (size_t i = 2; i < n; ++i)
+ {
+ while (u > 1 && !is_a_right_turn(P[u-2], P[u-1], P[i]))
+ {
+ --u;
+ }
+ std::swap(P[u], P[i]);
+ ++u;
+ }
+ std::sort(P.begin() + u, P.end(), lex_greater());
+ std::rotate(P.begin(), P.begin() + 1, P.end());
+ // lower hull
+ size_t l = u;
+ size_t k = u - 1;
+ for (size_t i = l; i < n; ++i)
+ {
+ while (l > k && !is_a_right_turn(P[l-2], P[l-1], P[i]))
+ {
+ --l;
+ }
+ std::swap(P[l], P[i]);
+ ++l;
+ }
+ P.resize(l);
+}
+
+
+////////////////////////////////////////////////////////////////////////////////
+// numerical routines
+
+/*
+ * Compute the binomial coefficient (n, k)
+ */
+inline
+double binomial(unsigned int n, unsigned int k)
+{
+ return choose<double>(n, k);
+}
+
+/*
+ * Compute the determinant of the 2x2 matrix with column the point P1, P2
+ */
+inline
+double det(Point const& P1, Point const& P2)
+{
+ return P1[X]*P2[Y] - P1[Y]*P2[X];
+}
+
+/*
+ * Solve the linear system [P1,P2] * P = Q
+ * in case there isn't exactly one solution the routine returns false
+ */
+inline
+bool solve(Point & P, Point const& P1, Point const& P2, Point const& Q)
+{
+ double d = det(P1, P2);
+ if (d == 0) return false;
+ d = 1 / d;
+ P[X] = det(Q, P2) * d;
+ P[Y] = det(P1, Q) * d;
+ return true;
+}
+
+////////////////////////////////////////////////////////////////////////////////
+// interval routines
+
+/*
+ * Map the sub-interval I in [0,1] into the interval J and assign it to J
+ */
+inline
+void map_to(Interval & J, Interval const& I)
+{
+ double length = J.extent();
+ J[1] = I.max() * length + J[0];
+ J[0] = I.min() * length + J[0];
+}
+
+/*
+ * The interval [1,0] is used to represent the empty interval, this routine
+ * is just an helper function for creating such an interval
+ */
+inline
+Interval make_empty_interval()
+{
+ Interval I(0);
+ I[0] = 1;
+ return I;
+}
+
+
+////////////////////////////////////////////////////////////////////////////////
+// bezier curve routines
+
+/*
+ * Return true if all the Bezier curve control points are near,
+ * false otherwise
+ */
+inline
+bool is_constant(std::vector<Point> const& A, double precision = EPSILON)
+{
+ for (unsigned int i = 1; i < A.size(); ++i)
+ {
+ if(!are_near(A[i], A[0], precision))
+ return false;
+ }
+ return true;
+}
+
+/*
+ * Compute the hodograph of the bezier curve B and return it in D
+ */
+inline
+void derivative(std::vector<Point> & D, std::vector<Point> const& B)
+{
+ D.clear();
+ size_t sz = B.size();
+ if (sz == 0) return;
+ if (sz == 1)
+ {
+ D.resize(1, Point(0,0));
+ return;
+ }
+ size_t n = sz-1;
+ D.reserve(n);
+ for (size_t i = 0; i < n; ++i)
+ {
+ D.push_back(n*(B[i+1] - B[i]));
+ }
+}
+
+/*
+ * Compute the hodograph of the Bezier curve B rotated of 90 degree
+ * and return it in D; we have N(t) orthogonal to B(t) for any t
+ */
+inline
+void normal(std::vector<Point> & N, std::vector<Point> const& B)
+{
+ derivative(N,B);
+ for (size_t i = 0; i < N.size(); ++i)
+ {
+ N[i] = rot90(N[i]);
+ }
+}
+
+/*
+ * Compute the portion of the Bezier curve "B" wrt the interval [0,t]
+ */
+inline
+void left_portion(Coord t, std::vector<Point> & B)
+{
+ size_t n = B.size();
+ for (size_t i = 1; i < n; ++i)
+ {
+ for (size_t j = n-1; j > i-1 ; --j)
+ {
+ B[j] = lerp(t, B[j-1], B[j]);
+ }
+ }
+}
+
+/*
+ * Compute the portion of the Bezier curve "B" wrt the interval [t,1]
+ */
+inline
+void right_portion(Coord t, std::vector<Point> & B)
