Common transversals and tangents in P3 Frank Sottile University of Massachusetts-Amherst

Abstract: The following geometric problem has its origins in computational vision: Determine the (degenerate) configurations of two lines l₁ and l₂ and two spheres in R³ for which there are infinitely many lines simultaneously transversal to l₁ and l₂ and tangent to both spheres. We generalize this, replacing the spheres by quadric surfaces in P³. In this setting, the question has an amazing answer. Fixing the two lines to be skew and one quadric, the set of degenerate second quadrics is a curve of degree 24 in the P⁹ of quadrics which is in fact the union of 12 plane conics! Moreover, there are examples where all 12 degenerate families are real. We describe the symbolic computation behind this result and give some vivid pictures.

This talk represents joint work with Gabor Megyesi and Thorsten Theobald