Robust Geometric Computation ( 2004 ; Li , Pion , Yap )

Algorithms in computational geometry are usually designed under the Real RAM model. In implementing these algorithms, however, fixed-precision arithmetic is used in place of exact arithmetic. This substitution introduces numerical errors in the computations that may lead to nonrobust behaviour in the implementation, such as infinite loops or segmentation faults. There are various approaches in the the literature addressing the problem of nonrobustness in geometric computations; see [1] for a survey. These approaches can be classified along two lines: the arithmetic approach and the geometric approach. The arithmetic approach tries to address nonrobustness in geometric algorithms by handling the numerical errors arising because of fixed-precision arithmetic; this can be done, for instance, by using multi-precision arithmetic [2], or by using rational arithmetic whenever possible. In general, all the arithmetic operations, including exact comparison, can be performed on algebraic quantities. The drawback of such a general approach is its inefficiency. The geometric approaches guarantee that certain geometric properties are maintained by the algorithm. For example, if the Voronoi diagram of a planar point set is being computed then it is desirable to ensure that the output is a planar graph as well. Other geometric approaches are finite resolution geometry [3], approximate predicates and fat geometry [4], consistency and topological approaches [5], and topology oriented approach [6]. The common drawback of these approaches is that they are problem or algorithm specific. In the past decade, a general approach called the Exact Geometric Computation (EGC) [7] has become very successful in handling the issue of nonrobustness in geometric computations; strictly speaking, this approach is subsumed in the arithmetic approaches. To understand the EGC approach, it helps to understand the two parts common to all geometric computations: a combi-natorial structure characterizing the discrete relations between geometric objects, e.g., whether a point is on a hyperplane or not; and a numerical part that consists of the numerical representation of the geometric objects, e.g. the coordinates of a point expressed as rational or floating-point numbers. Geometric algorithms characterize the combinatorial structure by numerically computing the discrete relations (that are embodied in geometric predicates) between geometric objects. Nonrobustness arises when numerical errors in the computations yield an incorrect characterization. The EGC approach ensures that all the geometric predicates are evaluated correctly thereby