Title: Common transversals and tangents in P3
Session Name:AMS Special Session on Computational Commutative Algebra
and Algebraic Geometry
Author: Gabor Megyesi
Author: Frank Sottile
Author: Thorsten Theobald
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.