Singularities of Generic Line Congruences

TL;DR

This paper provides a geometric description of the singularities such as folds, cusps, and swallowtails in generic line congruences in 3-space, using equiaffine pairs as the main analytical tool.

Contribution

It introduces a geometric framework for understanding singularities in line congruences through the use of equiaffine pairs, advancing the theoretical understanding.

Findings

Classification of singularities as folds, cusps, and swallowtails

Geometric description of these singularities in line congruences

Use of equiaffine pairs as a key analytical tool

Abstract

Line congruences are $2$-dimensional families of lines in $3$-space. The singularities that appear in generic line congruences are folds, cusps and swallowtails. In this paper we give a geometric description of these singularities. The main tool used is the existence of an equiaffine pair defining a generic line congruence.