ON THE GEOMETRY OF THE HOMOGENEOUS REPRESENTATION FOR THE GROUP OF PROPER RIGID-BODY DISPLACEMENTS
Journal Title: ROMANIAN JOURNAL OF TECHNICAL SCIENCES - APPLIED MECHANICS - Year 2013, Vol 58, Issue 1
Abstract
This work investigates the geometry of the homogeneous representation of the group of proper rigid-body displacements. In particular it is shown that there is a birationaltransformationfromtheStudyquadrictothevarietydeterminedbythehomogeneous representation. This variety is shown to be the join of a Veronese variety with a 2-plane. The rest of the paper looks at sub-varieties, first those which are sub-groups of the displacement group and then some examples defined by geometric constraints. In many cases the varieties are familiar as sub-varieties of the Study quadric, here their transforms to the homogeneous representation is considered. A final section deals with the map which sends each displacements to its inverse. This is shown to be a quadratic birational transformation.
Authors and Affiliations
J. M. SELIG
Keywords
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