A Robust Implementation for Three-Dimensional Delaunay Triangulations

Abstract

This paper presents an implementation for Delaunay triangulations of three-dimensional point sets. The code uses a variant of the randomized incremental flip algorithm and employs symbolic perturbation to achieve robustness. The algorithm's theoretical time complexity is quadratic in n, the number of input points, and this is optimal in the worst case. However, empirical running times are proportional to the number of triangles in the final triangulation, which is typically linear in n. Even though the symbolic perturbation scheme relies on exact arithmetic, the resulting code is efficient in practice. This is due to a careful implementation of the geometric primitives and the arithmetic module. The source code is available on the Internet.