On the geometry of the ricochet locus

TL;DR

This paper explores the geometric properties of the ricochet configuration related to Pascal's theorem, including symmetry, defining equations, and algebraic structure, advancing understanding of its geometric and algebraic nature.

Contribution

It provides a geometric proof of Pascal line coincidence, computes the symmetry group, and characterizes the ricochet locus as a complete intersection of invariant hypersurfaces.

Findings

Proves Pascal line coincidence in R-configuration

Calculates the symmetry group of R-configuration

Determines the defining equations 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.