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Stereographic projection of Study’s quadric. (English) Zbl 1079.53021

Weiß, Gunter (ed.), DSG CK 2003. Dresden Symposium Geometry: Constructive and kinematic/konstruktiv und kinematisch. Zum Gedenken an/in commemoration of Rudolf Bereis (1903–1966), Dresden, Germany, February 27–March 1, 2003. Proceedings. Dresden: Technische Universität Dresden (ISBN 3-86005-394-9/pbk). 82-89 (2003).
Study’s quadric is a hyperquadric in the 7-dimensional real projective space. It plays a paramount role within the kinematics of the Euclidean 3-space. Due to the nonlinear structure of the hyperquadric, which is basically the structure of the Euclidean motion group, there is often a need of linearising the manifold - at least locally. This need is being met by the stereographic projection described in this paper. Stereographic projection is the central projection of a quadric from one of its points onto a hyperplane. Not only do the authors present their fundamental idea, they also interpret the mapping in terms of the Euclidean motion group which is of course the paradigm of the whole matter. The bisecting line complex to a Euclidean displacement is also employed in order to shed some geometric light on the emerging coordinates of an image point. This paper will most likely turn out to be a considerable step forward in modern kinematics and robotics.
For the entire collection see [Zbl 1048.51001].
Reviewer: Johann Lang (Graz)

MSC:

53A17 Differential geometric aspects in kinematics
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