Unified approach to the Fuss relations in Poncelet's porism (Q6656088)

The paper studies relationships between the radii \(R\) and \(r\), respectively, of the circumscribed and inscribed circles of a polygon with \(n\) sides, and \(a\) which is the distance between the centers of the circles. The case \(n=3\) was famously studied by Euler: \(a^2=R^2-2Rr\). The cases \(n=4,\ldots,8\) were studied by his student Fuss. These cases also attracted the attention of Steiner, Poncelet, and Jacobi. The paper builds on the findings of the previous papers of the first and third authors [Comput. Aided Geom. Des. 66, 19--30 (2018; Zbl 1505.51022); Beitr. Algebra Geom. 55, No. 1, 301--309 (2014; Zbl 1298.53004)]. In contrast to the previously mentioned papers which were more analytic using ordinary differential equations, the current paper is more algebraic focusing on the properties of the polynomials involved in the expressions such as degree, being monic, the nature of the coefficients which are themselves polynomials, and homogeneity. Another novelty of the paper is considerable simplification and unification of the recurrence relations. The readable paper ends with the application of the obtained results to cases \(n=4,\ldots,10\).