+{
+ size_t n = B.size();
+ for (size_t i = 1; i < n; ++i)
+ {
+ for (size_t j = 0; j < n-i; ++j)
+ {
+ B[j] = lerp(t, B[j], B[j+1]);
+ }
+ }
+}
+
+/*
+ * Compute the portion of the Bezier curve "B" wrt the interval "I"
+ */
+inline
+void portion (std::vector<Point> & B , Interval const& I)
+{
+ if (I.min() == 0)
+ {
+ if (I.max() == 1) return;
+ left_portion(I.max(), B);
+ return;
+ }
+ right_portion(I.min(), B);
+ if (I.max() == 1) return;
+ double t = I.extent() / (1 - I.min());
+ left_portion(t, B);
+}
+
+
+////////////////////////////////////////////////////////////////////////////////
+// tags
+
+struct intersection_point_tag;
+struct collinear_normal_tag;
+template <typename Tag>
+void clip(Interval & dom,
+ std::vector<Point> const& A,
+ std::vector<Point> const& B);
+template <typename Tag>
+void iterate(std::vector<Interval>& domsA,
+ std::vector<Interval>& domsB,
+ std::vector<Point> const& A,
+ std::vector<Point> const& B,
+ Interval const& domA,
+ Interval const& domB,
+ double precision );
+
+
+////////////////////////////////////////////////////////////////////////////////
+// intersection
+
+/*
+ * Make up an orientation line using the control points c[i] and c[j]
+ * the line is returned in the output parameter "l" in the form of a 3 element
+ * vector : l[0] * x + l[1] * y + l[2] == 0; the line is normalized.
+ */
+inline
+void orientation_line (std::vector<double> & l,
+ std::vector<Point> const& c,
+ size_t i, size_t j)
+{
+ l[0] = c[j][Y] - c[i][Y];
+ l[1] = c[i][X] - c[j][X];
+ l[2] = cross(c[i], c[j]);
+ double length = std::sqrt(l[0] * l[0] + l[1] * l[1]);
+ assert (length != 0);
+ l[0] /= length;
+ l[1] /= length;
+ l[2] /= length;
+}
+
+/*
+ * Pick up an orientation line for the Bezier curve "c" and return it in
+ * the output parameter "l"
+ */
+inline
+void pick_orientation_line (std::vector<double> & l,
+ std::vector<Point> const& c)
+{
+ size_t i = c.size();
+ while (--i > 0 && are_near(c[0], c[i]))
+ {}
+ if (i == 0)
+ {
+ // this should never happen because when a new curve portion is created
+ // we check that it is not constant;
+ // however this requires that the precision used in the is_constant
+ // routine has to be the same used here in the are_near test
+ assert(i != 0);
+ }
+ orientation_line(l, c, 0, i);
+ //std::cerr << "i = " << i << std::endl;
+}
+
+/*
+ * Make up an orientation line for constant bezier curve;
+ * the orientation line is made up orthogonal to the other curve base line;
+ * the line is returned in the output parameter "l" in the form of a 3 element
+ * vector : l[0] * x + l[1] * y + l[2] == 0; the line is normalized.
+ */
+inline
+void orthogonal_orientation_line (std::vector<double> & l,
+ std::vector<Point> const& c,
+ Point const& p)
+{
+ if (is_constant(c))
+ {
+ // this should never happen
+ assert(!is_constant(c));
+ }
+ std::vector<Point> ol(2);
+ ol[0] = p;
+ ol[1] = (c.back() - c.front()).cw() + p;
+ orientation_line(l, ol, 0, 1);
+}
+
+/*
+ * Compute the signed distance of the point "P" from the normalized line l
+ */
+inline
+double distance (Point const& P, std::vector<double> const& l)
+{
+ return l[X] * P[X] + l[Y] * P[Y] + l[2];
+}
+
+/*
+ * Compute the min and max distance of the control points of the Bezier
+ * curve "c" from the normalized orientation line "l".
+ * This bounds are returned through the output Interval parameter"bound".
