Conic Sections in Space Defined by Intersection Conditions

This text generalizes and extends results on a certain class of incidence problems related to conic sections. We consider the set of planes in complex projective three-space P3 that intersect m ≤ 3 conic sections Ci and n = 6 − 2m straight lines Lj in six points of a conic section. The case ofm = 0 has been treated in (Sch04a) while m = 1 is the topic of (Sch04b). In this paper we also consider m = 2 and m = 3. Dual to the set of solution planes is the vertex locus of those quadratic cones that share two tangent planes with m given quadratic cones and have n given straight lines as tangents. This viewpoint relates the present article to a number of publications during the last twenty years. In (Sch85, Sch86, Str89, Str91, Wun93, Mic95, Zso97) similar problems were considered, usually with additional metric constraints while the purely projective viewpoint (that will also be taken in this text) dates back to the 19th century (Hie71). Of course, we also may consider P3 as projective extension of a euclidean space with the base conic C0 as absolute circle. Doing so, we contribute to the task of finding circles that intersect m− 1 conic sections in two points and n straight lines in one point. This approach allowed the advantageous use of the geometry of circles in space in (Sch04b) but seems inappropriate for the other cases. In general, we can expect a two-parametric set Sm of solution planes. For m ∈ {0, 1}, the following facts have been shown in (Sch04a) and (Sch04b):