Abstract
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 $\RR^n$.
Citation
Borut Jurčič Zlobec. Neža Mramor Kosta. "Geometric constructions on cycles in $\mathbb{R}^n$." Rocky Mountain J. Math. 45 (5) 1709 - 1753, 2015. https://doi.org/10.1216/RMJ-2015-45-5-1709
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