Elliptic-billiard invariant k_{807}

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix $a>b>0$ and the ellipse $E:x^2/a^2+y^2/b^2=1$, with foci $f_1,f_2$. Fix an inner confocal elliptic caustic $C$ admitting a Poncelet family of $N$-periodic billiard polygons $P=(P_i)_{i=1}^N$: vertices lie on $E$, sides are tangent to $C$, and consecutive sides satisfy the billiard reflection law. Indices are cyclic and polygons retain their cyclic order. The family and its caustic stay fixed while the starting vertex varies. All areas are signed: for $W=(W_i)$, \[\mathcal A(W)=\frac12\sum_{i=1}^N(x_i y_{i+1}-x_{i+1}y_i),\qquad W_i=(x_i,y_i).\] Write $A=\mathcal A(P)$. Unit-circle inversion about $F$ is $I_F(X)=F+(X-F)/\lVert X-F\rVert^2$; set $P_j^\dagger=(I_{f_j}(P_i))$ and $A_j^\dagger=\mathcal A(P_j^\dagger)$, for $j=1,2$. Quotients are considered where their denominators and the polygon constructions are defined. Let $P'_i$ be the intersection of the tangents to $E$ at $P_i$ and $P_{i+1}$, and let $P''_i$ be the contact point of $P_iP_{i+1}$ with $C$. Their polygons are $P',P''$, with signed areas $A',A''$. Write ${A'_j}^\dagger=\mathcal A(I_{f_j}(P'))$ and ${A''_j}^\dagger=\mathcal A(I_{f_j}(P''))$. Elliptic inversion of a point outside an ellipse sends it to the midpoint of the chord joining its two tangency points on that ellipse. Let $P^\otimes$ be the inversion of the vertices of $P$ in $C$, and let ${P'}^\ominus$ be the inversion of the vertices of $P'$ in $E$; their signed areas are $A^\otimes$ and ${A'}^\ominus$. Equivalently these are the polygons of side midpoints of $P''$ and $P$, respectively.

Statement recorded in the supplied source.

Prove that, for even $N$, the quantity $AA^\otimes$ is constant as $P$ varies through the family. [S2]

Short English statement

Show that, for even $N$, the quantity $AA^\otimes$ is constant as $P$ varies through the family. The constant may depend on the fixed ellipse and caustic. These are the experimentally recorded assertions of the cited 2020 version.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5100049 (AMR-050-0049). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Dan Reznik, Ronaldo Garcia and Jair Koiller, Eighty New Invariants in the Elliptic Billiard, arXiv:2004.12497v11 (29 October 2020), Section 2 and Section 3.9, Table 9, row $k_{807}$, pp. 3, 9--12. https://arxiv.org/abs/2004.12497v11.