Poncelet Curves

The paper proves that, given a positively oriented C^k vertex curve K with nonvanishing curvature and a torsion map f with f^n = id, the curve C defined by X = Y − (⟨Y′, J(Y∘f − Y)⟩/⟨(Y∘f − Y)′, J(Y∘f − Y)⟩)(Y∘f − Y) is the envelope of the sides of the associated f-polygons, so (K,C) is a Poncelet pair; it further shows regularity of C when its curvature never vanishes and, if K is simple, that the contact point lies in the open segment (0 < s < 1). These are exactly the three assertions the candidate establishes, via the standard envelope conditions F = 0 and ∂φF = 0 and the same geometric inequalities. The arguments, formulas (3.1)–(3.3), and the convexity-based sign analysis in the paper align essentially line-by-line with the candidate’s steps, so both are correct and use substantially the same proof .