Flip Algorithm for Segment Triangulations
Given a set S of disjoint line segments in the plane, which we call sites, a segment triangulation of S is a partition of the convex hull of S into sites, edges, and faces. The set of faces is a maximal set of disjoint triangles such that the vertices of each triangle are on three distinct sites. The segment Delaunay triangulation of S is the segment triangulation of S whose faces are inscribable in circles whose interiors do not intersect S. It is dual to the segment Voronoi diagram. The aim of this paper is to show that any given segment triangulation can be transformed by a finite sequence of local improvements in a segment triangulation that has the same topology as the segment Delaunay triangulation. The main difference with the classical flip algorithm for point set triangulations is that local improvements have to be computed on non convex regions. We overcome this difficulty by using locally convex functions.
KeywordsConvex Hull Voronoi Diagram Delaunay Triangulation Local Improvement Polyhedral Surface
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- 1.Aichholzer, O., Aurenhammer, F., Hackl, T.: Pre-triangulations and liftable complexes. In: Proc. 22th Annu. ACM Sympos. Comput. Geom., pp. 282–291 (2006)Google Scholar
- 6.Bronsted, A.: An Introduction to Convex Polytopes. Graduate Texts in Mathematics. Springer, New York (1983)Google Scholar
- 7.Cheng, S., Dey, T.K.: Delaunay edge flips in dense surface triangulations. In: Proceedings of the 24th European Workshop on Computational Geometry, pp. 1–4 (2008)Google Scholar
- 8.Chew, L.P., Kedem, K.: Placing the largest similar copy of a convex polygon among polygonal obstacles. In: Proc. 5th Annu. ACM Sympos. Comput. Geom., pp. 167–174 (1989)Google Scholar
- 9.Edelsbrunner, H.: Triangulations and meshes in computational geometry. In: Acta Numerica, pp. 133–213 (2000)Google Scholar
- 15.Karavelas, M.I.: A robust and efficient implementation for the segment Voronoi diagram. In: Proceedings of the International Symposium on Voronoi Diagrams in Science and Engineering, pp. 51–62 (2004)Google Scholar
- 16.Lawson, C.L.: Software for C 1 surface interpolation. In: Rice, J.R. (ed.) Math. Software III, pp. 161–194. Academic Press, New York, NY (1977)Google Scholar