Bipolar and Multipolar Coordinates

Abstract

Bipolar or multipolar coordinates offer useful insights and advantages over Cartesian coordinates in certain geometrical problems. In bipolar coordinates (r 1, r 2) the “simplest” curves are the conics, ovals of Cassini, Cartesian ovals, and their special cases, which are characterized by linear or hyperbolic relations in the (r 1, r 2) plane. As a natural extension of these classical examples, we consider the full range of curves characterized by conic (r 1, r 2) loci. A further useful generalization involves the extension of the curve equations to (redundant) multipolar coordinates (r 1,…, r n), taking the n—th roots of unity as “canonical” poles. We survey two key applications of these methods, in geometrical optics and the Minkowski geometric algebra of complex sets, and explore the formulation of geometric design schemes using planar and spatial bipolar or multipolar coordinates.