Minkowski geometric algebra of complex sets. (English) Zbl 0987.51012

Farouki, Rida T.; Pyo Moon, Hwan; Ravani, Bahram

Geom. Dedicata 85, No. 1-3, 283-315 (2001).

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.

Reviewer: Victor V.Pambuccian (Phoenix)

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

Keywords:

Minkowski geometric algebra; complex sets