Journal for Geometry and Graphics 22 (2018), No. 2, 229--244
Copyright Heldermann Verlag 2018
The Plücker Quadric's Ambient Space: Alternative Interpretation and its Application
Georg Nawratil
Inst. of Discrete Mathematics and Geometry, University of Technology, Wiedner Hauptstr. 8-10/104, 1040 Wien, Austria
nawratil@geometrie.tuwien.ac.at
It is well-known that there exists a bijection between the set of lines of the projective 3-dimensional space P3 and all real points of the so-called Plücker quadric Ψ. Moreover one can identify each point of the Plücker quadric's ambient space with a linear complex of lines in P3. Within this paper we give an alternative interpretation for the points of P5 as lines of an Euclidean 4-space E4, which are orthogonal to a fixed direction. By using the quaternionic notation for lines, we study straight lines in P5 which correspond in the general case to cubic 2-surfaces in E4. We show that these surfaces are geometrically connected with circular Darboux 2-motions in E4, as they are basic surfaces of the underlying line-symmetric motions. Moreover we extend the obtained results to line-elements of the Euclidean 3-space E3, which can be represented as points of a cone over Ψ sliced along the 2-dimensional generator space of ideal lines. We also study straight lines of its ambient space P6 and show that they correspond to ruled surface strips composed of the mentioned 2-surfaces with circles on it. Finally we present an application of this interpretation in the context of interactive design of ruled surfaces and ruled surface strips/patches based on the algorithm of De Casteljau.
Keywords: Pluecker quadric, line-element, Euclidean 4-space, circular Darboux 2-motion, De Casteljau algorithm.
MSC: 51M30; 53A17
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