Elliptic-billiard invariant k_{815}

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. For a polygon $W=(W_i)$ and point $F$, its antipedal polygon has as vertices the consecutive intersections of the lines through $W_i$ perpendicular to $W_i-F$. Write $A_{j,\mathrm{ant}}$ for the signed area of the antipedal polygon of $P$ with respect to $f_j$. The polar polygon with respect to $f_j$ is the antipedal of $P_j^\dagger$ with respect to $f_j$. Its signed area is $A_{j,\mathrm{pol}}$ and its internal angles are $\psi_{j,i}$. This polygon has a fixed circular caustic; let $O_{p,j}$ be its center. The dual polygon is the vertexwise inversion of the polar polygon about $O_{p,j}$, using a fixed unit radius. Denote its signed area by $A_{j,\mathrm{dual}}$ and its side lengths by $w_{j,i}$ (the source writes $w_i$ for a chosen focus). The pedal polygon of $W$ with respect to $F$ consists of the perpendicular feet from $F$ to the lines $W_iW_{i+1}$, in their cyclic order. Write $A_{j,\mathrm{ped}}^\dagger$ for the signed area of the pedal polygon of $P_j^\dagger$ with respect to $f_j$.

Statement recorded in the supplied source.

Prove that, for odd $N$, the quantity $A_{j,\mathrm{ped}}^\dagger A_{j,\mathrm{dual}}$ is constant as $P$ varies through the family. [S2]

Short English statement

Show that, for odd $N$, the quantity $A_{j,\mathrm{ped}}^\dagger A_{j,\mathrm{dual}}$ 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 5100058 (AMR-050-0058). 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 $k_{815}$, pp. 3, 9--12. https://arxiv.org/abs/2004.12497v11.