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 R-n) have been extensively studied, from both the theoretical and computational perspective, Minkowski products in R-2 (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.