Minkowski Geometric Algebra of Complex Sets: Theory, Algorithms, and Applications

University Of California-Davis, Davis CA

Investigators

Abstract

ABSTRACT 0202179 Rida Farouki U of Calif Berkeley Minkowski geometric algebra is concerned with the complex sets populated by the sums and products of all pairs of complex numbers drawn from given complex Cset operands.Whereas the Minkowski sum under vector addition in R n has been thoroughly studied,from both the theoretical and computational perspective,the Minkowski product in R 2 induced by the multiplication rule for complex numbers is a relatively unexplored oncept. Conceptually, Minkowski geometric algebra is the natural generalization of real interval arithmetic to complex Cnumber sets.With the transition from real to complex,however,the trivial geometry of real intervals and their consequent closure under addition and multiplication is relinquished. The two Cdimensional haracter of Minkowski geometric algebra endows it with a rich geometrical ontent,in which simple operands (e.g.,circular disks) yield subtle results described by analytic curves such as the Cartesian oval ovals of Cassini and their higher Corder generalizations,while sophisticated algorithms are required to approximate Minkowski combinations for general sets.Apart from being a basic tool to monitor the propagation of uncertainty in complex Cvariable computations,the algebra o .ers a versatile language for two Cdimensional shape operators and the description of (direct or inverse) wavefront re .ection/refraction problems in optics.It is also fundamental to the stability analysis of systems with uncertain parameters,in the context of the Routh CHurwitz criterion and the Kharitonov robustness theorem.