By Øyvind Hjelle

This ebook will function a useful resource of data approximately triangulations for the graduate scholar and researcher. With emphasis on computational matters, it offers the fundamental conception essential to build and manage triangulations. particularly, the booklet provides a journey during the idea in the back of the Delaunay triangulation, together with algorithms and software program matters. It additionally discusses quite a few info buildings used for the illustration of triangulations.

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3 Voronoi Diagrams 53 Fig. 4. The Voronoi diagram of a set of points in the plane. convex hull of P are unbounded and that the other regions are bounded. 2), a Voronoi region is the intersection of half-planes, the Voronoi regions are also convex. The collection of all Voronoi regions is called the Voronoi diagram, or the Dirichlet tessellation, of the given point set P . Further, the boundaries of the Voronoi regions are often referred to as Voronoi polygons, the edges of the polygons as Voronoi edges and the vertices of the Voronoi polygons as Voronoi points.

For in-depth theory, analysis and completeness, the interested reader is referred to literature on computational geometry, such as the book by Preparata & Shamos [68] and references given there. 5. It can be shown that this construction always results in a subdivision of the plane into a collection of non-overlapping adjacent triangles covering the convex hull of P . The line segments connecting Voronoi neighbors associated with unbounded Voronoi regions deﬁne the convex hull of P , and the other line segments deﬁne common edges of pairs of adjacent triangles.

The storage requirement for the topological information is twice as much as for the minimal data structure above. We get NP = 6|T | and thus, NP ≤ 12|V | − 30 and NP ≈ 12|V | in number of pointer ﬁelds when using the upper bound |T | ≤ 2|V | − 5 and the estimate |T | ≈ 2|V |. 4. Given a triangle ti,j,k we know exactly what triangles are adjacent to it and where they are positioned relative to the vertices and the edges of ti,j,k . The α0 - and α1 -iterators have the same complexity as in the minimal data structure, but α2 can be carried out in O(1) operations (constant time) as opposed to O(|T |) for the minimal structure.