+ */
+inline
+void fat_line_bounds (Interval& bound,
+ std::vector<Point> const& c,
+ std::vector<double> const& l)
+{
+ bound[0] = 0;
+ bound[1] = 0;
+ double d;
+ for (size_t i = 0; i < c.size(); ++i)
+ {
+ d = distance(c[i], l);
+ if (bound[0] > d) bound[0] = d;
+ if (bound[1] < d) bound[1] = d;
+ }
+}
+
+/*
+ * return the x component of the intersection point between the line
+ * passing through points p1, p2 and the line Y = "y"
+ */
+inline
+double intersect (Point const& p1, Point const& p2, double y)
+{
+ // we are sure that p2[Y] != p1[Y] because this routine is called
+ // only when the lower or the upper bound is crossed
+ double dy = (p2[Y] - p1[Y]);
+ double s = (y - p1[Y]) / dy;
+ return (p2[X]-p1[X])*s + p1[X];
+}
+
+/*
+ * Clip the Bezier curve "B" wrt the fat line defined by the orientation
+ * line "l" and the interval range "bound", the new parameter interval for
+ * the clipped curve is returned through the output parameter "dom"
+ */
+void clip_interval (Interval& dom,
+ std::vector<Point> const& B,
+ std::vector<double> const& l,
+ Interval const& bound)
+{
+ double n = B.size() - 1; // number of sub-intervals
+ std::vector<Point> D; // distance curve control points
+ D.reserve (B.size());
+ double d;
+ for (size_t i = 0; i < B.size(); ++i)
+ {
+ d = distance (B[i], l);
+ D.push_back (Point(i/n, d));
+ }
+ //print(D);
+
+ convex_hull(D);
+ std::vector<Point> & p = D;
+ //print(p);
+
+ bool plower, phigher;
+ bool clower, chigher;
+ double t, tmin = 1, tmax = 0;
+// std::cerr << "bound : " << bound << std::endl;
+
+ plower = (p[0][Y] < bound.min());
+ phigher = (p[0][Y] > bound.max());
+ if (!(plower || phigher)) // inside the fat line
+ {
+ if (tmin > p[0][X]) tmin = p[0][X];
+ if (tmax < p[0][X]) tmax = p[0][X];
+// std::cerr << "0 : inside " << p[0]
+// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
+ }
+
+ for (size_t i = 1; i < p.size(); ++i)
+ {
+ clower = (p[i][Y] < bound.min());
+ chigher = (p[i][Y] > bound.max());
+ if (!(clower || chigher)) // inside the fat line
+ {
+ if (tmin > p[i][X]) tmin = p[i][X];
+ if (tmax < p[i][X]) tmax = p[i][X];
+// std::cerr << i << " : inside " << p[i]
+// << " : tmin = " << tmin << ", tmax = " << tmax
+// << std::endl;
+ }
+ if (clower != plower) // cross the lower bound
+ {
+ t = intersect(p[i-1], p[i], bound.min());
+ if (tmin > t) tmin = t;
+ if (tmax < t) tmax = t;
+ plower = clower;
+// std::cerr << i << " : lower " << p[i]
+// << " : tmin = " << tmin << ", tmax = " << tmax
+// << std::endl;
+ }
+ if (chigher != phigher) // cross the upper bound
+ {
+ t = intersect(p[i-1], p[i], bound.max());
+ if (tmin > t) tmin = t;
+ if (tmax < t) tmax = t;
+ phigher = chigher;
+// std::cerr << i << " : higher " << p[i]
+// << " : tmin = " << tmin << ", tmax = " << tmax
+// << std::endl;
+ }
+ }
+
+ // we have to test the closing segment for intersection
+ size_t last = p.size() - 1;
+ clower = (p[0][Y] < bound.min());
+ chigher = (p[0][Y] > bound.max());
+ if (clower != plower) // cross the lower bound
+ {
+ t = intersect(p[last], p[0], bound.min());
+ if (tmin > t) tmin = t;
+ if (tmax < t) tmax = t;
+// std::cerr << "0 : lower " << p[0]
+// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
+ }
+ if (chigher != phigher) // cross the upper bound
+ {
+ t = intersect(p[last], p[0], bound.max());
+ if (tmin > t) tmin = t;
+ if (tmax < t) tmax = t;
+// std::cerr << "0 : higher " << p[0]
+// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
+ }
+
+ dom[0] = tmin;
+ dom[1] = tmax;
+}
+
+/*
+ * Clip the Bezier curve "B" wrt the Bezier curve "A" for individuating
+ * intersection points the new parameter interval for the clipped curve
+ * is returned through the output parameter "dom"
+ */
+template <>
+inline
+void clip<intersection_point_tag> (Interval & dom,
+ std::vector<Point> const& A,
+ std::vector<Point> const& B)
+{
+ std::vector<double> bl(3);
+ Interval bound;
+ if (is_constant(A))
+ {
+ Point M = middle_point(A.front(), A.back());
+ orthogonal_orientation_line(bl, B, M);
+ }
+ else
+ {
+ pick_orientation_line(bl, A);
+ }
+ fat_line_bounds(bound, A, bl);
+ clip_interval(dom, B, bl, bound);
+}
+
+
+///////////////////////////////////////////////////////////////////////////////
+// collinear normal
+
+/*
+ * Compute a closed focus for the Bezier curve B and return it in F
+ * A focus is any curve through which all lines perpendicular to B(t) pass.
