# Fast triangulation of simple polygons

## 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## Preview

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## References

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