# The Geometry of Billiards in Ellipses
and their Poncelet Grids

Hellmuth Stachel  Vienna University of Technology
Wiedner Hauptstr. 8-10/104
1040 Wien, Austria<EMAIL_ADDRESS>
(Date: April 2, 2021)

###### Abstract.

The goal of this paper is an analysis of the geometry of billiards in
ellipses, based on properties of confocal central conics. The extended sides
of the billiards meet at points which are located on confocal ellipses and
hyperbolas. They define the associated Poncelet grid. If a billiard is
periodic then it closes for any choice of the initial vertex on the ellipse.
This gives rise to a continuous variation of billiards which is called
billiard’s motion though it is neither a Euclidean nor a projective motion.
The extension of this motion to the associated Poncelet grid leads to new
insights and invariants.

###### Key words and phrases:

billiard in ellipse, caustic, Poncelet grid, confocal conics, billiard motion,
canonical parametrization

###### 1991 Mathematics Subject Classification:

Primary 51N35; Secondary 51N20, 52C30, 37D50

##  1\. Introduction

A billiard is the trajectory of a mass point within a domain with ideal
physical reflections in the boundary. Already for two centuries, billiards in
ellipses have attracted the attention of mathematicians, beginning with J.-V.
Poncelet and A. Cayley. One basis for the investigations was the theory of
confocal conics. In 2005 S. Tabachnikov published a book on billiards, based
on the theory of completely integrable systems [26]. In several publications
and in the book [13], V. Dragović and M. Radnović studied billiards, also in
higher dimensions, from the viewpoint of dynamical systems.

Computer animations of billiards in ellipses, which were carried out by D.
Reznik [21], stimulated a new vivid interest on this well studied topic, where
algebraic and analytic methods are meeting (see, e.g., [2, 3, 11, 19, 20] and
many further references in [21]). These papers focus on invariants of periodic
billiards when one vertex varies on the ellipse while the caustic remains
fixed. This variation is called billiard motion though neither angles nor side
lengths remain fixed and this is is also not a projective motion preserving
the circumscribed ellipse.

The goal of this paper is a geometric analysis of billiards in ellipses and
their associated Poncelet grid, starting from properties of confocal conics.
We concentrate on a certain symmetry between the vertices of any billiard and
the contact points with the caustic, which can be an ellipse or hyperbola.
Billiard motions induce motions of associated billiards with the same caustic
and circumscribed confocal ellipses.

##  2\. Metric properties of confocal conics

A family of confocal central conics (Figure 1) is given by

$\frac{x^{2}}{a^{2}+k}+\frac{y^{2}}{b^{2}+k}=1,\ \mbox{where}\
k\in{\mathbb{R}}\setminus\\{-a^{2},-b^{2}\\}$ (2.1)

serves as a parameter in the family. All these conics share the focal points

$F_{1,2}=(\pm d,0),\ \mbox{where}\ d^{2}:=a^{2}-b^{2}.$ (2.2)

The confocal family sends through each point $P$ outside the common axes of
symmetry two orthogonally intersecting conics, one ellipse and one hyperbola
[15, p. 38]. The parameters $(k_{e},k_{h})$ of these two conics define the
elliptic coordinates of $P$ with

$-a^{2}<k_{h}<-b^{2}<k_{e}\,.$

If $(x,y)$ are the cartesian coordinates of $P$, then $(k_{e},k_{h})$ are the
roots of the quadratic equation

$k^{2}+(a^{2}+b^{2}-x^{2}-y^{2})k+(a^{2}b^{2}-b^{2}x^{2}-a^{2}y^{2})=0,$ (2.3)

while conversely

$x^{2}=\frac{(a^{2}+k_{e})(a^{2}+k_{h})}{d^{2}}\,,\quad
y^{2}=-\frac{(b^{2}+k_{e})(b^{2}+k_{h})}{d^{2}}\,.$ (2.4)

Let $(a,b)=(a_{c},b_{c})$ be the semiaxes of the ellipse $c$ with $k=0$. Then,
for points $P$ on a confocal ellipse $e$ with semiaxes $(a_{e},b_{e})$ and
$k=k_{e}>0$, i.e., exterior to $c$, the standard parametrization yields

$\begin{array}[]{c}P=(x,y)=(a_{e}\cos t,\,b_{e}\sin t),\ 0\leq
t<2\pi,\\\\[2.84526pt] \mbox{with}\ a_{e}^{2}=a_{c}^{2}+k_{e},\
b_{e}^{2}=b_{c}^{2}+k_{e}\,.\end{array}$ (2.5)

For the elliptic coordinates $(k_{e},k_{h})$ of $P$ follows from (2.3) that

$k_{e}+k_{h}=a_{e}^{2}\cos^{2}t+b_{e}^{2}\sin^{2}t-a_{c}^{2}-b_{c}^{2}.$

After introducing the respective tangent vectors of $e$ and $c$, namely

$\begin{array}[]{rcl}{\mathbf{t}}_{e}(t)&:=&(-a_{e}\sin t,\,b_{e}\cos
t),\\\\[2.27621pt] {\mathbf{t}}_{c}(t)&:=&(-a_{c}\sin t,\,b_{c}\cos
t),\end{array}\ \mbox{where}\
\|{\mathbf{t}}_{e}\|^{2}=\|{\mathbf{t}}_{c}\|^{2}+k_{e}\,,$ (2.6)

we obtain111 The norm $\|{\mathbf{t}}_{e}\|$ equals half length of the
diameter of $e$ which is parallel to ${\mathbf{t}}_{e}\,$.

$k_{h}=k_{h}(t)=-(a_{c}^{2}\sin^{2}t+b_{c}^{2}\cos^{2}t)=-\|{\mathbf{t}}_{c}(t)\|^{2}=-\|{\mathbf{t}}_{e}(t)\|^{2}+k_{e}$
(2.7)

and

$\|{\mathbf{t}}_{e}(t)\|^{2}=k_{e}-k_{h}(t)\,.$ (2.8)

Note that points on the confocal ellipses $e$ and $c$ with the same parameter
$t$ have the same coordinate $k_{h}$. Consequently, they belong to the same
confocal hyperbola (Figure 5). Conversely, points of $e$ or $c$ on this
hyperbola have a parameter out of $\\{t,-t,\pi+t,\pi-t\\}$.

Figure 1. The tangents from the point $P$ to the conics of a confocal family
are symmetric w.r.t. the tangents at $P$ to the confocal conics passing
through $P$.

Normal vectors of $e$ and $c$ can be defined respectively as

$\begin{array}[]{rcl}{\mathbf{n}}_{e}(t)&:=&\Bigl{(}{\displaystyle\frac{\,\cos
t}{a_{e}}},\,{\displaystyle\frac{\sin t}{b_{e}}}\Bigr{)},\\\\[7.96677pt]
{\mathbf{n}}_{c}(t)&:=&\Bigl{(}{\displaystyle\frac{\,\cos
t}{a_{c}}},\,{\displaystyle\frac{\sin
t}{b_{c}}}\Bigr{)},\end{array}\quad\mbox{where}\
\|{\mathbf{n}}_{c}(t)\|={\displaystyle\frac{\|{\mathbf{t}}_{c}(t)\|}{a_{c}b_{c}}}\,.$
(2.9)

We complete with two useful relations between the parameter $t$ and the second
elliptic coordinate $k_{h}(t)$:

$\tan^{2}t=-\frac{b_{c}^{2}+k_{h}(t)}{a_{c}^{2}+k_{h}(t)}\quad\mbox{and}\quad\sin
t\cos t=\frac{a_{h}b_{h}}{d^{2}}$ (2.10)

with $a_{h}$ and $b_{h}$ as semiaxes of the hyperbola corresponding to the
parameter $t$. i.e., $a_{h}^{2}=a_{c}^{2}+k_{h}$ and
$b_{h}^{2}=-(b_{c}^{2}+k_{h})$.

Proof. From (2.7) follows

$k_{h}=-\frac{a_{c}^{2}\tan^{2}t+b_{c}^{2}}{1+\tan^{2}t},\quad\mbox{hence}\quad\tan^{2}t(a_{c}^{2}+k_{h})=-b_{c}^{2}-k_{h}$

and

$\sin t\cos t=\frac{\tan
t}{1+\tan^{2}t}=\frac{\sqrt{-(b_{c}^{2}+k_{h})(a_{c}^{2}+k_{h})}}{a_{c}^{2}-b_{c}^{2}}=\frac{a_{h}\,b_{h}}{d^{2}}\,.$
∎

Referring to Figure 1, the following lemma addresses an important property of
confocal conics (note, e.g., [15, p. 38, 309]).

######  Lemma 2.1.

The tangents drawn from any fixed point $P$ to the conics of a confocal family
share the axes of symmetry, which are tangent to the two conics passing
through $P$.

This means, if a ray is reflected at $P$ in one of the conics passing through,
then the incoming and the outgoing ray contact the same confocal ellipse or
hyperbola.

Figure 2. Periodic billiard $P_{1}P_{2}\dots P_{5}$ inscribed in the ellipse
$e$ with the caustic $c$.

Below, we report about results concerning a pair of confocal conics. Due to
their meaning for billiards in ellipses, we restrict ourselves to pairs
$(e,c)$ of confocal ellipses with $c$ in the interior $e$, and we call $c$ the
caustic (Figure 2).

