The product of three projectively connected conics. (Das Erzeugnis dreier projektiv gekoppelter Kegelschnitte.) (German) Zbl 1014.51013

Stachel, H. (ed.) et al., 25. Süddeutsches Differentialgeometrie-Kolloqium, Wien, Österreich, 2. Juni 2000. Wien: TU Wien, Institut für Geometrie. 79-94 (2001).
The author starts with 3 given conic sections \(X(s), Y(s), Z(s)\) in the projective 3-space, which are linked projectively via the common parameter \(s\). The straight line connections \(X(s), Y(s)\) span algebraic ruled surfaces (in general of degree 4), the planes \(X(s),Y(s),Z(s)\) determine developables (in general of degree 6).
The first chapters of the paper are devoted to possible degeneracies of these surfaces. The next chapters present the study of surfaces \(\Phi\) with a one-parametric set of conic sections. They can be generated by a projective motion of a starting conic. Its point paths determine a set of curves which intersect all conics in points with constant cross ratio (CR-curves on \(\Phi\)). With the help of considerations in a suitable image-space the author is able to characterize the surfaces \(\Phi\) with plane, projectively equivalent CR-curves.
For the entire collection see [Zbl 0980.00037].


51N15 Projective analytic geometry
53A25 Differential line geometry