diff options
| author | Peter Moulder <peter.moulder@monash.edu> | 2009-04-07 06:32:25 +0000 |
|---|---|---|
| committer | pjrm <pjrm@users.sourceforge.net> | 2009-04-07 06:32:25 +0000 |
| commit | 4f0165ebdaf2a07079d6b9f166a585fd89e8b064 (patch) | |
| tree | b56a880772fe7c03e4c9957d54da53bc9cfc85f5 /src/2geom | |
| parent | noop: svg/svg-path-geom-test.h: Change to consistent end-of-line separators, ... (diff) | |
| download | inkscape-4f0165ebdaf2a07079d6b9f166a585fd89e8b064.tar.gz inkscape-4f0165ebdaf2a07079d6b9f166a585fd89e8b064.zip | |
noop: Set svn:eol-style to native on all .cpp and .h files under src. (find \( -name '*.cpp' -o -name '*.h' \) -print0 | xargs -0 svn propset svn:eol-style native)
(bzr r7649)
Diffstat (limited to 'src/2geom')
| -rw-r--r-- | src/2geom/bezier-clipping.cpp | 2584 | ||||
| -rw-r--r-- | src/2geom/chebyshev.cpp | 252 | ||||
| -rw-r--r-- | src/2geom/chebyshev.h | 60 | ||||
| -rw-r--r-- | src/2geom/utils.cpp | 174 |
4 files changed, 1535 insertions, 1535 deletions
diff --git a/src/2geom/bezier-clipping.cpp b/src/2geom/bezier-clipping.cpp index c8d99c430..96a06376c 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 : diff --git a/src/2geom/chebyshev.cpp b/src/2geom/chebyshev.cpp index 4ba3e2325..447c5183f 100644 --- a/src/2geom/chebyshev.cpp +++ b/src/2geom/chebyshev.cpp @@ -1,126 +1,126 @@ -#include <2geom/chebyshev.h>
-
-#include <2geom/sbasis.h>
-#include <2geom/sbasis-poly.h>
-
-#include <vector>
-using std::vector;
-
-#include <gsl/gsl_math.h>
-#include <gsl/gsl_chebyshev.h>
-
-namespace Geom{
-
-SBasis cheb(unsigned n) {
- static std::vector<SBasis> basis;
- if(basis.empty()) {
- basis.push_back(Linear(1,1));
- basis.push_back(Linear(0,1));
- }
- for(unsigned i = basis.size(); i <= n; i++) {
- basis.push_back(Linear(0,2)*basis[i-1] - basis[i-2]);
- }
-
- return basis[n];
-}
-
-SBasis cheb_series(unsigned n, double* cheb_coeff) {
- SBasis r;
- for(unsigned i = 0; i < n; i++) {
- double cof = cheb_coeff[i];
- //if(i == 0)
- //cof /= 2;
- r += cheb(i)*cof;
- }
-
- return r;
-}
-
-SBasis clenshaw_series(unsigned m, double* cheb_coeff) {
- /** b_n = a_n
- b_n-1 = 2*x*b_n + a_n-1
- b_n-k = 2*x*b_{n-k+1} + a_{n-k} - b_{n - k + 2}
- b_0 = x*b_1 + a_0 - b_2
- */
-
- double a = -1, b = 1;
- SBasis d, dd;
- SBasis y = (Linear(0, 2) - (a+b)) / (b-a);
- SBasis y2 = 2*y;
- for(int j = m - 1; j >= 1; j--) {
- SBasis sv = d;
- d = y2*d - dd + cheb_coeff[j];
- dd = sv;
- }
-
- return y*d - dd + 0.5*cheb_coeff[0];
-}
-
-SBasis chebyshev_approximant (double (*f)(double,void*), int order, Interval in, void* p) {
- gsl_cheb_series *cs = gsl_cheb_alloc (order+2);
-
- gsl_function F;
-
- F.function = f;
- F.params = p;
-
- gsl_cheb_init (cs, &F, in[0], in[1]);
-
- SBasis r = compose(clenshaw_series(order, cs->c), Linear(-1,1));
-
- gsl_cheb_free (cs);
- return r;
-}
-
-struct wrap {
- double (*f)(double,void*);
- void* pp;
- double fa, fb;
- Interval in;
-};
-
-double f_interp(double x, void* p) {
- struct wrap *wr = (struct wrap *)p;
- double z = (x - wr->in[0]) / (wr->in[1] - wr->in[0]);
- return (wr->f)(x, wr->pp) - ((1 - z)*wr->fa + z*wr->fb);
-}
-
-SBasis chebyshev_approximant_interpolating (double (*f)(double,void*),
- int order, Interval in, void* p) {
- double fa = f(in[0], p);
- double fb = f(in[1], p);
- struct wrap wr;
- wr.fa = fa;
- wr.fb = fb;
- wr.in = in;
- printf("%f %f\n", fa, fb);
