da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Implement the Bezier clipping algorithm for finding
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Bezier curve intersection points and collinear normals
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Authors:
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Marco Cecchetti <mrcekets at gmail.com>
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Copyright 2008 authors
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * This library is free software; you can redistribute it and/or
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * modify it either under the terms of the GNU Lesser General Public
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * License version 2.1 as published by the Free Software Foundation
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * (the "LGPL") or, at your option, under the terms of the Mozilla
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Public License Version 1.1 (the "MPL"). If you do not alter this
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * notice, a recipient may use your version of this file under either
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * the MPL or the LGPL.
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * You should have received a copy of the LGPL along with this library
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * in the file COPYING-LGPL-2.1; if not, write to the Free Software
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * You should have received a copy of the MPL along with this library
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * in the file COPYING-MPL-1.1
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * The contents of this file are subject to the Mozilla Public License
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Version 1.1 (the "License"); you may not use this file except in
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * compliance with the License. You may obtain a copy of the License at
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * OF ANY KIND, either express or implied. See the LGPL or the MPL for
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * the specific language governing rights and limitations.
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm//#include <iomanip>
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm////////////////////////////////////////////////////////////////////////////////
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// for debugging
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid print(std::vector<Point> const& cp, const char* msg = "")
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmtemplate< class charT >
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmoperator<< (std::basic_ostream<charT> & os, const Interval & I)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm double a = std::atan2(A[n][Y] - A[0][Y], A[n][X] - A[0][X]);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm double d = I.extent();
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm int n = 0;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm while (n < 16 && d < e)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid range_assertion(int k, int m, int n, const char* msg)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm if ( k < m || k > n)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm << "value: " << k
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm assert (k >= m && k <= n);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm////////////////////////////////////////////////////////////////////////////////
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// numerical routines
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute the binomial coefficient (n, k)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute the determinant of the 2x2 matrix with column the point P1, P2
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Solve the linear system [P1,P2] * P = Q
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * in case there isn't exactly one solution the routine returns false
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmbool solve(Point & P, Point const& P1, Point const& P2, Point const& Q)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm if (d == 0) return false;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm d = 1 / d;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm return true;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm////////////////////////////////////////////////////////////////////////////////
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// interval routines
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Map the sub-interval I in [0,1] into the interval J and assign it to J
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński J.setEnds(J.valueAt(I.min()), J.valueAt(I.max()));
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm////////////////////////////////////////////////////////////////////////////////
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// bezier curve routines
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Return true if all the Bezier curve control points are near,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * false otherwise
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński// Bezier.isConstant(precision)
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosińskibool is_constant(std::vector<Point> const& A, double precision)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm return false;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm return true;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute the hodograph of the bezier curve B and return it in D
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński// derivative(Bezier)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid derivative(std::vector<Point> & D, std::vector<Point> const& B)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm if (sz == 0) return;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm for (size_t i = 0; i < n; ++i)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute the hodograph of the Bezier curve B rotated of 90 degree
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * and return it in D; we have N(t) orthogonal to B(t) for any t
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński// rot90(derivative(Bezier))
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid normal(std::vector<Point> & N, std::vector<Point> const& B)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm N[i] = rot90(N[i]);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute the portion of the Bezier curve "B" wrt the interval [0,t]
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński// portion(Bezier, 0, t)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute the portion of the Bezier curve "B" wrt the interval [t,1]
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński// portion(Bezier, t, 1)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm for (size_t j = 0; j < n-i; ++j)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute the portion of the Bezier curve "B" wrt the interval "I"
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński// portion(Bezier, I)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid portion (std::vector<Point> & B , Interval const& I)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm if (I.min() == 0)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm////////////////////////////////////////////////////////////////////////////////
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmtemplate <typename Tag>
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof KosińskiOptInterval clip(std::vector<Point> const& A,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmtemplate <typename Tag>
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm////////////////////////////////////////////////////////////////////////////////
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// intersection
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Make up an orientation line using the control points c[i] and c[j]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * the line is returned in the output parameter "l" in the form of a 3 element
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * vector : l[0] * x + l[1] * y + l[2] == 0; the line is normalized.
