New Invariants of Poncelet-Jacobi Bicentric Polygons

Abstract

The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville's theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family's limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive N-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the N=4 case).

Exploring the Steiner-Soddy Porism

“…In [11] a condition is derived which guarantees that a certain center of a triangle exists iff the sum of half-tangents is less than two; this is equivalent to whether the center of the outer Soddy circle is everted or not [9]. The Poncelet family which is the polar image of bicentrics was studied in [3], and that of harmonic polygons was studied in [16]. Seminal studies of loci of triangle centers over families of Poncelet triangles include [14,20,23,24].…”

“…Let their foci be called "inner" and "outer" ones. In [16] it was shown that the harmonic family is the polar image of the homothetic family with respect to an inversion circle centered on one of the outer foci. Referring to Figure 3:…”