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


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 $\RR^n$.


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Borut Jurčič Zlobec. Neža Mramor Kosta. "Geometric constructions on cycles in $\mathbb{R}^n$." Rocky Mountain J. Math. 45 (5) 1709 - 1753, 2015.


Published: 2015
First available in Project Euclid: 26 January 2016

zbMATH: 1343.51014
MathSciNet: MR3452236
Digital Object Identifier: 10.1216/RMJ-2015-45-5-1709

Primary: 15A63 , 51M04 , 51M15

Keywords: cycles , determinant , Lie form , Lie sphere geometry , projection , projective subspace

Rights: Copyright © 2015 Rocky Mountain Mathematics Consortium


Vol.45 • No. 5 • 2015
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