Minkowski Geometric Algebra of Complex Sets

Abstract

A geometric algebra of point sets in the complex plane is proposed, based on two fundamental operations: Minkowski sums and products. Although the (vector) Minkowski sum is widely known, the Minkowski product of two-dimensional sets (induced by the multiplication rule for complex numbers) has not previously attracted much attention. Many interesting applications, interpretations, and connections arise from the geometric algebra based on these operations. Minkowski products with lines and circles are intimately related to problems of wavefront reflection or refraction in geometrical optics. The Minkowski algebra is also the natural extension, to complex numbers, of interval-arithmetic methods for monitoring propagation of errors or uncertainties in real-number computations. The Minkowski sums and products offer basic 'shape operators' for applications such as computer-aided design and mathematical morphology, and may also prove useful in other contexts where complex variables play a fundamental role – Fourier analysis, conformal mapping, stability of control systems, etc.