Elliptic-billiard invariant k_{203,a}

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 $N\equiv0\pmod4$ and every fixed $M\in\mathbb R^2$, prove that $A A_M$ is constant on $\mathcal F$. The constant may depend on $M$. [S2]

Short English statement

When the period is divisible by four, prove for every fixed plane point that the signed orbit area times its signed pedal-polygon area stays constant.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5100011 (AMR-050-0011). 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 $k_{203,a}$; sections 1--2, pp. 1--3, and Appendix A, Table 12, p. 15. https://arxiv.org/abs/2004.12497v11.