######  Lemma 2.2.

Let $P=(a_{e}\cos t,\,b_{e}\sin t)$ with elliptic coordinates $(k_{e},k_{h})$
be a point on the ellipse $e$ with $k_{e}>0$ and $c$ be the confocal ellipse
with $k=0$. Then, the angle $\theta(t)/2$ between the tangent at $P$ to $e$
and any tangent from $P$ to $c$ satisfies

$\sin^{2}\frac{\theta}{2}=\frac{k_{e}}{\|{\mathbf{t}}_{e}(t)\|^{2}}=\frac{k_{e}}{k_{e}\\!-k_{h}}\,,\
\tan\frac{\theta}{2}=\pm\sqrt{-\frac{k_{e}}{k_{h}}}\,,$ (2.11)
$\cos\theta=1-\frac{2k_{e}}{\|{\mathbf{t}}_{e}(t)\|^{2}}=\frac{k_{h}\\!+k_{e}}{k_{h}\\!-k_{e}}\,,\quad\sin\theta=\pm\frac{2\sqrt{-k_{e}k_{h}}}{k_{e}-k_{h}}\,.$
(2.12)

Proof. The tangent $t_{P}$ to $e$ at $P=(a_{e}\cos t,\ b_{e}\sin t)$ in
direction of ${\mathbf{t}}_{e}$ has the slope

$f:=\tan\alpha_{1}=\frac{-b_{e}\cos t}{a_{e}\sin t}\,.$

If $s_{1}$ and $s_{2}$ denote the slopes of the tangents from $P$ to $c$, then
they satisfy

$y-b_{e}\sin t=s_{i}(x-a_{e}\cos t),\quad i=1,2.$

As tangents of $c$, their homogeneous line coordinates

$(u_{0}:u_{1}:u_{2})=\left((b_{e}\sin t-s_{i}\,a_{e}\cos t):s_{i}:-1\right)$

must satisfy the tangential equation
$-u_{0}^{2}+a_{c}^{2}u_{1}^{2}+b_{c}^{2}u_{2}^{2}=0$ of $c$. This results in a
quadratic equation for the unknown $s$, namely

$(a_{e}^{2}\sin^{2}t-k_{e})s^{2}+2a_{e}b_{e}s\sin t\cos
t+(b_{e}^{2}\cos^{2}t-k_{e})=0.$

We conclude

$s_{1}+s_{2}=\frac{-2a_{e}b_{e}\sin t\cos t}{a_{e}^{2}\sin^{2}t-k_{e}}\mskip
9.0mu\mbox{and}\mskip
9.0mus_{1}s_{2}=\frac{b_{e}^{2}\cos^{2}t-k_{e}}{a_{e}^{2}\sin^{2}t-k_{e}}.$

The slopes $f=\tan\alpha_{1}$ of $t_{P}$ and $\tan\alpha_{2}=s_{1}$ or $s_{2}$
of the tangents to $c$ imply for the enclosed signed angle $\theta(t)/2$ (for
brevity, we often suppress the parameter $t$)

$\tan\frac{\theta}{2}=\tan(\alpha_{1}-\alpha_{2})=\frac{s_{1}-f}{1+s_{1}f}=\frac{f-s_{2}}{1+s_{2}f}\,,$

hence

$\tan^{2}\frac{\theta}{2}=\frac{(s_{1}-f)(f-s_{2})}{(1+s_{1}f)(1+s_{2}f)}=\frac{f(s_{1}+s_{2})-s_{1}s_{2}-f^{2}}{f(s_{1}+s_{2})+1+f^{2}s_{1}s_{2}}\,.$

After some computations, we obtain

$\tan^{2}\frac{\theta}{2}=\frac{k_{e}}{a_{e}^{2}\sin^{2}t+b_{e}^{2}\cos^{2}t-k_{e}}=\frac{k_{e}}{\|{\mathbf{t}}_{e}\|^{2}-k_{e}}=\frac{k_{e}}{\|{\mathbf{t}}_{c}\|^{2}}\,,$

therefore

$\cot^{2}\frac{\theta}{2}=\frac{\|{\mathbf{t}}_{e}\|^{2}}{k_{e}}-1\mskip
9.0mu\mbox{and}\mskip
9.0mu\sin^{2}\frac{\theta}{2}=\frac{1}{1+\cot^{2}\frac{\theta}{2}}=\frac{k_{e}}{\|{\mathbf{t}}_{e}\|^{2}}\,,$

where $k_{e}=a_{e}^{2}-a_{c}^{2}=b_{e}^{2}-b_{c}^{2}$, and finally

$\cos\theta=1-2\sin^{2}\frac{\theta}{2}=1-\frac{2k_{e}}{\|{\mathbf{t}}_{c}\|^{2}+k_{e}}=\frac{\|{\mathbf{t}}_{c}\|^{2}-k_{e}}{\|{\mathbf{t}}_{c}\|^{2}+k_{e}}\,.$
∎

######  Remark 2.3.

A change of the origin $k=0$ for the elliptic coordinates in a family of
confocal conics corresponds to a shift of the coordinates. Hence, if in Lemma
2.2 the ellipse $c$ is replaced by another confocal conic with the coordinate
$k$, then the formulas (2.11) and (2.12) remain valid under the condition that
we replace $k_{e}$ by $k_{e}-k$ and $k_{h}$ by $k_{h}-k$.

######  Lemma 2.4.

Let $P_{1}P_{2}$ be a chord of the ellipse $e$, which contacts the caustic $c$
at the point $Q_{1}$. Then the signed distances of the line $[P_{1},P_{2}]$ to
the center $O$ and to the pole $R_{1}$ w.r.t. $e$ have the constant product
$-k_{e}$. The lines $[P_{1},P_{2}]$ and $[Q_{1},R_{1}]$ are orthogonal (Figure
2).

###### Proof.

Let the side $P_{1}P_{2}$ touch the caustic $c$ at the point $Q_{1}=(a_{c}\cos
t_{1}^{\prime},\,b_{c}\sin t_{1}^{\prime})$. Then the Hessian normal form of
the spanned line $t_{Q}=[P_{1},P_{2}]$ reads

$t_{Q}\\!:\ \frac{b_{c}\cos t_{1}^{\prime}\,x+a_{c}\sin
t_{1}^{\prime}\,y-a_{c}b_{c}}{\sqrt{b_{c}^{2}\cos^{2}t_{1}^{\prime}+a_{c}^{2}\sin^{2}t_{1}^{\prime}}}=0.$

Its pole w.r.t. $e$ has the coordinates

$R_{1}=\left(\frac{a_{e}^{2}\cos t_{1}^{\prime}}{a_{c}},\ \frac{b_{e}^{2}\sin
t_{1}^{\prime}}{b_{c}}\right).$ (2.13)

This yields for the signed distances to the line $t_{Q}$

$\overline{Ot_{Q}}=\frac{-a_{c}b_{c}}{\|{\mathbf{t}}_{c}(t_{1}^{\prime})\|}$
(2.14)

and

$\overline{R_{1}t_{Q}}=\overline{R_{1}Q_{1}}={\displaystyle\frac{k_{e}(b_{c}^{2}\cos^{2}t_{1}^{\prime}+a_{c}^{2}\sin^{2}t_{1}^{\prime})}{a_{c}b_{c}\|{\mathbf{t}}_{c}(t_{1}^{\prime})\|}}={\displaystyle\frac{k_{e}\|{\mathbf{t}}_{c}(t_{1}^{\prime})\|}{a_{c}b_{c}}}\,.$
(2.15)

Thus, we obtain a constant product
$\overline{Ot_{Q}}\cdot\overline{R_{1}t_{Q}}=-k_{e}\,$, as stated above.
The last statement holds since $R_{1}$ and $Q_{1}$ are the poles of
$[P_{1},P_{2}]$ w.r.t. the confocal conics $e$ and $c$. It is wellknown that
the poles of any line $\ell$ w.r.t. a family of confocal conics lie on a line
$\ell^{*}$ orthogonal to $\ell$ (see, e.g., [15, p. 340]). ∎

##  3\. Confocal conics and billiards

By virtue of Lemma 2.1, all sides of a billiard inscribed to the ellipse $e$
with parameter $k=k_{e}$ are tangent to a fixed conic $c$ confocal with $e$
(Figure 2). This conic $c$ with parameter $k_{c}$ is called caustic. It can be
a smaller ellipse with $-b^{2}<k_{c}<k_{e}$ or a hyperbola with
$-a^{2}<k_{c}<-b^{2}$ or, in the limiting case with $k_{c}=-b^{2}$, consist of
the pencils of lines with the focal points $F_{1},F_{2}$ of $c$ as carriers.
At the beginning, we confine ourselves to an ellipse as caustic with $k_{c}=0$
(Figure 2), and we speak of elliptic billiards. Only the Figures 10, 11 and 12
show examples of (periodic) billiards in $e$ with a hyperbola as caustic,
called hyperbolic billards.

For billiards $\dots P_{1}P_{2}P_{3}\dots$ in the ellipse $e$ and with the
ellipse $c$ as caustic, we assume from now on a counter-clockwise order and
signed exterior angles $\theta_{1},\theta_{2},\theta_{3},\dots$ (see Figure
2). The tangency points $Q_{1},Q_{2},\dots$ of the billiard’s sides
$P_{1}P_{2},P_{2}P_{3},\dots$ with the caustic $c$ subdivide the sides into
two segments. We denote the lengths of the segments adjacent to $P_{i}$ as

$l_{i}:=\overline{P_{i}Q_{i}}\mskip 9.0mu\mbox{and}\mskip
9.0mur_{i}:=\overline{P_{i}Q_{i-1}}.$ (3.1)

Based on the parametrizations $(a_{e}\cos t,\,b_{e}\sin t)$ of $e$ and
$(a_{c}\cos t^{\prime},\,b_{c}\sin t^{\prime})$ of $c$, we denote the
respective parameters of $P_{1}$, $Q_{1}$, $P_{2}$, $Q_{2}$, $P_{3},\dots$
with $t_{1}$, $t_{1}^{\prime}$, $t_{2}$, $t_{2}^{\prime}$, $t_{3},\dots$ in
strictly increasing order.

The following two lemmas deal with sides of billiards in the ellipse $e$.

