A Comparison of verified distance computation between implicit objects using different arithmetics for range enclosure

This paper describes a new algorithm for computing verified bounds on the distance between two arbitrary fat implicit objects. The algorithm dissects the objects into axis-aligned boxes by constructing an adaptive hierarchical decomposition during runtime. Actual distance computation is performed on the cubes independently of the original object’s complexity. As the whole decomposition process and the distance computation are carried out using verified techniques like interval arithmetic, the calculated bounds are rigorous. In the second part of the paper, we test our algorithm using 18 different test cases, split up into 5 groups. Each group represents a different level of complexity, ranging from simple surfaces like the sphere to more complex surfaces like the Kleins bottle. The algorithm is independent of the actual technique for range bounding, which allows us to compare different verified arithmetics. Using our newly developed uniform framework for verified computations, we perform tests with interval arithmetic, centered forms, affine arithmetic and Taylor models. Finally, we compare them based on the time needed for deriving verified bounds with a user defined accuracy.