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Minkowski geometric algebra of complex sets. (English) Zbl 0987.51012
For subsets $$A$$ and $$B$$ of the field of complex numbers two operations are defined, the sum of $$A$$ and $$B$$ as $$\{a+b : a\in A$$, $$b\in B\}$$, and their product (by replacing + with $$\times$$ in the definition). The sum has been introduced by H. Minkowski in 1903 and studied extensively.
The authors devote this paper to the study of the product operation, in particular to the effect multiplication by lines and circles has on various curves. The motivation behind this study lies in the potential applications to “a generalization of real interval arithmetic to the complex domain, reflection and refraction of wavefronts in geometrical optics, stability characterization of multi-parameter control systems,” mathematical morphology, and in the expectation that many more such applications will be found.

MSC:
 51N20 Euclidean analytic geometry 51M15 Geometric constructions in real or complex geometry 53A04 Curves in Euclidean and related spaces 65D18 Numerical aspects of computer graphics, image analysis, and computational geometry 65E05 General theory of numerical methods in complex analysis (potential theory, etc.) 65G40 General methods in interval analysis
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