Delaunay Triangulations

In Chapter 2 we have discussed triangulations of simple polygons. A triangulation nicely partitions a polygon into triangles, which allows, for instance, to easily compute the area or a guarding of the polygon. Another typical application scenario is to use a triangulation T for interpolation: Suppose a function f is defined on the vertices of the polygon P, and we want to extend it " reasonably " and continuously to P •. Then for a point p ∈ P • find a triangle t of T that contains p. As p can be written as a convex combination 3 i=1 λ i v i of the vertices v 1 , v 2 , v 3 of t, we just use the same coefficients to obtain an interpolation f(p) := 3 i=1 λ i f(v i) of the function values. If triangulations are a useful tool when working with polygons, they might also turn out useful to deal with other geometric objects, for instance, point sets. But what could be a triangulation of a point set? Polygons have a clearly defined interior, which naturally lends itself to be covered by smaller polygons such as triangles. A point set does not have an interior, except. .. Here the notion of convex hull comes handy, because it allows us to treat a point set as a convex polygon. Actually, not really a convex polygon, because points in the interior of the convex hull should not be ignored completely. But one way to think of a point set is as a convex polygon—its convex hull—possibly with some holes— which are points—in its interior. A triangulation should then partition the convex hull while respecting the points in the interior, as shown in the example in Figure 6.1b. (a) Simple polygon triangulation. (b) Point set triangulation. (c) Not a triangulation. Figure 6.1: Examples of (non-)triangulations.