Common Transversals and Tangents to Two Lines and Two Quadrics in P

Abstract

We solve the following geometric problem, which arises in several {\mbox three-dimensional} applications in computational geometry: For which arrangements of two lines and two spheres in ${\Bbb R}^3$ are there infinitely many lines simultaneously transversal to the two lines and tangent to the two spheres? We also treat a generalization of this problem to projective quadrics. Replacing the spheres in ${\bf R}^3$ by quadrics in projective space ${\Bbb P}^3$, and fixing the lines and one general quadric, we give the following complete geometric description of the set of (second) quadrics for which the two lines and two quadrics have infinitely many transversals and tangents: in the nine-dimensional projective space ${\Bbb P}^9$ of quadrics, this is a curve of degree 24 consisting of 12 plane conics, a remarkably reducible variety.