Delaunay triangulations in three dimensions with finite precision arithmetic

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Geometric algorithms, when implemented, often fail due to the degeneracies in the input data and numerical errors introduced by the finite precision arithmetic computations. In general, these algorithms deal with two types of data-numerical and combinatorial. Combinatorial inferences such as face adjacencies and vertex adjacencies are derived from the numerical data. Thus, inaccuracies in numerical computations may cause inconsistencies in combinatorial data which in effect either produces invalid output or makes the program fail. The ability of geometric algorithms to deal with the degeneracies and the inaccuracies during various numerical computations is referred to as their robustness. Several frameworks for achieving robustness have been proposed by different researchers. [Edelsbrunner & Miick ‘901 and [Yap ‘881 suggest using symbolic perturbation techniques to handle geometric degeneracies. [Sugihara & Iri ‘89a1 and [Dobkin & Silver ‘881 describe an approach to achieve consistent computations in solid modeling by ensuring that computations are carried out with sufficiently higher precision than that used for representing the numerical data. However, there are drawbacks as high precision routines are needed for all primitive numerical computations makin g algorithms highly machine dependent. Furthermore, it is difficult to a priori estimate the required precision for complex problems. Another approach is to live with the finite precision world and tune the arithmetic computations to satisfy certain topological constraints on the combinatorial structure of the problem. This reduces and in some cases completely avoids the chance of inconsistent results.