On the affine geometry of congruences of lines

TL;DR

This paper explores the affine-invariant geometry of line congruences in 3D space, analyzing their focal sets and singularities using singularity theory, with implications for higher-dimensional line geometries.

Contribution

It introduces a detailed analysis of affine-invariant properties of line congruences, identifying key singularities and geometric structures, including a projective quadric in tangent space.

Findings

Identified generic singularities of surfaces associated with line congruences.

Described the role of a projective quadric in understanding affine geometry.

Extended results to lines in higher-dimensional spaces.

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 $\mathbb R^n, n>3$.