######  Lemma 3.1.

The connecting line $[P_{i},P_{i+1}]$ of the vertices with respective
parameters $t_{1},t_{2}$ on $e$ contacts the caustic $c$ if and only if

$\frac{a_{c}^{2}}{a_{e}^{2}}\,\cos^{2}\frac{t_{1}+t_{2}}{2}+\frac{b_{c}^{2}}{b_{e}^{2}}\,\sin^{2}\frac{t_{1}+t_{2}}{2}=\cos^{2}\frac{t_{1}-t_{2}}{2}.$

This is equivalent to

$\sin^{2}\frac{t_{1}-t_{2}}{2}=\frac{k_{e}}{a_{e}b_{e}}\,\Bigl{\|}\,{\mathbf{t}}_{e}\Bigl{(}\mbox{\small$\displaystyle\frac{t_{1}+t_{2}}{2}$
}\Bigr{)}\Bigr{\|}^{2}.$

###### Proof.

The line connecting the points $(a_{e}\cos t_{i},\,b_{e}\sin t_{i})$, $i=1,2$,
has homogeneous line coordinates $(u_{0}:u_{1}:u_{2})$ equal to

$\left(a_{e}b_{e}(\cos t_{1}\sin t_{2}-\sin t_{1}\cos t_{2}):b_{e}(\sin
t_{1}-\sin t_{2}):a_{e}(\cos t_{2}-\cos t_{1})\right).$

It contacts the caustic $c$ if
$-u_{0}^{2}+a_{c}^{2}u_{1}^{2}+b_{c}^{2}u_{2}^{2}=0$, i.e.,

$\begin{array}[]{l}a_{c}^{2}b_{e}^{2}\sin^{2}{\displaystyle\frac{t_{1}-t_{2}}{2}}\cos^{2}{\displaystyle\frac{t_{1}+t_{2}}{2}}+b_{c}^{2}a_{e}^{2}\sin^{2}{\displaystyle\frac{t_{1}-t_{2}}{2}}\sin^{2}{\displaystyle\frac{t_{1}+t_{2}}{2}}\\\\[7.11317pt]
=a_{e}^{2}b_{e}^{2}\sin^{2}{\displaystyle\frac{t_{2}-t_{1}}{2}}\cos^{2}{\displaystyle\frac{t_{2}-t_{1}}{2}}\,.\end{array}$

Under the condition $\sin[(t_{1}-t_{2})/2]\neq 0$ we obtain the first claimed
equation. The second follows after the substitutions
$a_{c}^{2}=a_{e}^{2}-k_{e}$ and $b_{c}^{2}=b_{e}^{2}-k_{e}$ from

$1-\frac{k_{e}}{a_{e}^{2}b_{e}^{2}}\left(b_{e}^{2}\cos^{2}\frac{t_{1}+t_{2}}{2}+a_{e}^{2}\sin^{2}\frac{t_{1}+t_{2}}{2}\right)=\cos^{2}\frac{t_{2}-t_{1}}{2}\,.$

by (2.9). ∎

######  Lemma 3.2.

Referring to the notation in Lemma 3.1, if the side $P_{i}P_{i+1}$ contacts
the caustic $c$ at $Q_{i}$ with parameter $t_{i}^{\prime}$, then

$\sin
t_{i}^{\prime}=\frac{b_{c}}{b_{e}}\,\frac{\sin\frac{t_{i}+t_{i+1}}{2}}{\cos\frac{t_{i}-t_{i+1}}{2}}\,,\
\cos
t_{i}^{\prime}=\frac{a_{c}}{a_{e}}\,\frac{\cos\frac{t_{i}+t_{i+1}}{2}}{\cos\frac{t_{i}-t_{i+1}}{2}}\,,\
\tan
t_{i}^{\prime}=\frac{b_{c}a_{e}}{a_{c}b_{e}}\,\tan\frac{t_{i}+t_{i+1}}{2}\,.$

###### Proof.

The tangent to $c$ at $Q_{1}$ has the line coordinates

$(u_{0}:u_{1}:u_{2})=\left(-a_{c}b_{c}:b_{c}\cos t_{1}^{\prime}:a_{c}\sin
t_{1}^{\prime}\right),$

which must be proportional to those in the proof of Lemma 3.1. ∎

######  Remark 3.3.

The half-angle substitution

$\tau_{i}:=\tan\frac{t_{i}}{2}\quad\mbox{for}\ i=1,2$

allows to express the equation in Lemma 3.1 (for $i=1$) in projective
coordinates on $e$. We obtain a symmetric biquadratic condition

$b_{e}^{2}k_{e}\tau_{1}^{2}\tau_{2}^{2}-b_{c}^{2}a_{e}^{2}(\tau_{1}^{2}+\tau_{2}^{2})+2(a_{e}^{2}k_{e}+a_{c}^{2}b_{e}^{2})\tau_{1}\tau_{2}+b_{e}^{2}k_{e}=0,$

which defines a 2-2-correspondence on $e$ between the endpoints $P_{1},P_{2}$
of a chord which contacts $c$. This remains valid after iteration, i.e.,
between the initial point $P_{1}$ and the endpoint $P_{N+1}$ of a billiard
after $N$ reflections in $e$.
Now, we recall a classical argument for the underlying Poncelet porism (see
also [17] and the references there): A 2-2-correspondence different from the
identity keeps fixed at most four points. However, four fixed points on $e$
are already known as contact points between $e$ and the common complex
conjugate (isotropic) tangents222 They follow from the second equation in
Lemma 3.1 from $t_{1}=t_{2}$. with the caustic $c$, since tangents of $e$
remain fixed under the reflection in $e$. If therefore one $N$-sided billiard
in $e$ with caustic $c$ closes, then the correspondence is the identity and
all these billiards close.

Figure 3. The Joachimsthal integral
$J_{e}:=-\langle{\mathbf{u}}_{i},\,{\mathbf{n}}_{e|i}\rangle$ is constant
along $e$.

Given any billiard $P_{1}P_{2}\dots$ in the ellipse $e$, let
${\mathbf{p}}_{i}=(x_{i},\,y_{i})$ denote the position vector of $P_{i}$ for
$i=1,2,\dots$, while ${\mathbf{u}}_{1},{\mathbf{u}}_{2},\dots$ denote the unit
vectors of the oriented sides $P_{1}P_{2}$, $P_{2}P_{3}$, … . By (2.9), the
vector ${\mathbf{n}}_{e|i}:=(x_{i}/a_{e}^{2},\ y_{i}/b_{e}^{2})$ is orthogonal
to $e$ at $P_{i}$. According to [2, Proposition 2.1], the scalar product

$J_{e}:=-\langle{\mathbf{u}}_{i},\,{\mathbf{n}}_{e|i}\rangle$ (3.2)

is invariant along the billiard in $e$ and called Joachimsthal integral (note
also [26, p. 54]).

The invariance of the Joachimsthal integral, which also holds in higher
dimensions for billiards in quadrics, is the key result for the integrability
of billiards, i.e., in the planar case for the existence of a caustic [2, p.
3]. In our approach, the invariance of $J_{e}$ follows from Lemma 2.2.

######  Lemma 3.4.

The Joachimsthal integral
$J_{e}:=-\langle{\mathbf{u}}_{i},\,{\mathbf{n}}_{e|i}\rangle$ equals

$J_{e}=\frac{\sqrt{k_{e}}}{a_{e}b_{e}}$

with $k_{e}$ as elliptic coordinate of $e$ w.r.t. $c$, i.e.,
$k_{e}=a_{e}^{2}-a_{c}^{2}=b_{e}^{2}-b_{c}^{2}$.

###### Proof.

From (2.11) follows for the points $(a_{e}\cos t,\ b_{e}\sin t)$ of $e$

$J_{e}=-\langle{\mathbf{u}},\,{\mathbf{n}}_{e}\rangle=-\cos\Bigl{(}\frac{\pi}{2}+\frac{\theta}{2}\Bigr{)}\,\|{\mathbf{n}}_{e}\|=\sin\frac{\theta}{2}\,\|{\mathbf{n}}_{e}\|=\sin\frac{\theta}{2}\,\frac{\|{\mathbf{t}}_{e}\|}{a_{e}b_{e}}\,,$

hence by (2.11), (2.7), (2.6), and (2.9)

$J_{e}^{2}=\sin^{2}\frac{\theta}{2}\,\|{\mathbf{n}}_{e}\|^{2}=\frac{k_{e}}{\|{\mathbf{t}}_{e}\|^{2}}\,\|{\mathbf{n}}_{e}\|^{2}=\frac{k_{e}}{a_{e}^{2}\,b_{e}^{2}}\,.$

This confirms the claim. ∎

###  3.1. Poncelet grid

The following theorem is the basis for the Poncelet grid associated to each
billiard. We formulate and prove a projective version. The special case
dealing with confocal conics, has already been published by Chasles [9, p.
841] and later by Böhm in [8, p. 221]. The same theorem was studied in [22]
and in [1]. In [18], the authors proved it in a differential-geometric way.

In the theorem and proof below, the term conic stands for regular dual conics,
i.e., conics seen as the set of tangent lines, but also for pairs of line
pencils and for single line pencils with multiplicity two. Expressed in terms
of homogeneous line coordinates, the corresponding quadratic forms have rank
3, 2 or 1, respectively. Moreover, we use the term range for a pencil of dual
conics. The term net denotes a 2-parametric linear system of dual curves of
degree 2. Obviously, conics and ranges included in a net play the role of
points and lines of a projective plane within the 5-dimensional projective
space of dual conics. Any two ranges in a net share a conic (compare with [10,
Théorèmes I – IV]).

Figure 4. Left: Opposite vertices $A_{i},B_{i}$ of the quadrilateral
$t_{1}\dots t_{4}$ of tangents to $c$ belong to a conic $c_{i}$ out of the
range ${\mathcal{R}}_{c}$ (Theorem 3.5). Right: The net $\mathcal{N}$ and the
ranges ${\mathcal{R}}_{t}$, ${\mathcal{R}}_{c}$ in the projective space of
dual conics.

######  Theorem 3.5.

Let $c$ be a regular conic and $A_{1},B_{1}$ two points such that the tangents
$t_{1},\dots,t_{4}$ drawn from $A_{1}$ and $B_{1}$ to $c$ form a
quadrilateral. Its remaining pairs of opposite vertices are denoted by
$(A_{i},B_{i})$, $i=2,3$. Then,

  1. (1)

for each conic $c_{1}$ passing through $A_{1}$ and $B_{1}$, the range
${\mathcal{R}}_{c}$ spanned by $c$ and $c_{1}$ contains conics $c_{i}$ passing
through $A_{i}$ and $B_{i}$, simultaneously. The tangents at $A_{j}$ and
$B_{j}$ to $c_{j}$ for $j=1,2,3$ meet at a common point $R$. If $c_{i}$ has
rank 2, then we obtain, as the limit of $c_{i}\,$, the diagonal
$[A_{i},B_{i}]$ of the quadrilateral $t_{1},\dots,t_{4}$.

  2. (2)

This result holds also in the limiting case $t_{1}=t_{2}$, where the chord
$A_{1}B_{1}$ of $c_{1}$ contacts $c$ at $B_{2}$.