- wr.f = f;
- wr.pp = p;
- return compose(Linear(in[0], in[1]), Linear(fa, fb)) + chebyshev_approximant(f_interp, order, in, &wr) + Linear(fa, fb);
-}
-
-SBasis chebyshev(unsigned n) {
- static std::vector<SBasis> basis;
- if(basis.empty()) {
- basis.push_back(Linear(1,1));
- basis.push_back(Linear(0,1));
- }
- for(unsigned i = basis.size(); i <= n; i++) {
- basis.push_back(Linear(0,2)*basis[i-1] - basis[i-2]);
- }
-
- return basis[n];
-}
-
-};
-
-/*
- 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 :
+#include <2geom/chebyshev.h> + +#include <2geom/sbasis.h> +#include <2geom/sbasis-poly.h> + +#include <vector> +using std::vector; + +#include <gsl/gsl_math.h> +#include <gsl/gsl_chebyshev.h> + +namespace Geom{ + +SBasis cheb(unsigned n) { + static std::vector<SBasis> basis; + if(basis.empty()) { + basis.push_back(Linear(1,1)); + basis.push_back(Linear(0,1)); + } + for(unsigned i = basis.size(); i <= n; i++) { + basis.push_back(Linear(0,2)*basis[i-1] - basis[i-2]); + } + + return basis[n]; +} + +SBasis cheb_series(unsigned n, double* cheb_coeff) { + SBasis r; + for(unsigned i = 0; i < n; i++) { + double cof = cheb_coeff[i]; + //if(i == 0) + //cof /= 2; + r += cheb(i)*cof; + } + + return r; +} + +SBasis clenshaw_series(unsigned m, double* cheb_coeff) { + /** b_n = a_n + b_n-1 = 2*x*b_n + a_n-1 + b_n-k = 2*x*b_{n-k+1} + a_{n-k} - b_{n - k + 2} + b_0 = x*b_1 + a_0 - b_2 + */ + + double a = -1, b = 1; + SBasis d, dd; + SBasis y = (Linear(0, 2) - (a+b)) / (b-a); + SBasis y2 = 2*y; + for(int j = m - 1; j >= 1; j--) { + SBasis sv = d; + d = y2*d - dd + cheb_coeff[j]; + dd = sv; + } + + return y*d - dd + 0.5*cheb_coeff[0]; +} + +SBasis chebyshev_approximant (double (*f)(double,void*), int order, Interval in, void* p) { + gsl_cheb_series *cs = gsl_cheb_alloc (order+2); + + gsl_function F; + + F.function = f; + F.params = p; + + gsl_cheb_init (cs, &F, in[0], in[1]); + + SBasis r = compose(clenshaw_series(order, cs->c), Linear(-1,1)); + + gsl_cheb_free (cs); + return r; +} + +struct wrap { + double (*f)(double,void*); + void* pp; + double fa, fb; + Interval in; +}; + +double f_interp(double x, void* p) { + struct wrap *wr = (struct wrap *)p; + double z = (x - wr->in[0]) / (wr->in[1] - wr->in[0]); + return (wr->f)(x, wr->pp) - ((1 - z)*wr->fa + z*wr->fb); +} + +SBasis chebyshev_approximant_interpolating (double (*f)(double,void*), + int order, Interval in, void* p) { + double fa = f(in[0], p); + double fb = f(in[1], p); + struct wrap wr; + wr.fa = fa; + wr.fb = fb; + wr.in = in; + printf("%f %f\n", fa, fb); + wr.f = f; + wr.pp = p; + return compose(Linear(in[0], in[1]), Linear(fa, fb)) + chebyshev_approximant(f_interp, order, in, &wr) + Linear(fa, fb); +} + +SBasis chebyshev(unsigned n) { + static std::vector<SBasis> basis; + if(basis.empty()) { + basis.push_back(Linear(1,1)); + basis.push_back(Linear(0,1)); + } + for(unsigned i = basis.size(); i <= n; i++) { + basis.push_back(Linear(0,2)*basis[i-1] - basis[i-2]); + } + + return basis[n]; +} + +}; + +/* + 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 : diff --git a/src/2geom/chebyshev.h b/src/2geom/chebyshev.h index 309e09b3e..6de9e9cc0 100644 --- a/src/2geom/chebyshev.h +++ b/src/2geom/chebyshev.h @@ -1,30 +1,30 @@ -#ifndef _CHEBYSHEV
-#define _CHEBYSHEV
-
-#include <2geom/sbasis.h>
-#include <2geom/interval.h>
-
-/*** Conversion between Chebyshev approximation and SBasis.
