This paper surveys two methods for designing numerically robust geometric algorithms. The first method is the exact-arithmetic method, in which numerical computations are done in sufficiently high precision so that all the topological judgements can be done correctly. This method is usually accompanied with lazy evaluation and symbolic perturbation in order to reduce the computational cost and the implementation cost. The second method is the topology-oriented method, in which the consistency of the topological structure is considered as higher-priority information than numerical computation, and thus inconsistency is avoided. Both of the methods are described with the implementation examples.

A convex hull is one of the most fundamental and interesting geometric constructs in computational geometry. Considerable research effort has focused on developing algorithms, both in serial and in parallel, for computing convex hulls. In particular, there are few problems whose parallel algorithms are so thoroughly studied as convex hull problems. In this paper, we review the convex hull parallel algorithms and their paradigm. We provide a summary of results and introduce several interesting topics including typical techniques, output-size sensitive methods, randomized approaches, and robust algorithms for convex hull problems, with which we may see the highlights of the whole research for parallel algorithms. Most of our discussion uses the PRAM (Parallel Random Access Machine) computational model, but still we give a glance at the results of the other parallel computational models such as mesh, mesh-of-trees, hypercube, recofigurable array, and models of coarse grained multicomputers like BSP and LogP.