+ */
+inline
+void make_focus (std::vector<Point> & F, std::vector<Point> const& B)
+{
+ assert (B.size() > 2);
+ size_t n = B.size() - 1;
+ normal(F, B);
+ Point c(1, 1);
+#if VERBOSE
+ if (!solve(c, F[0], -F[n-1], B[n]-B[0]))
+ {
+ std::cerr << "make_focus: unable to make up a closed focus" << std::endl;
+ }
+#else
+ solve(c, F[0], -F[n-1], B[n]-B[0]);
+#endif
+// std::cerr << "c = " << c << std::endl;
+
+
+ // B(t) + c(t) * N(t)
+ double n_inv = 1 / (double)(n);
+ Point c0ni;
+ F.push_back(c[1] * F[n-1]);
+ F[n] += B[n];
+ for (size_t i = n-1; i > 0; --i)
+ {
+ F[i] *= -c[0];
+ c0ni = F[i];
+ F[i] += (c[1] * F[i-1]);
+ F[i] *= (i * n_inv);
+ F[i] -= c0ni;
+ F[i] += B[i];
+ }
+ F[0] *= c[0];
+ F[0] += B[0];
+}
+
+/*
+ * Compute the projection on the plane (t, d) of the control points
+ * (t, u, D(t,u)) where D(t,u) = <(B(t) - F(u)), B'(t)> with 0 <= t, u <= 1
+ * B is a Bezier curve and F is a focus of another Bezier curve.
+ * See Sederberg, Nishita, 1990 - Curve intersection using Bezier clipping.
+ */
+void distance_control_points (std::vector<Point> & D,
+ std::vector<Point> const& B,
+ std::vector<Point> const& F)
+{
+ assert (B.size() > 1);
+ assert (F.size() > 0);
+ const size_t n = B.size() - 1;
+ const size_t m = F.size() - 1;
+ const size_t r = 2 * n - 1;
+ const double r_inv = 1 / (double)(r);
+ D.clear();
+ D.reserve (B.size() * F.size());
+
+ std::vector<Point> dB;
+ dB.reserve(n);
+ for (size_t k = 0; k < n; ++k)
+ {
+ dB.push_back (B[k+1] - B[k]);
+ }
+ NL::Matrix dBB(n,B.size());
+ for (size_t i = 0; i < n; ++i)
+ for (size_t j = 0; j < B.size(); ++j)
+ dBB(i,j) = dot (dB[i], B[j]);
+ NL::Matrix dBF(n, F.size());
+ for (size_t i = 0; i < n; ++i)
+ for (size_t j = 0; j < F.size(); ++j)
+ dBF(i,j) = dot (dB[i], F[j]);
+
+ size_t k0, kn, l;
+ double bc, bri;
+ Point dij;
+ std::vector<double> d(F.size());
+ for (size_t i = 0; i <= r; ++i)
+ {
+ for (size_t j = 0; j <= m; ++j)
+ {
+ d[j] = 0;
+ }
+ k0 = std::max(i, n) - n;
+ kn = std::min(i, n-1);
+ bri = n / binomial(r,i);
+ for (size_t k = k0; k <= kn; ++k)
+ {
+ //if (k > i || (i-k) > n) continue;
+ l = i - k;
+#if CHECK
+ assert (l <= n);
+#endif
+ bc = bri * binomial(n,l) * binomial(n-1, k);
+ for (size_t j = 0; j <= m; ++j)
+ {
+ //d[j] += bc * dot(dB[k], B[l] - F[j]);
+ d[j] += bc * (dBB(k,l) - dBF(k,j));
+ }
+ }
+ double dmin, dmax;
+ dmin = dmax = d[m];
+ for (size_t j = 0; j < m; ++j)
+ {
+ if (dmin > d[j]) dmin = d[j];
+ if (dmax < d[j]) dmax = d[j];
+ }
+ dij[0] = i * r_inv;
+ dij[1] = dmin;
+ D.push_back (dij);
+ dij[1] = dmax;
+ D.push_back (dij);
+ }
+}
+
+/*
+ * Clip the Bezier curve "B" wrt the focus "F"; the new parameter interval for
+ * the clipped curve is returned through the output parameter "dom"
+ */
+void clip_interval (Interval& dom,
+ std::vector<Point> const& B,
+ std::vector<Point> const& F)
+{
+ std::vector<Point> D; // distance curve control points
