On tangents to quadric surfaces

TL;DR

This paper investigates the conditions under which up to four quadric surfaces in projective three-space share common tangents, with a focus on configurations that admit a continuum of such tangents, especially for spheres in Euclidean space.

Contribution

It provides a geometric characterization of when multiple quadrics are tangent along a curve to a common quadric, and fully answers the real case for four spheres with infinitely many common tangents.

Findings

Identifies conditions for a continuum of common tangents among quadrics.

Characterizes configurations of four spheres with infinitely many common tangents.

Provides a complete answer for the real case involving spheres.

Abstract

We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in the projective space defined by all complex quadric surfaces which express the fact that several quadrics are tangent along a curve to one and the same quadric of rank at least three, and called, for intuitive reasons: a basket. Lines in any ruling of the latter will be common tangents. These considerations are then restricted to spheres in Euclidean three-space, and result in a complete answer to the question over the reals: ``When do four spheres allow infinitely many common tangents?''.