New Properties of Triangular Orbits in Elliptic Billiards

New Invariants of Poncelet-Jacobi Bicentric Polygons

“…• Theorem 1: The sum of the cosines of bicentric polygons is invariant over the family. This mirrors an invariant recently proved for elliptic billiard N-periodics [16,10,1,4]. • Theorem 2 The perimeter of pedal polygons of the bicentrics with respect to its limiting points is invariant; see Figure 1.…”

Exploring self-intersected N-periodics in the elliptic billiard

“…A kinematic analysis of the geometry of 𝑁 =periodics using Jacobian elliptic functions is proposed in [24]. Works [13,19] derive explicit expressions for some invariants in the 𝑁 = 3 case (billiard triangles). Additional constructions derived from N-periodics (e.g., pedals, antipedals, etc.)…”

“…are considered in [20], augmenting the list of elliptic billiard invariants to 80. In recent publications [13,19,21] we have described several Euclidean quantities which remain invariant over a given family, some of which have been subsequently proved [2,5,8].…”

“…As before, let 𝛿 = √ 𝑎 4 − 𝑎 2 𝑏 2 + 𝑏 4 . For 𝑁 = 3 explicit expressions for 𝐽 and 𝐿 have been derived [13]:…”

“…Proof. We've shown ∑︀ 3 𝑖=1 cos 𝜃 𝑖 = 𝐽𝐿 − 3 is invariant for the 𝑁 = 3 family [13]. For any triangle ∑︀ 3 𝑖=1 cos 𝜃 𝑖 = 1 + 𝑟/𝑅 [28], so it follows that 𝑟/𝑅 = 𝐽𝐿 − 4 is also invariant.…”