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński// Line(c[i], c[j])
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm l[0] = c[j][Y] - c[i][Y];
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm l[1] = c[i][X] - c[j][X];
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Pick up an orientation line for the Bezier curve "c" and return it in
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * the output parameter "l"
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof KosińskiLine pick_orientation_line (std::vector<Point> const &c, double precision)
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński while (--i > 0 && are_near(c[0], c[i], precision))
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński // this should never happen because when a new curve portion is created
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński // we check that it is not constant;
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński // however this requires that the precision used in the is_constant
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński // routine has to be the same used here in the are_near test
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm //std::cerr << "i = " << i << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Make up an orientation line for constant bezier curve;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * the orientation line is made up orthogonal to the other curve base line;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * the line is returned in the output parameter "l" in the form of a 3 element
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * vector : l[0] * x + l[1] * y + l[2] == 0; the line is normalized.
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof KosińskiLine orthogonal_orientation_line (std::vector<Point> const &c,
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński // this should never happen
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński Line line(p, (c.back() - c.front()).cw() + p);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute the signed distance of the point "P" from the normalized line l
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosińskidouble signed_distance(Point const &p, Line const &l)
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński return a * p[X] + b * p[Y] + c;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute the min and max distance of the control points of the Bezier
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * curve "c" from the normalized orientation line "l".
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * This bounds are returned through the output Interval parameter"bound".
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof KosińskiInterval fat_line_bounds (std::vector<Point> const &c,
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński bound.expandTo(signed_distance(c[i], l));
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * return the x component of the intersection point between the line
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * passing through points p1, p2 and the line Y = "y"
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmdouble intersect (Point const& p1, Point const& p2, double y)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // we are sure that p2[Y] != p1[Y] because this routine is called
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // only when the lower or the upper bound is crossed
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Clip the Bezier curve "B" wrt the fat line defined by the orientation
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * line "l" and the interval range "bound", the new parameter interval for
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * the clipped curve is returned through the output parameter "dom"
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof KosińskiOptInterval clip_interval (std::vector<Point> const& B,
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński const double d = signed_distance(B[i], l);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm //print(D);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm //print(p);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "bound : " << bound << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "0 : inside " << p[0]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << i << " : inside " << p[i]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << " : tmin = " << tmin << ", tmax = " << tmax
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << i << " : lower " << p[i]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << " : tmin = " << tmin << ", tmax = " << tmax
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << i << " : higher " << p[i]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << " : tmin = " << tmin << ", tmax = " << tmax
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // we have to test the closing segment for intersection
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "0 : lower " << p[0]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "0 : higher " << p[0]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Clip the Bezier curve "B" wrt the Bezier curve "A" for individuating
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * intersection points the new parameter interval for the clipped curve
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * is returned through the output parameter "dom"
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmtemplate <>
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof KosińskiOptInterval clip<intersection_point_tag> (std::vector<Point> const& A,
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński bl = orthogonal_orientation_line(B, M, precision);
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński bl = pick_orientation_line(A, precision);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm///////////////////////////////////////////////////////////////////////////////
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// collinear normal
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute a closed focus for the Bezier curve B and return it in F
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * A focus is any curve through which all lines perpendicular to B(t) pass.
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid make_focus (std::vector<Point> & F, std::vector<Point> const& B)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << "make_focus: unable to make up a closed focus" << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "c = " << c << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // B(t) + c(t) * N(t)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm F[n] += B[n];
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm F[i] *= -c[0];
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm F[i] *= (i * n_inv);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm F[i] += B[i];
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm F[0] *= c[0];
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm F[0] += B[0];
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Compute the projection on the plane (t, d) of the control points
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * (t, u, D(t,u)) where D(t,u) = <(B(t) - F(u)), B'(t)> with 0 <= t, u <= 1
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * B is a Bezier curve and F is a focus of another Bezier curve.