+ distance_control_points(D, B, F);
+ //print(D, "D");
+// ConvexHull chD(D);
+// std::vector<Point>& p = chD.boundary; // convex hull vertices
+
+ convex_hull(D);
+ std::vector<Point> & p = D;
+ //print(p, "CH(D)");
+
+ bool plower, clower;
+ double t, tmin = 1, tmax = 0;
+
+ plower = (p[0][Y] < 0);
+ if (p[0][Y] == 0) // on the x axis
+ {
+ if (tmin > p[0][X]) tmin = p[0][X];
+ if (tmax < p[0][X]) tmax = p[0][X];
+// std::cerr << "0 : on x axis " << p[0]
+// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
+ }
+
+ for (size_t i = 1; i < p.size(); ++i)
+ {
+ clower = (p[i][Y] < 0);
+ if (p[i][Y] == 0) // on x axis
+ {
+ if (tmin > p[i][X]) tmin = p[i][X];
+ if (tmax < p[i][X]) tmax = p[i][X];
+// std::cerr << i << " : on x axis " << p[i]
+// << " : tmin = " << tmin << ", tmax = " << tmax
+// << std::endl;
+ }
+ else if (clower != plower) // cross the x axis
+ {
+ t = intersect(p[i-1], p[i], 0);
+ if (tmin > t) tmin = t;
+ if (tmax < t) tmax = t;
+ plower = clower;
+// std::cerr << i << " : lower " << p[i]
+// << " : tmin = " << tmin << ", tmax = " << tmax
+// << std::endl;
+ }
+ }
+
+ // we have to test the closing segment for intersection
+ size_t last = p.size() - 1;
+ clower = (p[0][Y] < 0);
+ if (clower != plower) // cross the x axis
+ {
+ t = intersect(p[last], p[0], 0);
+ if (tmin > t) tmin = t;
+ if (tmax < t) tmax = t;
+// std::cerr << "0 : lower " << p[0]
+// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
+ }
+ dom[0] = tmin;
+ dom[1] = tmax;
+}
+
+/*
+ * Clip the Bezier curve "B" wrt the Bezier curve "A" for individuating
+ * points which have collinear normals; the new parameter interval
+ * for the clipped curve is returned through the output parameter "dom"
+ */
+template <>
+inline
+void clip<collinear_normal_tag> (Interval & dom,
+ std::vector<Point> const& A,
+ std::vector<Point> const& B)
+{
+ std::vector<Point> F;
+ make_focus(F, A);
+ clip_interval(dom, B, F);
+}
+
+
+
+const double MAX_PRECISION = 1e-8;
+const double MIN_CLIPPED_SIZE_THRESHOLD = 0.8;
+const Interval UNIT_INTERVAL(0,1);
+const Interval EMPTY_INTERVAL = make_empty_interval();
+const Interval H1_INTERVAL(0, 0.5);
+const Interval H2_INTERVAL(0.5 + MAX_PRECISION, 1.0);
+
+/*
+ * iterate
+ *
+ * input:
+ * A, B: control point sets of two bezier curves
+ * domA, domB: real parameter intervals of the two curves
+ * precision: required computational precision of the returned parameter ranges
+ * output:
+ * domsA, domsB: sets of parameter intervals
+ *
+ * The parameter intervals are computed by using a Bezier clipping algorithm,
+ * in case the clipping doesn't shrink the initial interval more than 20%,
+ * a subdivision step is performed.
+ * If during the computation both curves collapse to a single point
+ * the routine exits indipendently by the precision reached in the computation
+ * of the curve intervals.