In Figure 4, the particular case is displayed where $c$ and $c_{1}=e$ span a
range $\mathcal{R}_{c}$ of confocal conics (note also [6, Fig. 19]). Then by
Lemma 2.1, the tangents at $A_{j}$ and $B_{j}$ to $c_{j}$ are angle bisectors
of the quadrilateral. In case of a rank deficiency of $c_{i}\,$, either one
axis of symmetry of the confocal family or the line at infinity shows up as
$c_{i}\,$.

###### Proof.

The conics tangent to $t_{1},\dots,t_{4}$ define a range $\mathcal{R}_{t}$,
which includes for $j=1,2,3$ the pairs of line pencils $(A_{j},B_{j})$ as well
as the initial conic $c$. On the other hand, $c$ and $c_{1}$ span a range
$\mathcal{R}_{c}$. Since both ranges share the conic $c$, they span a net
$\mathcal{N}$ of conics.

The pair $(A_{1},B_{1})$ of line pencils spans together with $c_{1}$ the range
of conics sharing the points $A_{1},B_{1}$ and the tangents there, which meet
at a point $R$. This range, which also belongs to $\mathcal{N}$, contains the
rank-1 conic with carrier $R$. Each pair of line pencils $(A_{i},B_{i})$,
$i=1,2\,$, spans with the pencil $R$ again a range within $\mathcal{N}$. This
range shares with the range $\mathcal{R}_{c}$ a conic $c_{i}$ passing through
$A_{i}$ and $B_{i}$ with respective tangent lines through $R$.333 An extended
version of this theorem in [23] addresses the symmetry between the ranges
$\mathcal{R}_{c}$ and $\mathcal{R}_{t}$. This generalizes the statement that
in the case of confocal conics $c$ and $c_{1}$ the quadrilateral
$A_{1}A_{2}B_{1}B_{2}$ has an incircle $d$ (Figures 4, 6 and 14, compare with
[1, 18]).

All these conclusions remain valid in the case, when $\mathcal{R}_{t}$
consists of conics which touch $c$ at $B_{2}$ and are tangent to $t_{3}$ and
$t_{4}\,$. ∎

As already indicated by the notation, we are interested in the particular case
of Theorem 3.5 where the conics $c$ and $e$ in the range $\mathcal{R}_{c}$ are
confocal. The following result follows directly from Theorem 3.5 and
summarizes properties of the Poncelet grid. For the points of intersection
between extended sides of a billiard $\dots P_{0}P_{1}P_{2}\dots$ we use the
notation

$S_{i}^{(j)}:=\left\\{\begin{array}[]{rl}[P_{i-k-1},P_{i-k}]\cap[P_{i+k},P_{i+k+1}]&\mbox{for}\mskip
9.0muj=2k,\\\\[1.70717pt]
[P_{i-k},P_{i-k+1}]\cap[P_{i+k},P_{i+k+1}]&\mbox{for}\mskip
9.0muj=2k-1\end{array}\right.$ (3.3)

where $i=\dots,0,1,2,\dots$ and $j=1,2,\dots$ Note that there are $j$ sides
between those which intersect at $S_{i}^{(j)}$, and ‘in the middle’ of these
$j$ sides there is for even $j$ the vertex $P_{i}$ and otherwise the point of
contact $Q_{i}\,$. At the same token, the point $S_{i}^{(j)}$ is the pole of
the diagonal $[Q_{i-k-1},Q_{i+k}]$ or $[Q_{i-k},Q_{i+k}]$ of the polygon
$\dots Q_{1}Q_{2}Q_{3}\dots$ of contact points w.r.t. the caustic.

######  Theorem 3.6.

Let $\dots P_{0}P_{1}P_{2}\dots$ be a billiard in the ellipse $e$ with sides
$P_{i}P_{i+1}$ which contact the ellipse $c$ at the respective points $Q_{i}$
for all $i\in{\mathbb{Z}}$. Then the vertices $S_{i}^{(j)}$ of the associated
Poncelet grid are distributed on the following conics.

  1. (1)

The points $S_{i}^{(1)}$, $S_{i}^{(3)},\dots$ are located on the confocal
hyperbola through $Q_{i}$, while the points $S_{i}^{(2)}$, $S_{i}^{(4)},\dots$
are located on the confocal hyperbola through $P_{i}$.

  2. (2)

For each $j\in\\{1,2,\dots\\}$, the points $\dots
S_{i}^{(j)}S_{i+(j+1)}^{(j)}S_{i+2(j+1)}^{(j)}\dots$ are vertices of another
billiard with the caustic $c$ inscribed in a confocal ellipse $e^{(j)}$,
provided that $e^{(j)}$ is regular. Otherwise $e^{(j)}$ coincides with an axis
of symmetry or with the line at infinity. The locus $e^{(j)}$ is independent
of the position of the initial vertex $P_{0}\in e$.

Figure 5. Periodic billiard $(N\\!=7)$ with extended sides.

###### Proof.

1\. The side lines $[P_{0},P_{1}]$ ($P_{0}\\!=\\!P_{7}$ in Figure 5) and
$[P_{2},P_{3}]$ meet at $S_{1}^{(1)}$, while $[P_{1},P_{2}]$ contacts $c$ at
$Q_{1}$. By Theorem 3.5, 2. the points $Q_{1}$ and $S_{1}^{(1)}$ belong to the
same confocal hyperbola.
Now we go one step away from $Q_{1}$: the tangents from $P_{0}$ and $P_{3}$ to
$c$ intersect at $S_{1}^{(1)}$ and
$S_{1}^{(3)}=[P_{-1},P_{0}]\cap[P_{3},P_{4}]$. The confocal conic through
$S_{1}^{(1)}$ and $S_{1}^{(3)}$ must again be the hyperbola through $Q_{1}$.
This follows by continuity after choosing $Q_{1}$ on one axis of symmetry.
Iteration confirms the first claim in Theorem 3.6.
The tangents to $c$ from $P_{0}$ and $P_{2}$ form a quadrilateral with $P_{1}$
and $S_{1}^{(2)}$ as opposite vertices. Therefore, there exists a confocal
conic passing through both points. This conic must be a hyperbola, as can be
concluded by continuity: If $P_{1}$ is specified at a vertex of $e$, then due
to symmetry the points $P_{1}$ and $S_{1}^{(2)}$ are located on an axis of
symmetry.
The tangents to $c$ from $P_{-1}$ and $P_{3}$ form a quadrilateral with
$S_{1}^{(2)}$ and $S_{1}^{(4)}$ as opposite vertices. Theorem 3.5 and
continuity guarantee that this is again the confocal hyperbola through
$P_{1}$. Iteration shows the same of $S_{1}^{(6)}$ etc. However, the points
$P_{1}$, $S_{1}^{(2)}$, $S_{1}^{(4)},\dots$ need not belong to the same branch
of the hyperbola.

2\. The tangents through $P_{2}$ and $S_{2}^{(2)}$ (note Figure 5) form a
quadrilateral with $S_{1}^{(1)}$ and $S_{2}^{(1)}$ as opposite vertices. This
time, continuity shows that the two points belong to the same confocal ellipse
$e^{(1)}$. The same holds for the tangents through $P_{3}$ and $S_{3}^{(2)}$
etc.
Similarily, starting with the points $P_{0}$ and $P_{3}$, we find the ellipse
$e^{(2)}$ through $S_{1}^{(2)}$ and $S_{2}^{(2)}$, and so on.

In order to prove that these ellipses $e^{(1)},e^{(2)},\dots$ are independent
of the choice of the initial point $P_{1}\in e$, we follow an argument from
[2, proof of Corollary 2.2]: The claim holds for all confocal ellipses $e$
where billiards with the same caustic $c$ are aperiodic and traverse $e$
infinitely often. Since these ellipses form a dense set, the claim holds also
for those with periodic billiards. The invariance of the ellipses
$e^{(1)},e^{(2)},\dots$ is already mentioned in [2, Theorem 7].

An alternative proof consists in demonstrating that the elliptic coordinate
$k_{e}^{(j)}$ of $e^{(j)}$ for all $j$ does not depend on the parameter $t$.
As one example, we present below in (3.7), (3.6) and (3.8) formulas for the
semiaxes $a_{e|1}$, $b_{e|1}$ and the elliptic coordinate $k_{e|1}$ of
$e^{(1)}$. ∎

######  Remark 3.7.

Figure 5 reveals, that the polygons $P_{1}S_{1}^{(1)}P_{2}S_{2}^{(1)}\dots$ as
well as $P_{1}S_{2}^{(2)}P_{3}S_{4}^{(2)}\dots$ and
$S_{1}^{(2)}S_{1}^{(1)}S_{2}^{(2)}S_{2}^{(1)}\dots$ are zigzag billiards in
rings bounded by two confocal ellipses. However, we find also zigzag billiards
between two confocal hyperbolas, e.g., $\dots
S_{1}^{(2)}P_{2}P_{1}S_{2}^{(2)}\dots$ or the twofold covered $\dots
S_{1}^{(2)}$ $S_{1}^{(1)}P_{1}Q_{1}P_{1}S_{1}^{(1)}S_{1}^{(2)}\dots\,$.
Billiards between other combinations of confocal conics can be found in [12].

Figure 6. The power $w^{2}$ of $S_{2}^{(1)}$ w.r.t. the incircle of the
triangle $S_{2}^{(1)}P_{2}P_{3}$ shows up at all $S_{i}^{(1)}$ and equals the
power of $P_{i}$ w.r.t. the incircle of the quadrangle
$P_{i}S_{i-1}^{(1)}S_{i}^{(2)}S_{i}^{(1)}$.

The coming lemma addresses invariants related to the incircles of
quadrilaterals built from the tangents to $c$ from any two vertices $P_{i}$
and $P_{j}$ of a billiard in $e$ (note circle $d$ in Figure 4 and [1, 5, 18]).

######  Lemma 3.8.

Referring to Figure 6, the power $w^{2}$ of the point $S_{i}^{(1)}$ w.r.t. the
incircle of the triangle $P_{i}P_{i+1}S_{i}^{(1)}$ is the same for all $i$.
Similarly, the power $w_{1}^{2}$ of $S_{i}^{(2)}$ w.r.t. the incircle of the
quadrangle $P_{i}S_{i-1}^{(1)}S_{i}^{(2)}S_{i}^{(1)}$ is constant.

