Cubic Ruled Surfaces with Constant Distribution Parameter in E4

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 P3 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 P3. Within this paper we give an alternative interpretation for the points of P5 as lines of an Euclidean 4-space E4, ...

Characterizations of Slant Ruled Surfaces in the Euclidean 3-space

In this study, we give the relationships between the conical curvatures of ruled surfaces generated by the unit vectors of the ruling, central normal and central tangent of a ruled surface in the Euclidean 3-space E^3. We obtain differential equations characterizing slant ruled surfaces and if the reference ruled surface is a slant ruled surface, we give the conditions for the surfaces generate...

Homology of Singular Cubic Surfaces in P(3,C)

An 1871 paper by Diekmann contains all the information needed to compute the integer singular homology for all classes of not ruled singular cubic surfaces. Three linearly independent quadrics in projective threespace deene a cubic Cremona transformation that associates to any point x the intersection of the polar planes of x with respect to the three quadrics. That intersection is a point y ex...

Higher Order Contact on Cayley’s Ruled Cubic Surface

It is well known that Cayley’s ruled cubic surface carries a three-parameter family of twisted cubics sharing a common point, with the same tangent and the same osculating plane. We report on various results and open problems with respect to contact of higher order and dual contact of higher order for these curves.

A Series of Involutorial Cremona Space Transformations D E F I N E D by a Pencil of Ruled Cubic Surfaces* by Amos Black