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * See Sederberg, Nishita, 1990 - Curve intersection using Bezier clipping.
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm for (size_t k = 0; k < n; ++k)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm for (size_t i = 0; i < n; ++i)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm for (size_t i = 0; i < n; ++i)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm for (size_t i = 0; i <= r; ++i)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm for (size_t j = 0; j <= m; ++j)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm //if (k > i || (i-k) > n) continue;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm l = i - k;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm for (size_t j = 0; j <= m; ++j)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm //d[j] += bc * dot(dB[k], B[l] - F[j]);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm for (size_t j = 0; j < m; ++j)
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Clip the Bezier curve "B" wrt the focus "F"; the new parameter interval for
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * the clipped curve is returned through the output parameter "dom"
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof KosińskiOptInterval clip_interval (std::vector<Point> const& B,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm //print(D, "D");
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// ConvexHull chD(D);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::vector<Point>& p = chD.boundary; // convex hull vertices
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm //print(p, "CH(D)");
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm plower = (p[0][Y] < 0);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm if (p[0][Y] == 0) // on the x axis
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "0 : on x axis " << p[0]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm clower = (p[i][Y] < 0);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm if (p[i][Y] == 0) // on x axis
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << i << " : on x axis " << p[i]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << " : tmin = " << tmin << ", tmax = " << tmax
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << i << " : lower " << p[i]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << " : tmin = " << tmin << ", tmax = " << tmax
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // we have to test the closing segment for intersection
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm clower = (p[0][Y] < 0);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "0 : lower " << p[0]
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// << " : tmin = " << tmin << ", tmax = " << tmax << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * Clip the Bezier curve "B" wrt the Bezier curve "A" for individuating
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * points which have collinear normals; the new parameter interval
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * for the clipped curve is returned through the output parameter "dom"
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmtemplate <>
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof KosińskiOptInterval clip<collinear_normal_tag> (std::vector<Point> const& A,
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński double /*precision*/)
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosińskiconst Interval H2_INTERVAL(nextafter(0.5, 1.0), 1.0);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * A, B: control point sets of two bezier curves
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * domA, domB: real parameter intervals of the two curves
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * precision: required computational precision of the returned parameter ranges
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * domsA, domsB: sets of parameter intervals
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * The parameter intervals are computed by using a Bezier clipping algorithm,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * in case the clipping doesn't shrink the initial interval more than 20%,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * a subdivision step is performed.
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * If during the computation both curves collapse to a single point
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * the routine exits indipendently by the precision reached in the computation
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * of the curve intervals.
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmtemplate <>
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid iterate<intersection_point_tag> (std::vector<Interval>& domsA,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // in order to limit recursion
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm if (domA.extent() == 1 && domB.extent() == 1) counter = 0;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << ">> curve subdision performed <<" << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "angle(A) : " << angle(A) << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "angle(B) : " << angle(B) << std::endl;
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński if ( is_constant(A, precision) && is_constant(B, precision) ){
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm && (dompA.extent() >= precision || dompB.extent() >= precision))
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński dom = clip<intersection_point_tag>(*C1, *C2, precision);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // all other cases where dom[0] > dom[1] are invalid
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński if (is_constant(*C2, precision) && is_constant(*C1, precision))
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm break; // append the new interval
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm return; // exit without appending any new interval
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // if we have clipped less than 20% than we need to subdive the curve
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // with the largest domain into two sub-curves
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński if (dom->extent() > MIN_CLIPPED_SIZE_THRESHOLD)
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński std::cerr << "clipped less than 20% : " << dom->extent() << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * A, B: control point sets of two bezier curves
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * domA, domB: real parameter intervals of the two curves
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * precision: required computational precision of the returned parameter ranges
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * domsA, domsB: sets of parameter intervals
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * The parameter intervals are computed by using a Bezier clipping algorithm,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * in case the clipping doesn't shrink the initial interval more than 20%,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * a subdivision step is performed.