###### Proof.

According to Graves’s construction [15, p. 47], an ellipse $e$ can be
constructed from a smaller ellipse $c$ in the following way: Let a closed
piece of string strictly longer than the perimeter of $c$ be posed around $c$.
If point $P$ is used to pull the string taut, then $P$ traces a confocal
ellipse $e$. Consequently, for each vertex $P_{i}$ and neighboring tangency
points $Q_{i-1}$ and $Q_{i}$ of a billiard in $e$ with caustic $c$, the sum of
the lengths $\overline{Q_{i-1}P_{i}}$ and $\overline{P_{i}Q_{i}}$ minus the
length of the elliptic arc between $Q_{i-1}$ and $Q_{i}$, i.e.,

$D_{e}:=\overline{Q_{i-1}P_{i}}+\overline{P_{i}Q_{i}}-\overset{\mbox{\raisebox{-2.27621pt}{\Large$\frown$}}}{Q_{i-1}Q_{i}}$
(3.4)

is constant (Figure 6).

The incircle of $P_{2}P_{3}S_{2}^{(1)}$ has the center $R_{2}$ and the radius
$\overline{Q_{2}R_{2}}$ by (2.15). The power of $P_{2}$ w.r.t. this circle is
$l_{2}^{2}$, that of $P_{3}$ is $r_{3}^{2}$. From Graves’ construction follows
for the ellipse $e$ that

$D_{e}=r_{2}+l_{2}-\overset{\mbox{\raisebox{-2.27621pt}{\Large$\frown$}}}{Q_{1}Q_{2}}=r_{3}+l_{3}-\overset{\mbox{\raisebox{-2.27621pt}{\Large$\frown$}}}{Q_{2}Q_{3}}=\mbox{const.}$

is the same for all $P_{i}$, provided that
$\overset{\mbox{\raisebox{-2.27621pt}{\Large$\frown$}}}{Q_{i}Q_{j}}$ denotes
the length of the (shorter) arc along $c$ between $Q_{i}$ and $Q_{j}$. For the
analogue invariant at $e^{(1)}$ follows (Figure 6)

$\begin{array}[]{rcl}D_{e|1}&:=&\overline{Q_{1}S_{2}^{(1)}}+\overline{S_{2}^{(1)}Q_{3}}-\overset{\mbox{\raisebox{-2.27621pt}{\Large$\frown$}}}{Q_{1}Q_{3}}\\\\[1.42262pt]
&=&(r_{2}+l_{2}+w)+(r_{3}+l_{3}+w)-\overset{\mbox{\raisebox{-2.27621pt}{\Large$\frown$}}}{Q_{1}Q_{3}}=2D_{e}+2w=\mbox{const.},\end{array}$
(3.5)

hence $w=\mbox{const.}$, where $w^{2}$ is the power of $S_{2}^{(1)}$ w.r.t.
the said incircle.

Since the incircle of the quadrangle $P_{2}S_{1}^{(1)}S_{2}^{(2)}S_{2}^{(1)}$
is an excircle of the triangle $P_{2}P_{3}S_{2}^{(1)}$ (Figure 6), the power
of $P_{2}$ w.r.t. the excircle equals $w^{2}$, too. This follows from
elementary geometry.

As an alternative, the constancy of $w$ can also be concluded from the fact,
that neighboring circles with centers $R_{i}^{1},R_{i}$ or $R_{i},R_{i+1}^{1}$
share three tangents, and one circle is an incircle, the other an excircle of
the triangle. Therefore, on the common side the same lengths $w$ shows up
twice and also at the adjacent pairs of neighboring circles. Since the
distance $w$ is constant for all aperiodic billiards, it reveals also for
periodic billiards that $w$ is independent of the choice of the initial
vertex. In a similar way follows the invariance of the lengths $w_{1}$, as
shown in Figure 6. ∎

It needs to be noted that $S_{2}^{(1)}$ or $S_{2}^{(2)}$ can be located on the
other branch of the related hyperbola. Then the said ‘incircle’ of the
triangle $P_{2}P_{3}S_{2}^{(1)}$ has to be replaced by the excircle which
contacts $c$ at $Q_{2}$ and is tangent to the side lines $[P_{1},P_{2}]$ and
$[P_{3},P_{4}]$. Similarly, the said ’incircle’ of the quadrangle becomes an
excircle. In all these cases, Lemma 3.8 and the proof given above have to be
adapted.

Figure 7. Periodic billiard $P_{1}P_{2}\dots P_{7}$ on the sphere with
extended sides and their contact points with the incircles. All circular arcs
marked in black have the same length, as well as those marked in green, and
both lengths are invariant against changes of $P_{1}$ on $e$.

######  Remark 3.9.

The Theorems 3.5 and 3.6 as well as the constancy of the length $w$ according
to Lemma 3.8 are also valid in spherical geometry (note [23]) and in
hyperbolic geometry. On the sphere (see Figure 7), the caustic consists of a
pair of opposite components, and for $N$-periodic billiards the confocal
spherical ellipse $e^{(j)}$ coincides with $e^{(N-2-j)}$ w.r.t. the opposite
caustic.
We obtain other families of incircles when we focus on pairs of consecutive
sides of the billiards in $e^{(1)}$, $e^{(2)}$ and so on. However, these
circles are not mutually disjoint. By the way, the centers $R_{i}^{(j)}$ of
all these circles are the poles of diagonals of $\dots P_{1}P_{2}P_{3}\dots$
w.r.t. the ellipse $e$.

For the sake of completeness, we express below in (3.9) the distance $w$ in
terms of the semiaxes of $e$ and $c\,$. For this purpose, we compute first the
semiaxes $a_{e|1}$ an $b_{e|1}$ of $e^{(1)}$, since we need the coordinates of
$S_{2}^{(1)}$. From (2.13) and (2.15) follows

$R_{2}=\left(\frac{a_{e}^{2}\cos t_{2}^{\prime}}{a_{c}},\ \frac{b_{e}^{2}\sin
t_{2}^{\prime}}{b_{c}}\right),\quad\overline{Q_{2}R_{2}}=\frac{k_{e}\|{\mathbf{t}}_{c}(t_{2}^{\prime})\|}{a_{c}b_{c}}\,.$

By virtue of Theorem 3.5, the tangents from $R_{2}$ to the confocal hyperbola
through $Q_{2}$ contact at

$Q_{2}=(a_{c}\cos t_{2}^{\prime},\ b_{c}\sin t_{2}^{\prime})\in
c\quad\mbox{and}\quad S_{2}^{(1)}=(a_{e|1}\cos t_{2}^{\prime},\ b_{e|1}\sin
t_{2}^{\prime})\in e^{(1)}.$

Both points lie on the polar of $R_{2}$ w.r.t. the hyperbola in question with
the elliptic coordinate $k_{h}=-\|{\mathbf{t}}_{e}(t_{2}^{\prime})\|^{2}$.
This yields the condition

$\frac{a_{e}^{2}\cos
t_{2}^{\prime}}{a_{c}(a_{c}^{2}+k_{h}(t_{2}^{\prime}))}\,a_{e|1}\cos
t_{2}^{\prime}+\frac{b_{e}^{2}\sin
t_{2}^{\prime}}{b_{c}(b_{c}^{2}+k_{h}(t_{2}^{\prime}))}\,b_{e|1}\sin
t_{2}^{\prime}=1$

or, by virtue of (2.7),

$a_{e}^{2}b_{c}a_{e|1}-b_{e}^{2}a_{c}b_{e|1}=a_{c}b_{c}d^{2},\quad\mbox{where}\quad
a_{e|1}^{2}-b_{e|1}^{2}=d^{2}.$

We eliminate $a_{e|1}$ and obtain after some computation the quadratic
equation

$(b_{e}^{4}-2b_{c}^{2}b_{e}^{2}-b_{c}^{2}d^{2})\,b_{e|1}^{2}+2a_{c}^{2}b_{c}b_{e}^{2}\,b_{e|1}+b_{c}^{2}(a_{c}^{2}d^{2}-a_{e}^{4})=0.$

The second solution besides $b_{e|1}=b_{c}$ is

$b_{e|1}=\frac{b_{c}(a_{c}^{2}d^{2}-a_{e}^{4})}{b_{e}^{4}-2b_{c}^{2}b_{e}^{2}-b_{c}^{2}d^{2}}=\frac{b_{c}(a_{e}^{2}b_{e}^{2}+d^{2}k_{e})}{a_{c}^{2}b_{c}^{2}-k_{e}^{2}}.$
(3.6)

This implies

$a_{e|1}=\frac{a_{c}}{a_{e}^{2}b_{c}}\left(b_{c}d^{2}+b_{e}^{2}b_{e|1}\right)=\frac{a_{c}(a_{e}^{2}b_{e}^{2}-d^{2}k_{e})}{a_{c}^{2}b_{c}^{2}-k_{e}^{2}}$
(3.7)

and

$k_{e|1}=b_{e|1}^{2}-b_{c}^{2}=k_{e}\left(\frac{2a_{c}b_{c}a_{e}b_{e}}{a_{c}^{2}b_{c}^{2}-k_{e}^{2}}\right)^{\\!\\!2}.$
(3.8)

This leads finally to

$w:=\frac{2a_{e}b_{e}\sqrt{k_{e}^{3}}}{a_{c}^{2}b_{c}^{2}-k_{e}^{2}}\,.$ (3.9)

Negative semiaxes $a_{e|1},b_{e|1}$ and a negative $w$ in the formulas above
mean that the points $S_{i}^{(1)}$ are located on the respective second
branches of the hyperbolas and the incircles of the triangles
$P_{i}P_{i+1}S_{i}^{(1)}$ become excircles. In the case of a vanishing
denominator for $k_{e}=a_{c}b_{c}$ (periodic four-sided billiard) the ellipse
$e^{(1)}$ is the line at infinity.

If on the right-hand side of the formulas (3.7), (3.6) and (3.8) we replace
$a_{e},b_{e},k_{e}$ respectively by $a_{e|1},b_{e|1},k_{e|1}$, then we obtain
expressions for $a_{e|3},b_{e|3},k_{e|3}$, i.e.,

$k_{e|3}=k_{e|1}\left(\frac{2a_{c}b_{c}a_{e|1}b_{e|1}}{a_{c}^{2}b_{c}^{2}-k_{e|1}^{2}}\right)^{\\!\\!2},$
(3.10)

and this can be iterated.

