A THEORY FOR LOCUS ELLIPTICITY OVER PONCELET 3-PERIODICS

The paper proves that for a generic nested ellipse pair admitting a Poncelet 3‑periodic family, any fixed linear combination X = αX2 + βX3 has an elliptic locus via the parametrization X(λ) = uλ + v/λ + w, and therefore any translate by a stationary center Xk is also elliptic (Corollary 1) . It also gives precise degeneracy and circularity conditions for such loci in the confocal case . By contrast, the model asserts that a 3‑Poncelet family exists only when the outer ellipse is the 2‑homothetic of the inner one and that the tangency parameters must be equally spaced, which is contradicted by the paper’s treatment of generic, non‑homothetic, non‑concentric pairs that admit 3‑periodics (see Figure 1 and Theorem 1) . The model also claims X2 is stationary in general, while the paper recalls that over generic Poncelet pairs the vertex centroid traces an ellipse (not a point) , and it oversimplifies degeneracy to “β=0 or a=b,” missing the paper’s exact α/β thresholds .