Alternative interpretation of the Plücker quadric's ambient space and its application
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It is well-known that there exists a bijection between the set of lines of the projective 3-dimensional space P^3 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 P^3. Within this paper we give an alternative interpretation for the points of P^5 as lines of an Euclidean 4-space 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 2-surfaces in E^4. We show that these surfaces are geometrically connected with circular Darboux 2-motions in E^4, 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 E^3, 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 P^6 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.
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