A Complete Implementation for Computing General Dimensional Convex Hulls

We present a robust implementation of the Beneath-Beyond algorithm for computing convex hulls in arbitrary dimension. Certain techniques used are of independent interest in the implementation of geometric algorithms. In particular, two important, and often complementary, issues are studied, namely exact arithmetic and degeneracy. We focus on integer arithmetic and propose a general and eecient method for its implementation based on modular arithmetic. We suggest that probabilistic modular arithmetic may be of wide interest, as it combines the advantages of modular arithmetic with the speed of randomization. The use of perturbations as a method to cope with input degeneracy is also illustrated. A computationally eecient scheme is implemented which, moreover, greatly simpliies the task of programming. We concentrate on postprocessing, often perceived as the Achilles' heel of perturbations. Experimental results illustrate the dependence of running time on the various input parameters and attempt a comparison with existing programs. Lastly, we discuss the visualization capabilities of our software and illustrate them for problems in computational algebraic geometry. All code is publicly available.