Algorithms for Minkowski products and implicitly-defined complex sets

Rida T. Farouki, Hwan Pyo Moon, Bahram Ravani

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)199-229
Number of pages31
JournalAdvances in Computational Mathematics
Volume13
Issue number3
StatePublished - 2000

Keywords

  • Boundary evaluation algorithms
  • Envelopes
  • Families of curves
  • Implicitly-defined complex sets
  • Logarithmic curvature
  • Logarithmic Gauss map
  • Minkowski geometric algebra
  • Minkowski sums and products
  • Set inclusion

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