Algorithms for Minkowski products and implicitly‐defined complex sets

Minkowski geometric algebra is concerned with the complex sets populated by the sums and products of all pairs of complex numbers selected from given complex‐set operands. Whereas Minkowski sums (under vector addition in Rⁿ have been extensively studied, from both the theoretical and computational perspective, Minkowski products in R² (induced by the multiplication of complex numbers) have remained relatively unexplored. The complex logarithm reveals a close relation between Minkowski sums and products, thereby allowing algorithms for the latter to be derived through natural adaptations of those for the former. A novel concept, the logarithmic Gauss maps of plane curves, plays a key role in this process, furnishing geometrical insights that parallel those associated with the “ordinary” Gauss map. As a natural generalization of Minkowski sums and products, the computation of “implicitly‐defined” complex sets (populated by general functions of values drawn from given sets) is also considered. By interpreting them as one‐parameter families of curves, whose envelopes contain the set boundaries, algorithms for evaluating such sets are sketched. This revised version was published online in June 2006 with corrections to the Cover Date.