+ */
+template <>
+void iterate<intersection_point_tag> (std::vector<Interval>& domsA,
+ std::vector<Interval>& domsB,
+ std::vector<Point> const& A,
+ std::vector<Point> const& B,
+ Interval const& domA,
+ Interval const& domB,
+ double precision )
+{
+ // in order to limit recursion
+ static size_t counter = 0;
+ if (domA.extent() == 1 && domB.extent() == 1) counter = 0;
+ if (++counter > 100) return;
+#if VERBOSE
+ std::cerr << std::fixed << std::setprecision(16);
+ std::cerr << ">> curve subdision performed <<" << std::endl;
+ std::cerr << "dom(A) : " << domA << std::endl;
+ std::cerr << "dom(B) : " << domB << std::endl;
+// std::cerr << "angle(A) : " << angle(A) << std::endl;
+// std::cerr << "angle(B) : " << angle(B) << std::endl;
+#endif
+
+ if (precision < MAX_PRECISION)
+ precision = MAX_PRECISION;
+
+ std::vector<Point> pA = A;
+ std::vector<Point> pB = B;
+ std::vector<Point>* C1 = &pA;
+ std::vector<Point>* C2 = &pB;
+
+ Interval dompA = domA;
+ Interval dompB = domB;
+ Interval* dom1 = &dompA;
+ Interval* dom2 = &dompB;
+
+ Interval dom;
+
+ if ( is_constant(A) && is_constant(B) ){
+ Point M1 = middle_point(C1->front(), C1->back());
+ Point M2 = middle_point(C2->front(), C2->back());
+ if (are_near(M1,M2)){
+ domsA.push_back(domA);
+ domsB.push_back(domB);
+ }
+ return;
+ }
+
+ size_t iter = 0;
+ while (++iter < 100
+ && (dompA.extent() >= precision || dompB.extent() >= precision))
+ {
+#if VERBOSE
+ std::cerr << "iter: " << iter << std::endl;
+#endif
+ clip<intersection_point_tag>(dom, *C1, *C2);
+
+ // [1,0] is utilized to represent an empty interval
+ if (dom == EMPTY_INTERVAL)
+ {
+#if VERBOSE
+ std::cerr << "dom: empty" << std::endl;
+#endif
+ return;
+ }
+#if VERBOSE
+ std::cerr << "dom : " << dom << std::endl;
+#endif
+ // all other cases where dom[0] > dom[1] are invalid
+ if (dom.min() > dom.max())
+ {
+ assert(dom.min() < dom.max());
+ }
+
+ map_to(*dom2, dom);
+
+ portion(*C2, dom);
+ if (is_constant(*C2) && is_constant(*C1))
+ {
+ Point M1 = middle_point(C1->front(), C1->back());
+ Point M2 = middle_point(C2->front(), C2->back());
+#if VERBOSE
+ std::cerr << "both curves are constant: \n"
+ << "M1: " << M1 << "\n"
+ << "M2: " << M2 << std::endl;
+ print(*C2, "C2");
+ print(*C1, "C1");
+#endif
+ if (are_near(M1,M2))
+ break; // append the new interval
+ else
+ return; // exit without appending any new interval
+ }
+
+
+ // if we have clipped less than 20% than we need to subdive the curve
+ // with the largest domain into two sub-curves
+ if ( dom.extent() > MIN_CLIPPED_SIZE_THRESHOLD)
+ {
+#if VERBOSE
+ std::cerr << "clipped less than 20% : " << dom.extent() << std::endl;
+ std::cerr << "angle(pA) : " << angle(pA) << std::endl;
+ std::cerr << "angle(pB) : " << angle(pB) << std::endl;
+#endif
+ std::vector<Point> pC1, pC2;
+ Interval dompC1, dompC2;
+ if (dompA.extent() > dompB.extent())
+ {
+ pC1 = pC2 = pA;
+ portion(pC1, H1_INTERVAL);
+ portion(pC2, H2_INTERVAL);
+ dompC1 = dompC2 = dompA;
+ map_to(dompC1, H1_INTERVAL);
+ map_to(dompC2, H2_INTERVAL);
+ iterate<intersection_point_tag>(domsA, domsB, pC1, pB,
+ dompC1, dompB, precision);
+ iterate<intersection_point_tag>(domsA, domsB, pC2, pB,
+ dompC2, dompB, precision);
+ }
+ else
+ {
+ pC1 = pC2 = pB;
+ portion(pC1, H1_INTERVAL);
+ portion(pC2, H2_INTERVAL);
+ dompC1 = dompC2 = dompB;
+ map_to(dompC1, H1_INTERVAL);
+ map_to(dompC2, H2_INTERVAL);
+ iterate<intersection_point_tag>(domsB, domsA, pC1, pA,
+ dompC1, dompA, precision);
+ iterate<intersection_point_tag>(domsB, domsA, pC2, pA,
+ dompC2, dompA, precision);
+ }
+ return;
+ }
+
+ std::swap(C1, C2);
+ std::swap(dom1, dom2);
+#if VERBOSE
+ std::cerr << "dom(pA) : " << dompA << std::endl;
+ std::cerr << "dom(pB) : " << dompB << std::endl;
+#endif
+ }
+ domsA.push_back(dompA);
+ domsB.push_back(dompB);
+}
+
+
+/*
+ * iterate
+ *
+ * input:
+ * A, B: control point sets of two bezier curves
+ * domA, domB: real parameter intervals of the two curves
+ * precision: required computational precision of the returned parameter ranges
+ * output:
+ * domsA, domsB: sets of parameter intervals
+ *
+ * The parameter intervals are computed by using a Bezier clipping algorithm,
+ * in case the clipping doesn't shrink the initial interval more than 20%,
+ * a subdivision step is performed.
+ * If during the computation one of the two curve interval length becomes less
+ * than MAX_PRECISION the routine exits indipendently by the precision reached
+ * in the computation of the other curve interval.
