On the affine geometry of congruences of lines

Abstract

Congruences, or $2$-parameter families of lines in $3$-space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geometry which are invariant under affine changes of co-ordinates, for example that associated with their focal sets, and less well studied focal planes. We use tools from singularity theory to describe some generic phenomena. In particular we determine the generic singularities of various surfaces in affine 3-space associated to these congruences. We identify a projective quadric in the projectivised tangent space to the manifold of lines which plays a key role in understanding the affine geometry of ruled surfaces, congruences and $3$-parameter families or complexes. Many of the results generalise to lines in $\mathbb R^n, n>3$.