Elliptic-billiard invariant k_{108}

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 noncircular ellipse \[ E=\{(x,y)\in\mathbb R^2:x^2/a^2+y^2/b^2=1\},\qquad a>b>0, \] with center $O=(0,0)$ and foci $f_1=(\sqrt{a^2-b^2},0)$ and $f_2=-f_1$. Fix an inner confocal ellipse $C$, and an integer $N\geq3$, for which the pair $(E,C)$ admits a Poncelet family $\mathcal F$ of $N$-periodic billiard trajectories. A member $P=(P_1,\ldots,P_N)$ has vertices on $E$, each segment $P_iP_{i+1}$ is tangent to $C$, and incoming and outgoing segments satisfy the usual specular reflection law at $P_i$. Indices are cyclic: $P_{N+1}=P_1$. For any ordered polygon $W=(W_1,\ldots,W_N)$ define its signed area by \[ S(W)=\frac12\sum_{i=1}^N\det(W_i,W_{i+1}),\qquad \det((x,y),(u,v))=xv-yu. \] Let $t_i$ be the tangent line to $E$ at $P_i$. The outer polygon $P'$ has vertices $P_i'=t_i\cap t_{i+1}$, and the inner polygon $P''$ has vertices $P_i''$, the contact points of $P_iP_{i+1}$ with $C$. Write $A=S(P)$, $A'=S(P')$ and $A''=S(P'')$. These areas retain their signs. Constancy below means constancy as $P$ varies in the fixed family $\mathcal F$; its value may depend on $E$, $C$ and $N$. Let $\theta_i$ be the angle between $P_{i-1}-P_i$ and $P_{i+1}-P_i$, so that $\theta_i\in[0,\pi]$ and \[ \cos\theta_i=\frac{(P_{i-1}-P_i)\cdot(P_{i+1}-P_i)}{|P_{i-1}-P_i|\,|P_{i+1}-P_i|}. \] Write $H(P)=\prod_{i=1}^N\sin(\theta_i/2)$. Here $|\cdot|$ and $\cdot$ denote Euclidean norm and inner product. A displayed quotient is evaluated where its denominator is nonzero; its constancy means that its values at any two members where it is defined agree.

Statement recorded in the supplied source.

For every such family with $N\equiv2\pmod4$, prove that $\displaystyle \frac{A'}{A\prod_{i=1}^N\sin(\theta_i/2)}$ is constant on $\mathcal F$. [S2]

Short English statement

For every family with two more sides than a multiple of four, prove that the signed outer-to-orbit area ratio divided by the product of half-angle sines stays constant.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5100002 (AMR-050-0002). 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 (2020), arXiv version 11, section 3.1, Table 2, p. 5, invariant $k_{108}$; sections 1--2, pp. 1--3, and Appendix A, Table 12, p. 15. https://arxiv.org/abs/2004.12497v11.