Three-dimensional alpha shapes

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.Comment: 32 page

Kinetic and Dynamic Delaunay tetrahedralizations in three dimensions

We describe the implementation of algorithms to construct and maintain three-dimensional dynamic Delaunay triangulations with kinetic vertices using a three-simplex data structure. The code is capable of constructing the geometric dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points is triangulated. Time evolution of the triangulation is not only governed by kinetic vertices but also by a changing number of vertices. We use three-dimensional simplex flip algorithms, a stochastic visibility walk algorithm for point location and in addition, we propose a new simple method of deleting vertices from an existing three-dimensional Delaunay triangulation while maintaining the Delaunay property. The dual Dirichlet tessellation can be used to solve differential equations on an irregular grid, to define partitions in cell tissue simulations, for collision detection etc.Comment: 29 pg (preprint), 12 figures, 1 table Title changed (mainly nomenclature), referee suggestions included, typos corrected, bibliography update