Alternative interpretation of the Pl\"ucker quadric's ambient space and its application

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 Pl\"ucker quadric $\Psi$. Moreover one can identify each point of the Pl\"ucker 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 $\Psi$ 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.