  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.

Computational Geometry