FCT 1983: Foundations of Computation Theory pp 207-218

Fast triangulation of simple polygons

• Stefan Hertel
• Kurt Mehlhorn
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 158)

Abstract

We present a new algorithm for triangulating simple polygons that has four advantages over previous solutions [GJPT, Ch].

a) It is faster: Whilst previous solutions worked in time O(nlogn), the new algorithm only needs time O(n+rlogr) where r is the number of concave angles of the polygon.

b) It works for a larger class of inputs: Whilst previous solutions worked for simple polygons, the new algorithm handles simple polygons with polygonal holes.

c) It does more: Whilst previous solutions only triangulated the interior of a simple polygon, the new algorithm triangulates both the interior and the exterior region.

d) It is simpler: The algorithm is based on the plane-sweep paradigm and is — at least in its O(nlogn) version — very simple.

In addition to the new triangulation algorithm, we present two new applications of triangulation.

a) We show that one can compute the intersection of a convex m-gon Q and a triangulated simple n-gon P in time O(n+m). This improves a result by Shamos [Sh] stating that the intersection of two convex polygons can be computed in time O(n).

b) Given the triangulation of a simple n-gon P, we show how to compute in time O(n) a convex decomposition of P into at most 4·OPT pieces. Here OPT denotes the minimum number of pieces in any convex decomposition. The best factor known so far was 4.333 (Chazelle[Ch]).

Keywords

Convex Polygon Simple Polygon Polygonal Region Triangulation Algorithm Sweeping Line
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

1. [AHU]
A.V.Aho/J.E.Hopcroft/J.D.Ullman: The Design and Analysis of Computer Algorithms, Addison-Wesley Publ. Comp., Reading, Mass., 1974.Google Scholar
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J.L. Bentley/T.A. Ottmann: Algorithms for Reporting and Counting Geometric Intersections, IEEE Trans. on Comp., Vol. C-28, No. 9 (1975), pp. 643–647.Google Scholar
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B.Chazelle: A Theorem on Polygon Cutting with Applications, Proc. 23rd IEEE FOCS Symp. (1982), pp. 339–349.Google Scholar
4. [GJPT]
M.R. Garey/D.S. Johnson/F.P. Preparata/R.E. Tarjan: Triangulating a Simple Polygon, Info. Proc. Letters, Vol. 7(4), June 1978, pp. 175–179.
5. [LP]
D.T. Lee/F.P. Preparata: Location of a Point in a Planar Subdivision and its Applications, SIAM J. Comp., Vol. 6(1977), pp. 594–606.Google Scholar
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R.J.Lipton/R.E.Tarjan: Applications of a Planar Separator Theorem, Proc. 18th IEEE FOCS Symp. (1977), pp. 162–170.Google Scholar
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J. Nievergelt/F.P. Preparata: Plane-Sweep Algorithms for Intersecting Geometric Figures, CACM 25, 10(Oct. 1982), pp. 739–747.Google Scholar
8. [Sh]
M.I.Shamos: Geometric Complexity, Proc. 7th ACM STOC (1975), pp. 224–233.Google Scholar

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

• Stefan Hertel
• 1
• Kurt Mehlhorn
• 1
1. 1.Universität des SaarlandesSaarbrücken

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