The Plücker Quadric's Ambient Space: Alternative Interpretation and its Application

G. Nawratil

Journal for geometry and graphics, Tome 22 (2018) no. 2, pp. 229-244.

Résumé

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 Pl�cker quadric Ψ. Moreover one can identify each point of the Pl�cker 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.

Détail

Classification : 51M30, 53A17
Mots-clés : Pluecker quadric, line-element, Euclidean 4-space, circular Darboux 2-motion, De Casteljau algorithm