Explicit solutions to Poncelet’s porism

Solution to the following problem is considered: for given conics C and K and an integer N⩾3, determine whether there exists a closed N-sided polygon inscribed in C and circumscribed about K. The case of C and K being circles is considered in detail. Equations are proposed with a relatively small number of arithmetic operations – near log2N. Along the way, the following result is obtained: for circles with rational coefficients, the polygons can only have the following number of sides N = 3, 4, 5, 6, 8, 10 and 12 (a subset of the Mazur’s set of integers for rational elliptic curves). The proposed solution may also be applied to determine whether a Hankel determinant of order N/2 having special form (used in the classical Cayley criterion) is equal to zero, and for related problems. Possible generalizations for ellipses and hyperbolas are also presented. Particularly, equations are proposed for parameters of concentric Poncelet’s ellipses (billiard case).