- *
- */
-
-namespace Geom{
-
-SBasis chebyshev_approximant (double (*f)(double,void*), int order, Interval in, void* p=0);
-SBasis chebyshev_approximant_interpolating (double (*f)(double,void*), int order, Interval in, void* p=0);
-SBasis chebyshev(unsigned n);
-
-};
-
-/*
- 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 :
-
-#endif
+#ifndef _CHEBYSHEV +#define _CHEBYSHEV + +#include <2geom/sbasis.h> +#include <2geom/interval.h> + +/*** Conversion between Chebyshev approximation and SBasis. + * + */ + +namespace Geom{ + +SBasis chebyshev_approximant (double (*f)(double,void*), int order, Interval in, void* p=0); +SBasis chebyshev_approximant_interpolating (double (*f)(double,void*), int order, Interval in, void* p=0); +SBasis chebyshev(unsigned n); + +}; + +/* + 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 : + +#endif diff --git a/src/2geom/utils.cpp b/src/2geom/utils.cpp index f0f42ce01..579718553 100644 --- a/src/2geom/utils.cpp +++ b/src/2geom/utils.cpp @@ -1,87 +1,87 @@ -/** Various utility functions.
- *
- * Copyright 2008 Marco Cecchetti <mrcekets at gmail.com>
- * Copyright 2007 Johan Engelen <goejendaagh@zonnet.nl>
- * Copyright 2006 Michael G. Sloan <mgsloan@gmail.com>
- *
- * 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/utils.h>
-
-
-namespace Geom
-{
-
-// return a vector that contains all the binomial coefficients of degree n
-void binomial_coefficients(std::vector<size_t>& bc, size_t n)
-{
- size_t s = n+1;
- bc.clear();
- bc.resize(s);
- bc[0] = 1;
- size_t k;
- for (size_t i = 1; i < n; ++i)
- {
- k = i >> 1;
- if (i & 1u)
- {
- bc[k+1] = bc[k] << 1;
- }
- for (size_t j = k; j > 0; --j)
- {
- bc[j] += bc[j-1];
- }
- }
- s >>= 1;
- for (size_t i = 0; i < s; ++i)
- {
- bc[n-i] = bc[i];
- }
-}
-
-} // 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 :
+/** Various utility functions. + * + * Copyright 2008 Marco Cecchetti <mrcekets at gmail.com> + * Copyright 2007 Johan Engelen <goejendaagh@zonnet.nl> + * Copyright 2006 Michael G. Sloan <mgsloan@gmail.com> + * + * 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/utils.h> + + +namespace Geom +{ + +// return a vector that contains all the binomial coefficients of degree n +void binomial_coefficients(std::vector<size_t>& bc, size_t n) +{ + size_t s = n+1; + bc.clear(); + bc.resize(s); + bc[0] = 1; + size_t k; + for (size_t i = 1; i < n; ++i) + { + k = i >> 1; + if (i & 1u) + { + bc[k+1] = bc[k] << 1; + } + for (size_t j = k; j > 0; --j) + { + bc[j] += bc[j-1]; + } + } + s >>= 1; + for (size_t i = 0; i < s; ++i) + { + bc[n-i] = bc[i]; + } +} + +} // 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 : |