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| -rw-r--r-- | src/2geom/orphan-code/intersection-by-bezier-clipping.cpp | 560 |
1 files changed, 560 insertions, 0 deletions
diff --git a/src/2geom/orphan-code/intersection-by-bezier-clipping.cpp b/src/2geom/orphan-code/intersection-by-bezier-clipping.cpp new file mode 100644 index 0000000..c55f623 --- /dev/null +++ b/src/2geom/orphan-code/intersection-by-bezier-clipping.cpp @@ -0,0 +1,560 @@ + +/* + * Find intersecions between two Bezier curves. + * The intersection points are found by using Bezier clipping. + * + * 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/bezier.h> +#include <2geom/interval.h> +#include <2geom/convex-hull.h> + + +#include <vector> +#include <utility> +#include <iomanip> + + +namespace Geom { + +namespace detail { namespace bezier_clipping { + + +//////////////////////////////////////////////////////////////////////////////// +// for debugging +// + +inline +void print(std::vector<Point> const& cp) +{ + 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 = 1, p = 1; + size_t n = 0; + while (n < 16 && (are_near(d, 0, e))) + { + p *= 10; + e = 1 /p; + ++n; + } + return n; +} + +//////////////////////////////////////////////////////////////////////////////// + + +/* + * 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; +} + +/* + * 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; +} + +/* + * 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& 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); + ConvexHull chD(D); + std::vector<Point> & p = chD.boundary; // convex hull vertices + + //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; +} + +/* + * Compute the portion of the Bezier curve "B" wrt the interval "I" + */ +void portion (std::vector<Point> & B, Interval const& I) +{ + Bezier::Order bo(B.size()-1); + Bezier Bx(bo), By(bo); + for (size_t i = 0; i < B.size(); ++i) + { + Bx[i] = B[i][X]; + By[i] = B[i][Y]; + } + Bx = portion(Bx, I.min(), I.max()); + By = portion(By, I.min(), I.max()); + assert (Bx.size() == By.size()); + B.resize(Bx.size()); + for (size_t i = 0; i < Bx.size(); ++i) + { + B[i][X] = Bx[i]; + B[i][Y] = By[i]; + } +} + +/* + * 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; +} + + + + +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); + +/* + * intersection + * + * 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 describing an intersection point + * + * 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 independently by the precision reached + * in the computation of the other curve interval. + */ +void intersection (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) +{ +// 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; + + + 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; + + std::vector<double> bl(3); + Interval bound, dom; + + + size_t iter = 0; + while (++iter < 100 + && (dompA.extent() >= precision || dompB.extent() >= precision)) + { +// std::cerr << "iter: " << iter << std::endl; + + pick_orientation_line(bl, *C1); + fat_line_bounds(bound, *C1, bl); + clip(dom, *C2, bl, bound); + + // [1,0] is utilized to represent an empty interval + if (dom == EMPTY_INTERVAL) + { +// std::cerr << "dom: empty" << std::endl; + return; + } +// std::cerr << "dom : " << dom << std::endl; + + // 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 (dom2->extent() <= MAX_PRECISION) + { +// std::cerr << "beyond max precision limit" << std::endl; + break; + } + + portion(*C2, dom); + if (is_constant(*C2)) + { +// std::cerr << "new curve portion is constant" << std::endl; + 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) + { +// 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; + + 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); + portion(pC2, H2_INTERVAL); + dompC1 = dompC2 = dompA; + map_to(dompC1, H1_INTERVAL); + map_to(dompC2, H2_INTERVAL); + intersection(domsA, domsB, pC1, pB, dompC1, dompB, precision); + intersection(domsA, domsB, pC2, pB, dompC2, dompB, precision); + } + else + { + if ((dompB.extent() / 2) < MAX_PRECISION) + { + break; + } + pC1 = pC2 = pB; + portion(pC1, H1_INTERVAL); + portion(pC2, H2_INTERVAL); + dompC1 = dompC2 = dompB; + map_to(dompC1, H1_INTERVAL); + map_to(dompC2, H2_INTERVAL); + intersection(domsB, domsA, pC1, pA, dompC1, dompA, precision); + intersection(domsB, domsA, pC2, pA, dompC2, dompA, precision); + } + return; + } + + using std::swap; + swap(C1, C2); + swap(dom1, dom2); +// std::cerr << "dom(pA) : " << dompA << std::endl; +// std::cerr << "dom(pB) : " << dompB << std::endl; + } + domsA.push_back(dompA); + domsB.push_back(dompB); +} + +} /* end namespace bezier_clipping */ } /* end namespace detail */ + + +/* + * find_intersection + * + * 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 - Computer Aided Geometric Design + */ +void find_intersections (std::vector< std::pair<double, double> > & xs, + std::vector<Point> const& A, + std::vector<Point> const& B, + double precision) +{ + std::cout << "find_intersections: intersection-by-clipping.cpp version\n"; +// std::cerr << std::fixed << std::setprecision(16); + + using detail::bezier_clipping::get_precision; + using detail::bezier_clipping::operator<<; + using detail::bezier_clipping::intersection; + using detail::bezier_clipping::UNIT_INTERVAL; + + std::pair<double, double> ci; + std::vector<Interval> domsA, domsB; + intersection (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) + { +// std::cerr << i << " : domA : " << domsA[i] << std::endl; +// std::cerr << "precision A: " << get_precision(domsA[i]) << std::endl; +// std::cerr << i << " : domB : " << domsB[i] << std::endl; +// std::cerr << "precision B: " << get_precision(domsB[i]) << std::endl; + + ci.first = domsA[i].middle(); + ci.second = domsB[i].middle(); + xs.push_back(ci); + } +} + +} // 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:fileencoding=utf-8:textwidth=99 : |