Uncertain Geometry with Circles, Spheres and Conics

Abstract

In this text the description of uncertain geometric entities is extended from points, lines and planes to circles, spheres and 2D-conic sections. While the former has been treated previously by Kanatani (Kanatani, 1996), F¨orstner (F¨orstner et al., 2000) and Heuel (Heuel, 2004) in matrix spaces, the latter can be treated advantageously in a multilinear setting using Clifford algebra. It is shown how error propagation can be applied to Clifford algebra operations in general, and specifically for the construction of circles, spheres and 2D-conic sections. While circles and spheres are treated in the Clifford algebra of conformal space (Hestenes, 1991; Li et al., 2001), the construction of uncertain 2D-conic sections is treated in the Clifford algebra of a specially developed vector space. Some results on synthetic data are presented.