A Bi-Involutive Approach to the Euler-Chasles Theorem and Poncelet's Porism

Poncelet's Porism is traditionally established via synthetic methods, elliptic integrals, or the group law on elliptic curves. This article presents a purely projective framework for the foundational Euler-Chasles theorem by connecting Poncelet dynamics to the bi-involutive generation of plane cubics. By mapping the underlying $(2,2)$ algebraic correspondence onto the residual intersection locus of two homographic line pencils, the abstract coordinate involutions are realized as geometric reflections through the pencil centers. This structural link unifies the classical closure condition (nT = O) with the intrinsic projective geometry of the dynamically generated cubic curve.