
Journal for Geometry and Graphics 22 (2018), No. 2, 229244 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. 810/104, 1040 Wien, Austria nawratil@geometrie.tuwien.ac.at It is wellknown that there exists a bijection between the set of lines of the projective 3dimensional space P^{3} and all real points of the socalled Plücker quadric Ψ. Moreover one can identify each point of the Plücker quadric's ambient space with a linear complex of lines in P^{3}. Within this paper we give an alternative interpretation for the points of P^{5} as lines of an Euclidean 4space E^{4}, which are orthogonal to a fixed direction. By using the quaternionic notation for lines, we study straight lines in P^{5} which correspond in the general case to cubic 2surfaces in E^{4}. We show that these surfaces are geometrically connected with circular Darboux 2motions in E^{4}, as they are basic surfaces of the underlying linesymmetric motions. Moreover we extend the obtained results to lineelements of the Euclidean 3space E^{3}, which can be represented as points of a cone over Ψ sliced along the 2dimensional generator space of ideal lines. We also study straight lines of its ambient space P^{6} and show that they correspond to ruled surface strips composed of the mentioned 2surfaces 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, lineelement, Euclidean 4space, circular Darboux 2motion, De Casteljau algorithm. MSC: 51M30; 53A17 [ Fulltextpdf (1299 KB)] 