On the geometry of the ricochet locus

Abstract

This paper is a study of the so-called `ricochet configuration' (or $R$-configuration) which arises in the context of Pascal's theorem. We give a geometric proof of the fact that a specific pair of Pascal lines is coincident for a sextuple in $R$-configuration. We calculate the symmetry group of a generic $R$-configuration, as well as the degree of the subvariety $\mathcal{R}\subseteq {\mathbb{P}}^6$ of all such configurations. We also determine the $SL(2)$-equivariant defining equations for $\mathcal{R}$, and show that it is an ideal-theoretic complete intersection of two invariant hypersurfaces.