+ */
+template <>
+void iterate<collinear_normal_tag> (std::vector<Interval>& domsA,
+ std::vector<Interval>& domsB,
+ std::vector<Point> const& A,
+ std::vector<Point> const& B,
+ Interval const& domA,
+ Interval const& domB,
+ double precision)
+{
+ // in order to limit recursion
+ static size_t counter = 0;
+ if (domA.extent() == 1 && domB.extent() == 1) counter = 0;
+ if (++counter > 100) return;
+#if VERBOSE
+ std::cerr << std::fixed << std::setprecision(16);
+ std::cerr << ">> curve subdision performed <<" << std::endl;
+ std::cerr << "dom(A) : " << domA << std::endl;
+ std::cerr << "dom(B) : " << domB << std::endl;
+// std::cerr << "angle(A) : " << angle(A) << std::endl;
+// std::cerr << "angle(B) : " << angle(B) << std::endl;
+#endif
+
+ if (precision < MAX_PRECISION)
+ precision = MAX_PRECISION;
+
+ std::vector<Point> pA = A;
+ std::vector<Point> pB = B;
+ std::vector<Point>* C1 = &pA;
+ std::vector<Point>* C2 = &pB;
+
+ Interval dompA = domA;
+ Interval dompB = domB;
+ Interval* dom1 = &dompA;
+ Interval* dom2 = &dompB;
+
+ Interval dom;
+
+ size_t iter = 0;
+ while (++iter < 100
+ && (dompA.extent() >= precision || dompB.extent() >= precision))
+ {
+#if VERBOSE
+ std::cerr << "iter: " << iter << std::endl;
+#endif
+ clip<collinear_normal_tag>(dom, *C1, *C2);
+
+ // [1,0] is utilized to represent an empty interval
+ if (dom == EMPTY_INTERVAL)
+ {
+#if VERBOSE
+ std::cerr << "dom: empty" << std::endl;
+#endif
+ return;
+ }
+#if VERBOSE
+ std::cerr << "dom : " << dom << std::endl;
+#endif
+ // all other cases where dom[0] > dom[1] are invalid
+ if (dom.min() > dom.max())
+ {
+ assert(dom.min() < dom.max());
+ }
+
+ map_to(*dom2, dom);
+
+ // it's better to stop before losing computational precision
+ if (iter > 1 && (dom2->extent() <= MAX_PRECISION))
+ {
+#if VERBOSE
+ std::cerr << "beyond max precision limit" << std::endl;
+#endif
+ break;
+ }
+
+ portion(*C2, dom);
+ if (iter > 1 && is_constant(*C2))
+ {
+#if VERBOSE
+ std::cerr << "new curve portion pC1 is constant" << std::endl;
+#endif
+ break;
+ }
+
+
+ // if we have clipped less than 20% than we need to subdive the curve
+ // with the largest domain into two sub-curves
+ if ( dom.extent() > MIN_CLIPPED_SIZE_THRESHOLD)
+ {
+#if VERBOSE
+ std::cerr << "clipped less than 20% : " << dom.extent() << std::endl;
+ std::cerr << "angle(pA) : " << angle(pA) << std::endl;
+ std::cerr << "angle(pB) : " << angle(pB) << std::endl;
+#endif
+ std::vector<Point> pC1, pC2;
+ Interval dompC1, dompC2;
+ if (dompA.extent() > dompB.extent())
+ {
+ if ((dompA.extent() / 2) < MAX_PRECISION)
+ {
+ break;
+ }
+ pC1 = pC2 = pA;
+ portion(pC1, H1_INTERVAL);
+ if (false && is_constant(pC1))
+ {
+#if VERBOSE
+ std::cerr << "new curve portion pC1 is constant" << std::endl;
+#endif
+ break;
+ }
+ portion(pC2, H2_INTERVAL);
+ if (is_constant(pC2))
+ {
+#if VERBOSE
+ std::cerr << "new curve portion pC2 is constant" << std::endl;
+#endif
+ break;
+ }
+ dompC1 = dompC2 = dompA;
+ map_to(dompC1, H1_INTERVAL);
+ map_to(dompC2, H2_INTERVAL);
+ iterate<collinear_normal_tag>(domsA, domsB, pC1, pB,
+ dompC1, dompB, precision);
+ iterate<collinear_normal_tag>(domsA, domsB, pC2, pB,
+ dompC2, dompB, precision);
+ }
+ else
+ {
+ if ((dompB.extent() / 2) < MAX_PRECISION)
+ {
+ break;
+ }
+ pC1 = pC2 = pB;
+ portion(pC1, H1_INTERVAL);
+ if (is_constant(pC1))
+ {
+#if VERBOSE
+ std::cerr << "new curve portion pC1 is constant" << std::endl;
+#endif
+ break;
+ }
+ portion(pC2, H2_INTERVAL);
+ if (is_constant(pC2))
+ {
+#if VERBOSE
+ std::cerr << "new curve portion pC2 is constant" << std::endl;
+#endif
+ break;
+ }
