Geometric constructions on cycles in $\mathbb{R}^n$

Abstract

In Lie sphere geometry, a cycle in R n is either a point or an oriented sphere or plane of codimension 1, and it is represented by a point on a projective surface Ω ⊂ P n+2 . The Lie product, a bilinear form on the space of homogeneous coordinates R n+3 , provides an algebraic description of geometric properties of cycles and their mutual position in R n . In this paper we discuss geometric objects which correspond to the intersection of Ω with projective subspaces of P n+2 . Examples of such objects are spheres and planes of codimension 2 or more, cones and tori. The algebraic framework which Lie geometry provides gives rise to simple and efficient computation of invariants of these objects, their properties and their mutual position in R n .2000 Mathematics Subject Classification. 51M04,51M15,15A63.