Three-dimensional alpha shapes

TL;DR

This paper introduces the concept of alpha shapes in three dimensions, providing a formal definition, an algorithm for their construction, and discussing applications in scientific computing.

Contribution

It formalizes 3D alpha shapes, presents an efficient algorithm for their construction, and discusses practical implementation and applications.

Findings

Algorithm constructs alpha shapes in O(n^2) time.

Provides a robust implementation of the alpha shape algorithm.

Demonstrates applications in scientific computing.

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 $\alpha$-shapes of a finite point set in $\Real^3$. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter $\alpha \in \Real$ 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 time $O(n^2)$, worst case. A robust implementation of the algorithm is discussed and several applications in the area of scientific computing are mentioned.