A unified algebraic framework for fast and precise planar swept volumes and Minkowski sums

Abstract

The topic of swept-volumes is well-studied. Yet, the high degree of underlying mathematical and computational complexity has prevented a general implementation which accepts freeform input, even for the planar case. Likewise for Minkowski sums. This paper proposes a unified approach to the two related problems, in 2D, via algebraic modeling of the underlying mathematical constraints. Handling self-intersections in the envelope has been the special focus of this work. Algebraic modeling results in a high degree of numerical precision and guarantee of topological correctness. A top-down computational strategy leads to a numerically robust algorithm. Further, an auxiliary point-cloud representation of the swept-volume yields a significant boost in computational efficiency. The robustness and numerical stability of our approach are demonstrated by tens of thousands of examples, generated from an implementation of our algorithm.