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It is well-known that there exists a bijection between the set of lines ofthe projective 3-dimensional space $P^3$ and all real points of the so-calledPl\"ucker quadric $\Psi$. Moreover one can identify each point of the Pl\"uckerquadric's ambient space with a linear complex of lines in $P^3$. Within thispaper we give an alternative interpretation for the points of $P^5$ as lines ofan Euclidean 4-space $E^4$, which are orthogonal to a fixed direction. By usingthe quaternionic notation for lines, we study straight lines in $P^5$ whichcorrespond in the general case to cubic 2-surfaces in $E^4$. We show that thesesurfaces are geometrically connected with circular Darboux 2-motions in $E^4$,as they are basic surfaces of the underlying line-symmetric motions. Moreoverwe 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 the2-dimensional generator space of ideal lines. We also study straight lines ofits ambient space $P^6$ and show that they correspond to ruled surface stripscomposed of the mentioned 2-surfaces with circles on it. Finally we present anapplication of this interpretation in the context of interactive design ofruled surfaces and ruled surface strips/patches based on the algorithm of DeCasteljau.

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