###  3.2. Conjugate billiards

For two confocal ellipses $c$ and $e$, there exists an axial scaling

$\alpha\\!:\,(x,y)\mapsto\Bigl{(}\mbox{\small$\displaystyle\frac{a_{e}}{a_{c}}$
}x,\,\mbox{\small$\displaystyle\frac{b_{e}}{b_{c}}$
}y\Bigr{)}\quad\mbox{with}\quad c\to e\,.$ (3.11)

Corresponding points share the parameter $t$. Hence, they belong to the same
confocal hyperbola (Figure 8). The affine transformation $\alpha$ maps the
tangency point $Q_{i}\in c$ of the side $P_{i}P_{i+1}$ to a point
$P_{i}^{\prime}\in e$, while $\alpha^{-1}$ maps $P_{i}$ to the tangency point
$Q_{i-1}^{\prime}$ of $P_{i-1}^{\prime}P_{i}^{\prime}$, i.e.,

$\alpha\\!:\,Q_{i}\mapsto P_{i}^{\prime},\quad Q_{i-1}^{\prime}\mapsto
P_{i}\,.$

This results from the symmetry between $t_{i}$ and $t_{i}^{\prime}$ in the
equation

$b_{c}a_{e}\cos t_{i}\cos t_{i}^{\prime}+a_{c}b_{e}\sin t_{i}\sin
t_{i}^{\prime}=a_{c}b_{c}$ (3.12)

which expresses that $P_{i}\in e$ with parameter $t_{i}$ lies on the tangent
to $c$ at $Q_{i}$ with parameter $t_{i}^{\prime}$. Referring to Figure 8,
$\alpha$ sends the tangent $[P_{i-1}^{\prime},P_{i}^{\prime}]$ to $c$ at
$Q_{i-1}^{\prime}$ to the tangent $[R_{i-1},R_{i}]$ to $e$ at $P_{i}\,$.
Hence, by $\alpha$ the polygon $Q_{1}Q_{2}\dots$ is mapped to
$P_{1}^{\prime}P_{2}^{\prime}\dots$ and futhermore to that of the poles
$R_{1}R_{2}\dots$ of the billiard’s sides $P_{1}P_{2},\,P_{2}P_{3},\dots\,$.

Figure 8. The periodic billiard $P_{1}P_{2}\dots P_{5}$ in $e$ with the
caustic $c$ and the conjugate billiard $P_{1}^{\prime}P_{2}^{\prime}\dots
P_{5}^{\prime}\,$.

######  Definition 3.10.

Referring to Figure 8, the billiard $\dots
P_{0}^{\prime}P_{1}^{\prime}P_{2}^{\prime}\dots$ is called conjugate to the
billiard $\dots P_{0}P_{1}P_{2}\dots$ in the ellipse $e$ with the ellipse $c$
as caustic, when the axial scaling $\alpha\\!:\,c\to e$ defined in (3.11) maps
the tangency point $Q_{i}$ of the side $P_{i}P_{i+1}$ to the vertex
$P_{i}^{\prime}$.

######  Lemma 3.11.

For each billiard $\dots P_{0}P_{1}P_{2}\dots$ in the ellipse $e$ with the
ellipse $c$ as caustic, there exists a unique conjugate billiard $\dots
P_{0}^{\prime}P_{1}^{\prime}P_{2}^{\prime}\dots$, and the relation between the
two billiards in $e$ is symmetric. Moreover,

$l_{i}=\overline{P_{i}Q_{i}}=\overline{P_{i}^{\prime}Q_{i-1}^{\prime}}=r_{i}^{\prime}\quad\mbox{and}\quad
r_{i}=\overline{P_{i}Q_{i-1}}=\overline{P_{i-1}^{\prime}Q_{i-1}^{\prime}}=l_{i-1}^{\prime}.$
(3.13)

###### Proof.

From the symmetry in (3.12) follows for $\alpha\\!:\,c\to e$ that $P_{i}$ is
the preimage of $P_{i}$ is the tangency point $Q_{i-1}$ of
$P_{i-1}^{\prime}P_{i}^{\prime}$. The congruences stated in (3.13) follow from
Ivory’s Theorem for the two diagonals in the curvilinear quadrangle
$P_{i}P_{i}^{\prime}Q_{i}Q_{i-1}^{\prime}$. In view of the sequence of
parameters $t_{1},t_{1}^{\prime},t_{2},t_{2}^{\prime},t_{3},\dots$ of the
vertices $P_{1}$, $P_{1}^{\prime}$, $P_{2}$, $P_{2}^{\prime}$, $P_{3},\dots$
on $e$, the switch between the original billiard and its conjugate corresponds
the interchange of $t_{i}$ with $t_{i}^{\prime}$ for $i=1,2,\dots$ . ∎

Finally we recall that, based on the Arnold-Liouville theorem from the theory
of completely integrable systems, it is proved in [18] that there exist
canonical coordinates $u$ on the ellipses $e$ and $c$ such that for any
billiard the transitions from $P_{i}\to P_{i+1}$ and $Q_{i}\to Q_{i+1}$
correspond to shifts of the respective canonical coordinates $u_{i}$ and
$u_{i+1}$ by $2\mskip 1.0mu\varDelta u$. Explicit formulas for the parameter
transformation $t\mapsto u$ are provided in [24].

Figure 9. An example of canonical coordinates on $c$, $e$ and $e^{(1)}$, this
time with origin $Q_{1}$ and unit point $Q_{3}$

Figure 9 shows how on $c$ such coordinates can be constructed by iterated
subdivision, provided that $Q_{N}$ and $Q_{2}$ get the respective canonical
coordinates $u=0$ and $1$. A comparison with Figure 8 reveals that, in the
sense of a canonical parametrization, the contact point $Q_{i}$ is exactly
halfway from the $P_{i}$ to $P_{i+1}$, i.e.,

$u_{i}^{\prime}=u_{i}+\varDelta u,\quad u_{i+1}=u_{i}+2\mskip 1.0mu\varDelta
u\,.$ (3.14)

Hence, the transition from a billiard to its conjugate is equivalent to a
shift of canonical coordinates by $\varDelta u$.

###  3.3. Billiards with a hyperbola as caustic

As illustrated in Figure 10, billiards in ellipses $e$ with a confocal
hyperbola $c$ as caustic are zig-zags between an upper and lower subarc of
$e$. If the initial point $P_{1}$ is chosen at any point of intersection
between $e$ and the hyperbola $c$, then the billiard is twofold covered, and
the first side $P_{1}P_{2}$ is tangent to $c$ at $P_{1}$.

Here we report briefly, in which way these billiard differ from those with an
elliptic caustic. Proofs are left to the readers. In view of the associated
Poncelet grid, we start with the analogue to Theorem 3.6 (see Figures 10, 11
and 12).

Figure 10. Periodic billiard $P_{1}P_{2}\dots P_{12}$ in the ellipse $e$ with
the hyperbola $c$ as caustic, together with the hyperbolas $e^{(1)}$,
$e^{(3)}$ and the ellipse $e^{(2)}$ with the inscribed billiard consisting of
three quadrangles $S_{i}^{(2)}S_{i+3}^{(2)}S_{i+6}^{(2)}S_{i+9}^{(2)}$.

######  Theorem 3.12.

Let $\dots P_{0}P_{1}P_{2}\dots$ be a billiard in the ellipse $e$ with the
hyperbola $c$ as caustic.

  1. (1)

Then the points $S_{i}^{(1)}$, $S_{i}^{(3)},\dots$ are located on confocal
ellipses through the contact point $Q_{i}$ of $[P_{i},P_{i+1}]$ with $c$,
while the points $S_{i}^{(2)}$, $S_{i}^{(4)},\dots$ are located on the
confocal hyperbola through $P_{i}$.

  2. (2)

For even $j$, the points $\dots
S_{i}^{(j)}S_{i+(j+1)}^{(j)}S_{i+2(j+1)}^{(j)}\dots$ are vertices of another
billiard with the caustic $c$ inscribed in a confocal ellipse $e^{(j)}$,
provided that the $S_{i}^{(j)}$ are finite.
For odd $j$, the points $S_{i}^{(j)}$ are located on confocal hyperbolas
$e^{(j)}$ or an axis of symmetry. At each vertex of $\dots
S_{i}^{(j)}S_{i+(j+1)}^{(j)}S_{i+2(j+1)}^{(j)}\dots$, one angle bisector is
tangent to $e^{(j)}$.
All conics $e^{(j)}$ are independent of the position of the initial vertex
$P_{1}\in e$.

At the 12-periodic billiard depicted in Figure 10, the billiard inscribed to
$e^{(2)}$ splits into three quadrangles (dashed). Note that for odd $j$ there
are some points $S_{i}^{(j)}$ where the tangent to $e^{(j)}$ is the interior
bisector of the angle $\angle\,S_{i-(j+1)}^{(j)}S_{i}^{(j)}S_{i+(j+1)}^{(j)}$.
Hence, we obtain no billiards inscribed to hyperbolas $e^{(j)}$ with the
hyperbola $c$ as caustic. In Figure 12, the 6-periodic billiard
$P_{1}P_{2}\dots P_{6}$ yields two triangles
$S_{i}^{(1)}S_{i+2}^{(1)}S_{i+4}^{(1)}$ inscribed to $e^{(1)}$; one of them is
shaded in green.

Lemma 3.8 is also valid for hyperbolas as caustic. An example is depicted in
Figure 11: The power $w^{2}$ of $P_{9}$ w.r.t. the circle tangent to the four
consecutive sides $[P_{7},P_{8}]$, $[P_{8},P_{9}]$,$[P_{9},P_{10}]$, and
$[P_{10},P_{11}]$ equals that of $P_{12}$ w.r.t. the incircle of the
quadrilateral with sides $[P_{10},P_{11}]$, $[P_{11},P_{12}]$,
$[P_{12},P_{1}]$, and $[P_{1},P_{2}]$.

