2D Intersection of Curves

29.1 | Introduction | ||||

29.2 | Example | ||||

29.3 | Design and Implementation History |

Let C = {C_{1}, C_{2}, … , C_{n}} be a set of curves.
We wish to compute all
intersection
points between
two curves in the set in an output-sensitive manner, without having to
go over all O(n^{2}) curve pairs. To this end, we sweep an imaginary line l
from x = -∞ to x = ∞ over the plane. While sweeping
the plane, we keep track of the order of curves intersecting it.
This order changes at a finite number of *event points*, such that
we only have to calculate the
intersection
points
between two curves when they become contiguous. For more details on the
*sweep-line algorithm* see, for example, [dBvKOS00, Chapter 2].

This chapter describes three functions implemented using the sweep-line algorithm: given a collection of input curves, compute all intersection points, compute the set of subcurves that are pairwise interior-disjoint induced by them, and checking whether there is at least one pair of curves among them that intersect in their interior.

The implementation is robust. It supports general
curves and handles all degenerate cases, including overlapping curves,
vertical segments, and tangency between curves. The robustness of the
algorithm is guaranteed if the functions are instantiated with a traits
class that employs certified computations. This traits class must be a model
of the *ArrangementTraits_2* concept - see the
Chapter 28 for more details.

The complexity of the sweep-line algorithm is O((n + k)log n) where n is the number of the input curves and k is the number of intersection points induced by these curves.

File:examples/Arrangement_on_surface_2/sweep_line.cpp

#include <CGAL/Cartesian.h> #include <CGAL/MP_Float.h> #include <CGAL/Quotient.h> #include <CGAL/Arr_segment_traits_2.h> #include <CGAL/Sweep_line_2_algorithms.h> #include <list> typedef CGAL::Quotient<CGAL::MP_Float> NT; typedef CGAL::Cartesian<NT> Kernel; typedef Kernel::Point_2 Point_2; typedef CGAL::Arr_segment_traits_2<Kernel> Traits_2; typedef Traits_2::Curve_2 Segment_2; int main() { // Construct the input segments. Segment_2 segments[] = {Segment_2 (Point_2 (1, 5), Point_2 (8, 5)), Segment_2 (Point_2 (1, 1), Point_2 (8, 8)), Segment_2 (Point_2 (3, 1), Point_2 (3, 8)), Segment_2 (Point_2 (8, 5), Point_2 (8, 8))}; // Compute all intersection points. std::list<Point_2> pts; CGAL::compute_intersection_points (segments, segments + 4, std::back_inserter (pts)); // Print the result. std::cout << "Found " << pts.size() << " intersection points: " << std::endl; std::copy (pts.begin(), pts.end(), std::ostream_iterator<Point_2>(std::cout, "\n")); // Compute the non-intersecting sub-segments induced by the input segments. std::list<Segment_2> sub_segs; CGAL::compute_subcurves(segments, segments + 4, std::back_inserter(sub_segs)); std::cout << "Found " << sub_segs.size() << " interior-disjoint sub-segments." << std::endl; CGAL_assertion (CGAL::do_curves_intersect (segments, segments + 4)); return 0; }

Next: Reference Manual

CGAL Open Source Project.
Release 3.5.
1 October 2009.