Computing the Umbrella Neighbourhood of a Vertex in the Delaunay Triangulation and a Single Voronoi Cell in Arbitrary Dimension

Delaunay triangulations and their geometric dual Voronoi diagrams are carefully studied topics in computational geometry and have numerous applications spanning fields such as physics, engineering, geographic information systems, and computer graphics. There are numerous efficient algorithms for computing both in two and three dimensional real space, but for higher dimensional space, the computational complexity grows exponentially. For many applications, it is only necessary to compute the star of simplices incident to (often called the umbrella neighbourhood of) a single vertex of the Delaunay triangulation, or equivalently, the Voronoi cell of a point. In practice, this may be a relatively small subset of the total Delaunay triangulation or Voronoi diagram. In this paper, an algorithm is proposed for computing the umbrella neighbourhood of a single vertex in the Delaunay triangulation.