Figure 11. The power $w^{2}$ of $P_{9}$ w.r.t. the circle tangent to the sides
$[P_{7},P_{8}]$, $[P_{8},P_{9}]$,$[P_{9},P_{10}]$ equals that of $P_{12}$
w.r.t. the circle tangent to $[P_{10},P_{11}]$, $[P_{11},P_{12}]$,
$[P_{12},P_{1}]$.

Also for billiards $P_{1}P_{2}\dots$ in $e$ with a hyperbola $c$ as caustic,
there exists a conjugate billiard $P_{1}^{\prime}P_{2}^{\prime}\dots$, and the
relation is symmetric. However, the definition is different. It uses the
singular affine transformation

$\alpha_{h}\\!:\,e\to F_{1}F_{2}\mskip 9.0mu\mbox{with}\mskip
9.0muP_{i}\mapsto T_{i}=[P_{i},P_{i+1}]\cap[F_{1},F_{2}],$ (3.15)

with $F_{1}$ and $F_{2}$ as the focal points of $e$ and $c$.

Figure 12. Periodic billiard $P_{1}P\dots P_{6}$ in the ellipse $e$ with the
hyperbola $c$ as caustic, together with the conjugate billiard
$P_{1}^{\prime}P_{2}^{\prime}\dots P_{6}^{\prime}$. The associated polygon
with vertices on $e^{(1)}$ splits into two triangles
$S_{1}^{(1)}S_{3}^{(1)}S_{5}^{(1)}$ (green shaded) and
$S_{2}^{(1)}S_{4}^{(1)}S_{6}^{(1)}$.

######  Definition 3.13.

Referring to Figure 12, the billiard $\dots
P_{0}^{\prime}P_{1}^{\prime}P_{2}^{\prime}\dots$ in the ellipse $e$ with the
hyperbola $c$ as caustic is called conjugate to the billiard $\dots
P_{0}P_{1}P_{2}\dots$ in $e$ with the same caustic $c$ if the axial scaling
$\alpha_{h}$ defined in (3.15) maps the point $P_{i}^{\prime}$ to the
intersection $T_{i}$ of $P_{i}P_{i+1}$ with the principal axis.

######  Lemma 3.14.

Let $\dots P_{0}P_{1}P_{2}\dots$ be a billiard in the ellipse $e$ with the
hyperbola $c$ as caustic. Then to this billiard and to its mirror w.r.t. the
principal axis exists a conjugate billiard $\dots
P_{0}^{\prime}P_{1}^{\prime}P_{2}^{\prime}\dots$, and it is unique up to a
reflection in the principal axis. The relation between the two conjugate
billiards in $e$ is symmetric. Moreover, if $T_{i}^{\prime}$ denotes the
intersection of $P_{i}^{\prime}P_{i+1}^{\prime}$ with the principal axis, then

$\overline{P_{i}T_{i}}=\overline{P_{i}^{\prime}T_{i-1}^{\prime}}\mskip
9.0mu\mbox{and}\mskip
9.0mu\overline{P_{i}T_{i-1}}=\overline{P_{i-1}^{\prime}T_{i-1}^{\prime}}.$
(3.16)

###### Proof.

The singular affine transformation $\alpha_{h}$ maps $P_{1}^{\prime}$ to
$T_{1}$ and $P_{1}$ to a point $T^{\prime}$ (= $T_{6}^{\prime}$ in Figure 12).
We assume that $P_{1}$ and $P_{1}^{\prime}$ lie on the same side of the
principal axis, since otherwise we apply a reflection in the axis. Then we
obtain a curvilinear Ivory quadrangle $P_{1}^{\prime}T_{1}T^{\prime}P_{1}$
with diagonals of equal lengths. On the other hand, the lines
$[P_{1}^{\prime},T^{\prime}]$ and $[P_{1},T_{1}]$ must contact the same
confocal conic (see, e.g., [7, p. 153] or [23, Lemma 1]). Hence, the billiard
through the point $P_{1}^{\prime}\in e$ with caustic $c$ contains one side on
the line $[P_{1}^{\prime},T^{\prime}]$. Iteration confirms the claim. ∎

Let $\overline{P}_{1}$ and $\overline{P}_{2}$ be the images of $P_{1}$ and
$P_{2}$ under reflection in the principal axis of $e$ (Figure 12). Then, a
comparison with Figure 9 reveals that the confocal hyperbola through the
intersection $T_{1}=[P_{1},P_{2}]\cap[\overline{P}_{1},\overline{P}_{1}]$ lies
‘in the middle’ between the hyperbolas through $P_{1}$ and $P_{2}$.

######  Remark 3.15.

If $\dots P_{0}P_{1}P_{2}\dots$ and $\dots
P_{0}^{\prime}P_{1}^{\prime}P_{2}^{\prime}\dots$ is a pair of conjugate
billiards in an ellipse $e$, then the points of intersection
$[P_{i},P_{i+1}]\cap[P_{i}^{\prime},P_{i-1}^{\prime}]$ are located on a
confocal ellipse $e^{\prime}$ inside $e$. This holds for ellipses and
hyperbolas as caustics. If the billiards are $N$-periodic, then in the
elliptical case, the restriction of the two billiards to the interior of
$e^{\prime}$ is $2N$-periodic; conversely, $e$ plays the role of $e^{(1)}$
w.r.t. $e^{\prime}$ (Figure 8). In the hyperbolic case, the restriction to the
interior of $e^{\prime}$ gives two symmetric $2N$-periodic billiards, provided
that also the reflected billiards are involved (Figure 12).

We conclude with citing a result from [25] about billiards in ellipses. It
states that for each billiard $\dots P_{1}P_{2}P_{3}\dots$ in $e$ with a
hyperbola as caustic there exists a billiard $\dots
P_{1}^{\ast}P_{2}^{\ast}P_{3}^{\ast}\dots$ in $e^{\ast}$ with an ellipse as
caustic such that corresponding sides $P_{i}P_{i+1}$ and
$P_{i}^{\ast}P_{i+1}^{\ast}$ are congruent.

##  4\. Periodic $\boldsymbol{N}$-sided billiards

Let the billiard $P_{1}P_{2}\dots P_{N}$ in the ellipse $e$ be periodic with
an ellipse $c$ as caustic. Then, the sequence of parameters
$t_{1},t_{1}^{\prime},t_{2},\dots,t_{N},t_{N}^{\prime}$ of the vertices
$P_{i}$ and the intermediate contact points $Q_{1},\dots,Q_{N}$ with $c$ is
cyclic. Each side line intersects only a finite number of other side lines.
Hence, the corresponding Poncelet grid contains a finite number of confocal
ellipses $e^{(j)}$ through the points $S_{i}^{(j)}$, namely $[\frac{N-2}{2}]$
(including possibly the line at infinity), provided that $N\geq 5$ (Figure 5).
The sequence of ellipses $e,e^{(1)},e^{(2)},\dots$ is cyclic, and

$e^{(j)}=e^{(N-2-j)}.$ (4.1)

For example, in the case $N=7$ (Figure 6), the ellipse $e^{(2)}$ coincides
with $e^{(3)}$.

######  Definition 4.1.

The sum of exterior angles $\theta_{i}$ of a periodic billiard in an ellipse
$e$ is an integer multiple of $2\pi$, namely $2\tau\pi$. We call
$\tau\in\mathbb{N}$ the turning number of the billiard. It counts the loops of
the billiard around the center $O$ of $e$, anti-clockwise or clockwise.

If the periodic billiard $P_{1}P_{2}\dots P_{N}$ has the turning number
$\tau=1$ (Figure 13), then the billiard
$S_{1}^{(1)}S_{3}^{(1)}S_{5}^{(1)}\dots$ in $e^{(1)}$ has $\tau=2$, that of
$S_{1}^{(2)}S_{4}^{(2)}\dots$ in $e^{(2)}$ the turning number $\tau=3$, and so
on. In cases with $g=\mathrm{gcd}(N,\tau)>1$ the corresponding billiard splits
into $g$ $\frac{N}{g}$-sided billiards, each with turning number $\tau/g$
(note [22, Theorem 1.1]).

###  4.1. Symmetries of periodic billiards

The following is a corollary to Theorem 3.6.

######  Corollary 4.2.

Let $P_{1}P_{2}\dots P_{N}$ be an $N$-sided periodic billiard in the ellipse
$e$ with the ellipse $c$ as caustic.

  1. (i)

For even $N$ and odd $\tau$, the billiard is centrally symmetric.

  2. (ii)

For odd $N=2n+1$ and odd $\tau$, the billiard is centrally symmetric to the
conjugate billiard, where $P_{i}$ corresponds to $P_{i+n}^{\prime}$.444 All
subscripts in this section are understood modulo $N$.

  3. (iii)

If $N$ is odd and $\tau$ is even, then the conjugate billiard coincides with
the original one, and $P_{i}=P^{\prime}_{i+n}$.

###### Proof.

By virtue of Theorem 3.6, the lines $[P_{i-j-1},P_{i-j},]$ and
$[P_{i+j},P_{i+j+1}]$ for $j=1,2,\dots$ meet at the point $S_{i}{(j)}$ on the
confocal hyperbola through $P_{i}$.
(i) This means for even $N=2n$, odd $\tau$ and $j=n-1$, that also the opposite
vertex $S_{i}^{(n)}=P_{i-n}=P_{i+n}$ belongs to this hyperbola. If $P_{i}$ is
specified at a vertex on the minor axis of the ellipse $e$, then $P_{i+n}$ is
the opposite vertex. Continuity implies that the two points belong to
different branches of the hyperbola and are symmetric w.r.t. the center $O$ of
$e$.
(ii), (iii): If $N$ is odd, say $N=2n+1$, then for $j=n-1$ the sides
$[P_{i-n+1},P_{i-n}]$ and $[P_{i+n-1},P_{i+n}]$ intersect at a point on the
hyperbola through $Q_{i+n}$ and $P^{\prime}_{i+n}$. For odd $\tau$ (Figures 8
and 13), the same continuity argument as before proves that $P_{i}$ and
$P^{\prime}_{i+n}$ are opposite w.r.t. $O$.
If $\tau$ is even (Figure 14), then the choice of $P_{i}\in e$ on an axis of
symmetry shows the coincidence with $P^{\prime}_{i+n}\in e$, and this must be
preserved, when $P_{i}$ varies continuously on $e$. In the case of even $\tau$
and $N$ the billiard splits. ∎

Figure 13. Periodic billiard $P_{1}P_{2}\dots P_{9}$ with $\tau=1$. Note that
$l_{2}=\overline{P_{2}Q_{2}}=\overline{Q_{6}P_{7}}=r_{7}$ and
$\overline{S_{2}^{(1)}P_{3}}=\overline{P_{7}S_{7}^{(1)}}$. The associated
billiard in $e^{(2)}$ splits into three triangles.