+ dompC1 = dompC2 = dompB;
+ map_to(dompC1, H1_INTERVAL);
+ map_to(dompC2, H2_INTERVAL);
+ iterate<collinear_normal_tag>(domsB, domsA, pC1, pA,
+ dompC1, dompA, precision);
+ iterate<collinear_normal_tag>(domsB, domsA, pC2, pA,
+ dompC2, dompA, precision);
+ }
+ return;
+ }
+
+ std::swap(C1, C2);
+ std::swap(dom1, dom2);
+#if VERBOSE
+ std::cerr << "dom(pA) : " << dompA << std::endl;
+ std::cerr << "dom(pB) : " << dompB << std::endl;
+#endif
+ }
+ domsA.push_back(dompA);
+ domsB.push_back(dompB);
+}
+
+
+/*
+ * get_solutions
+ *
+ * input: A, B - set of control points of two Bezier curve
+ * input: precision - required precision of computation
+ * input: clip - the routine used for clipping
+ * output: xs - set of pairs of parameter values
+ * at which the clipping algorithm converges
+ *
+ * This routine is based on the Bezier Clipping Algorithm,
+ * see: Sederberg - Computer Aided Geometric Design
+ */
+template <typename Tag>
+void get_solutions (std::vector< std::pair<double, double> >& xs,
+ std::vector<Point> const& A,
+ std::vector<Point> const& B,
+ double precision)
+{
+ std::pair<double, double> ci;
+ std::vector<Interval> domsA, domsB;
+ iterate<Tag> (domsA, domsB, A, B, UNIT_INTERVAL, UNIT_INTERVAL, precision);
+ if (domsA.size() != domsB.size())
+ {
+ assert (domsA.size() == domsB.size());
+ }
+ xs.clear();
+ xs.reserve(domsA.size());
+ for (size_t i = 0; i < domsA.size(); ++i)
+ {
+#if VERBOSE
+ std::cerr << i << " : domA : " << domsA[i] << std::endl;
+ std::cerr << "extent A: " << domsA[i].extent() << " ";
+ std::cerr << "precision A: " << get_precision(domsA[i]) << std::endl;
+ std::cerr << i << " : domB : " << domsB[i] << std::endl;
+ std::cerr << "extent B: " << domsB[i].extent() << " ";
+ std::cerr << "precision B: " << get_precision(domsB[i]) << std::endl;
+#endif
+ ci.first = domsA[i].middle();
+ ci.second = domsB[i].middle();
+ xs.push_back(ci);
+ }
+}
+
+} /* end namespace bezier_clipping */ } /* end namespace detail */
+
+
+/*
+ * find_collinear_normal
+ *
+ * input: A, B - set of control points of two Bezier curve
+ * input: precision - required precision of computation
+ * output: xs - set of pairs of parameter values
+ * at which there are collinear normals
+ *
+ * This routine is based on the Bezier Clipping Algorithm,
+ * see: Sederberg, Nishita, 1990 - Curve intersection using Bezier clipping
+ */
+void find_collinear_normal (std::vector< std::pair<double, double> >& xs,
+ std::vector<Point> const& A,
+ std::vector<Point> const& B,
+ double precision)
+{
+ using detail::bezier_clipping::get_solutions;
+ using detail::bezier_clipping::collinear_normal_tag;
+ get_solutions<collinear_normal_tag>(xs, A, B, precision);
+}
+
+
+/*
+ * find_intersections_bezier_clipping
+ *
+ * input: A, B - set of control points of two Bezier curve
+ * input: precision - required precision of computation
+ * output: xs - set of pairs of parameter values
+ * at which crossing happens
+ *
+ * This routine is based on the Bezier Clipping Algorithm,
+ * see: Sederberg, Nishita, 1990 - Curve intersection using Bezier clipping
+ */
+void find_intersections_bezier_clipping (std::vector< std::pair<double, double> >& xs,
+ std::vector<Point> const& A,
+ std::vector<Point> const& B,
+ double precision)
+{
+ using detail::bezier_clipping::get_solutions;
+ using detail::bezier_clipping::intersection_point_tag;
+ get_solutions<intersection_point_tag>(xs, A, B, precision);
+}
+
+} // end namespace Geom
+
+
+
+
+/*
+ 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:encoding=utf-8:textwidth=99 :
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