Binary differential equations associated to congruences of lines in Euclidean 3-space

TL;DR

This paper explores classical and new binary differential equations linked to line congruences in Euclidean 3-space, revealing new singular surfaces and local configurations through quadratic form analysis.

Contribution

It introduces a novel binary differential equation related to line congruences, expanding the understanding of their geometric and differential properties.

Findings

Identification of a new binary differential equation for line congruences.

Discovery of a new singular surface associated with this BDE.

Characterization of generic local configurations of BDEs on congruences.

Abstract

We study quotients of quadratic forms and associated polar lines in the projective plane. Our results, applied pointwise to quadratic differential forms, shed some light on classical binary differential equations (BDEs) associated to congruences of lines in Euclidean 3-space and allows us to introduce a new one. The new BDE yields a new singular surface in the Euclidean 3-space associated to a congruence of lines. We determine the generic local configurations of the above BDEs on congruences.