Geometric constructions on cycles in $\rr^n$

In Lie sphere geometry, a cycle in $\RR^n$ is either a point or an oriented sphere or plane of codimension $1$, and it is represented by a point on a projective surface $\Omega\subset \PP^{n+2}$. The Lie product, a bilinear form on the space of homogeneous coordinates $\RR^{n+3}$, provides an algebraic description of geometric properties of cycles and their mutual position in $\RR^n$. In this paper we discuss geometric objects which correspond to the intersection of $\Omega$ with projective subspaces of $\PP^{n+2}$. 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 $\RR^n$.