Binary Differential Equations Associated to Congruences of Lines in Euclidean 3-Space

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 lead us to construct a new one. The new BDE yields a new singular surface in the Euclidean 3-space associated to the congruence of lines. We determine the generic local configurations of the new and the classical BDEs.