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * If during the computation one of the two curve interval length becomes less
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * than MAX_PRECISION the routine exits indipendently by the precision reached
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * in the computation of the other curve interval.
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmtemplate <>
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid iterate<collinear_normal_tag> (std::vector<Interval>& domsA,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // in order to limit recursion
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm if (domA.extent() == 1 && domB.extent() == 1) counter = 0;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << ">> curve subdision performed <<" << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "angle(A) : " << angle(A) << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm// std::cerr << "angle(B) : " << angle(B) << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm && (dompA.extent() >= precision || dompB.extent() >= precision))
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński dom = clip<collinear_normal_tag>(*C1, *C2, precision);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // it's better to stop before losing computational precision
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << "beyond max precision limit" << std::endl;
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński if (iter > 1 && is_constant(*C2, precision))
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << "new curve portion pC1 is constant" << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // if we have clipped less than 20% than we need to subdive the curve
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm // with the largest domain into two sub-curves
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński if ( dom->extent() > MIN_CLIPPED_SIZE_THRESHOLD)
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński std::cerr << "clipped less than 20% : " << dom->extent() << std::endl;
76addc201c409e81eaaa73fe27cc0f79c4db097cKrzysztof Kosiński if (false && is_constant(pC1, precision))
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << "new curve portion pC1 is constant" << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << "new curve portion pC2 is constant" << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << "new curve portion pC1 is constant" << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << "new curve portion pC2 is constant" << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * get_solutions
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * input: A, B - set of control points of two Bezier curve
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * input: precision - required precision of computation
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * input: clip - the routine used for clipping
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * output: xs - set of pairs of parameter values
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * at which the clipping algorithm converges
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * This routine is based on the Bezier Clipping Algorithm,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * see: Sederberg - Computer Aided Geometric Design
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmtemplate <typename Tag>
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid get_solutions (std::vector< std::pair<double, double> >& xs,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm iterate<Tag> (domsA, domsB, A, B, UNIT_INTERVAL, UNIT_INTERVAL, precision);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << i << " : domA : " << domsA[i] << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << "precision A: " << get_precision(domsA[i]) << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << i << " : domB : " << domsB[i] << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm std::cerr << "precision B: " << get_precision(domsB[i]) << std::endl;
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm} /* end namespace bezier_clipping */ } /* end namespace detail */
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * find_collinear_normal
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * input: A, B - set of control points of two Bezier curve
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * input: precision - required precision of computation
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * output: xs - set of pairs of parameter values
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * at which there are collinear normals
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * This routine is based on the Bezier Clipping Algorithm,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * see: Sederberg, Nishita, 1990 - Curve intersection using Bezier clipping
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid find_collinear_normal (std::vector< std::pair<double, double> >& xs,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm get_solutions<collinear_normal_tag>(xs, A, B, precision);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * find_intersections_bezier_clipping
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * input: A, B - set of control points of two Bezier curve
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * input: precision - required precision of computation
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * output: xs - set of pairs of parameter values
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * at which crossing happens
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * This routine is based on the Bezier Clipping Algorithm,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm * see: Sederberg, Nishita, 1990 - Curve intersection using Bezier clipping
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrmvoid find_intersections_bezier_clipping (std::vector< std::pair<double, double> >& xs,
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm get_solutions<intersection_point_tag>(xs, A, B, precision);
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm} // end namespace Geom
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm Local Variables:
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm c-file-style:"stroustrup"
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm indent-tabs-mode:nil
da58597f9f9ecb17c4f545c4483a844a363bcc27pjrm fill-column:99
a4030d5ca449e7e384bc699cd249ee704faaeab0Chris Morgan// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 :