Elliptic-billiard invariant k_{203,b}

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 $N$. Fix $M\in\mathbb R^2$. Let $Q_i$ be the orthogonal projection of $M$ onto the supporting line of the side $P_iP_{i+1}$, and let $Q_i'$ be its orthogonal projection onto the supporting line of $P_i'P_{i+1}'$. Define signed pedal areas \[ A_M=S(Q_1,\ldots,Q_N),\qquad A_M'=S(Q_1',\ldots,Q_N'). \] The point $M$ remains fixed while $P$ varies; it need not lie inside either polygon.

Statement recorded in the supplied source.

For every such family with $N\not\equiv2\pmod4$, set $M=O$ and prove that $A A_O$ is constant on $\mathcal F$. This includes every odd $N$ and every $N\equiv0\pmod4$. [S2]

Short English statement

With the pedal point at the ellipse center, prove that the signed orbit area times its signed pedal area stays constant for every period except those congruent to two modulo four.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5100012 (AMR-050-0012). 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 3.3, Table 3, p. 6, invariant $k_{203,b}$; sections 1--2, pp. 1--3, and Appendix A, Table 12, p. 15. https://arxiv.org/abs/2004.12497v11.