Three-dimensional alpha shapes

Abstract

Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the “shape” of the set. For that purpose this paper introduces the formal notion of the family of r-x-shapes of a finite point set in IR3. Each shape is a polytope, derived from the Delaunay triangulation of the point set, with a parameter cy E R controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size n in worst-case time O(n’). A robust implementation of the algorithm is discussed and seveml applications in the area of scientific computing are mentioned.