The billiards with a hyperbola $c$ as caustic (see Figures 10 and 12)
oscillate between the upper and lower section of $e$. Therefore, only
billiards with an even $N$ can be periodic. Also for billiards of this type,
it possible to define a turning number $\tau$ which counts how often the
points $P_{1},\overline{P}_{2},P_{3},\dots,P_{N}$ (Figure 12) run to and fro
along the upper component of $e$.555 The turning number of hyperbolic
billiards becomes more intuitive when the billiard is seen as the limit of a
focal billiard in the sense of [25, Theorem 2]. The symmetry properties of
these periodic $N$-sided billiards differ from those in Corollary 4.2. They
follow from Theorem 3.6, since opposite vertices $P_{i}$ and $P_{i+N/2}$
belong to the same confocal hyperbola.

######  Corollary 4.3.

Let $P_{1}P_{2}\dots P_{N}$ be an $N$-sided periodic billiard in the ellipse
$e$ with the hyperbola $c$ as caustic.

  1. (i)

For $N\equiv 0\pmod{4}$, the billiard is symmetric w.r.t. the secondary axis
of $e$ and $c$.

  2. (ii)

For $N\equiv 2\pmod{4}$ and odd turning number $\tau$, the billiards are
centrally symmetric. For even $\tau$, each billiard is symmetric w.r.t. the
principal axis of $e$ and $c$.

###  4.2. Some invariants

As a direct consequence of the results so far, we present new proofs for the
invariants k101, k118 and k119 listed in [21, Table 2], though this table
refers already to proofs for some of them in [2] and [11].

We begin with a result that has first been proved for a much more general
setting in [26, p. 103].

######  Lemma 4.4.

The length $L_{e}$ of a periodic $N$-sided billiard in the ellipse $e$ with
the ellipse $c$ as caustic is independent of the position of the initial
vertex $P_{1}\in e$.

###### Proof.

We refer to Graves’s construction [15, p. 47]. According to (3.4) holds

$D_{e}:=\overline{Q_{i-1}P_{i}}+\overline{P_{i}Q_{i}}-\overset{\mbox{\raisebox{-2.27621pt}{\Large$\frown$}}}{Q_{i-1}Q_{i}}\,.$

This yields for an $N$-sided billiard with turning number $\tau$ the total
length

$L_{e}=N\cdot D_{e}-\tau\cdot P_{c}\,,$ (4.2)

where $P_{c}$ denotes the perimeter of the caustic $c$. Thus, $L_{e}$ does not
depend of the choice of the initial vertex $P_{1}\in e$. ∎

If the billiard in $e$ has the turning number $\tau$, then its extension in
$e^{(1)}$ has the turning number $2\tau$, and from (3.5) and (3.9) follows

$L_{e|1}=ND_{e|1}-2\tau P_{c}=2N(D_{e}+w)-2\tau
P_{c}=2L_{e}+2N\,\frac{2a_{e}b_{e}\sqrt{k_{e}^{3}}}{a_{c}^{2}b_{c}^{2}-k_{e}^{2}}\,.$
(4.3)

Figure 14. Periodic billiard with $N=7$ and $\tau=2$.

The following theorem on the invariant k118 in [21] deals with the lengths
$r_{i}$ and $l_{i}$ of the segments $Q_{i-1}P_{i}$ and $P_{i}Q_{i}$, as
defined in (3.1).

######  Theorem 4.5.

In each $N$-sided periodic billiard opposite segments are congruent, i.e., if
$N=2n$, then $r_{i+n}=r_{i}$ and $l_{i+n}=l_{i}\,$, and if $N=2n+1$, then
$r_{i+n}=l_{i-1}$ and $l_{i+n}=r_{i}\,$. Thus, for odd $N$ holds

$\sum_{i=1}^{N}l_{i}=\sum_{i=1}^{N}r_{i}=\frac{L_{e}}{2}\,.$

###### Proof.

By Corollary 4.2, for even $N=2n$ the central symmetry implies for opposite
segments $r_{i}=r_{i+n}$ and $l_{i}=l_{i+n}$.
If $N=2n+1$, then $l_{i}=\overline{P_{i}Q_{i}}$ shows up as
$l_{i+n}^{\prime}=\overline{P_{i+n}^{\prime}Q_{i+n}^{\prime}}$ at the
conjugate billiard and, by virtue of (3.13), this equals
$r_{i+n+1}=\overline{P_{i+n+1}Q_{i+n}}$ (Figure 13). Similarly follows
$r_{i}=l_{i+n}$. ∎

######  Remark 4.6.

In the particular case $N=3$ the two segments adjacent to any side are
congruent (note in Figure 13 the triangular billiards in $e^{(2)}$).
Therefore, the Cevians $[P_{i},Q_{i+1}]$ are concurrent and meet at the Nagel
point of the triangle. This has already been proved in [20] and agrees with
the circles through $Q_{i}$ and centered at $R_{i}$ (see Figures 6 and 14),
which for $N=3$ are excircles of the triangle $P_{1}P_{2}P_{3}\,$.

The following theorem has first been proved in [2, p. 4]. Another proof can be
found in [3, Cor. 3.2]. We give below a new proof.

######  Theorem 4.7.

For the exterior angles $\theta_{1},\dots,\theta_{N}$ of the periodic
$N$-sided elliptic billiard in the ellipse $e\,$, the sum of cosines is
independent of the initial vertex, namely

$\sum_{i=1}^{N}\cos\theta_{i}=N-J_{e}L_{e}=N-\frac{\sqrt{k_{e}}}{a_{e}b_{e}}\,L_{e}\,,$

where $L_{e}$ is the common perimeter of these billiards in $e\,$.

Figure 15. Billiard $P_{1}P_{2}\dots$ with pedal points w.r.t. $O$.

###### Proof.

The pedal points $F_{i}$ and $F_{i-1}$ on the sides $P_{i}P_{i+1}$ and
$P_{i-1}P_{i}$ w.r.t. the center $O$ have the position vectors

$\begin{array}[]{c}{\mathbf{f}}_{i,i-1}={\mathbf{p}}+\mbox{\small$\displaystyle\frac{\lambda_{i,i-1}}{\|{\mathbf{t}}_{e}\|}$
}\left(\cos\mbox{\small$\displaystyle\frac{\theta_{i}}{2}$
}\,{\mathbf{t}}\pm\sin\mbox{\small$\displaystyle\frac{\theta_{i}}{2}$
}\,{\mathbf{t}}^{\perp}\right),\\\\[5.69054pt] \mbox{where}\quad
0=\Bigl{\langle}{\mathbf{f}}_{i,i-1},\,\Bigl{(}\cos\mbox{\small$\displaystyle\frac{\theta_{i}}{2}$
}\,{\mathbf{t}}\pm\sin\mbox{\small$\displaystyle\frac{\theta_{i}}{2}$
}\,{\mathbf{t}}^{\perp}\Bigr{)}\Bigr{\rangle}.\end{array}$

Here, $\lambda_{i}$ and $\lambda_{i+1}$ denote the signed distances from the
vertex $P_{i}$ in one case towards $P_{i+1}$, in the other opposite to
$P_{i-1}$. From

$\langle{\mathbf{p}}_{i},\,{\mathbf{t}}\rangle=(-a_{e}^{2}+b_{e}^{2})\cos
t\sin t\mskip 9.0mu\mbox{and}\mskip
9.0mu\langle{\mathbf{p}}_{i},\,{\mathbf{t}}^{\perp}\rangle=-a_{e}b_{e}$

follows

$\lambda_{i,i-1}=\frac{1}{\|{\mathbf{t}}_{e}\|}\left((-a_{e}^{2}+b_{e}^{2})\cos
t\sin t\cos\mbox{\small$\displaystyle\frac{\theta_{i}}{2}$ }\pm
a_{e}b_{e}\sin\mbox{\small$\displaystyle\frac{\theta_{i}}{2}$ }\right)$

This implies by virtue of (2.11) and after reversing the orientation for
$\lambda_{i-1}$,

$\lambda_{i}-\lambda_{i-1}=\overline{P_{i}F_{i}}+\overline{P_{i}F_{i-1}}=\frac{2a_{e}b_{e}}{\|{\mathbf{t}}_{e}\|^{2}}\,\sqrt{k_{e}}\,.$

Since the sum over all signed lengths between $P_{i}$ and the adjacent pedal
points gives the total perimeter $L_{e}$ of the billiard, we obtain by (2.12)

$L_{e}=\sum_{i=1}^{N}\left(\,\overline{P_{i}F_{i}}+\overline{P_{i}F_{i-1}}\,\right)=\frac{a_{e}b_{e}}{\sqrt{k_{e}}}\,\sum_{i=1}^{N}\frac{2k_{e}}{\|{\mathbf{t}}_{e}\|^{2}}=\frac{a_{e}b_{e}}{\sqrt{k_{e}}}\,\sum_{i=1}^{N}(1-\cos\theta_{i}),$
(4.4)

hence

$\sum_{i=1}^{N}\frac{1}{\|{\mathbf{t}}_{e}\|^{2}}=\frac{L_{e}}{2a_{e}b_{e}\sqrt{k_{e}}}\mskip
9.0mu\mbox{and also}\mskip
9.0mu\sum_{i=1}^{N}\cos\theta_{i}=N-\frac{\sqrt{k_{e}}}{a_{e}b_{e}}\,L_{e}\,,$
(4.5)

as stated. ∎

######  Remark 4.8.

Note that the result in [2] relates to the interior angles of the billiard. As
already mentioned in [2, Theorem 7], the constant sum of cosines holds also
for the ‘extended’ billiards in $e^{(j)}$, where the exterior angles are
$\theta_{i}+\theta_{i+1}+\dots+\theta_{i+j}$ (note Figure 5).

Similar to (2.14), the distance of $O$ to the tangent $t_{P}$ to $e$ at $P$
equals

$\overline{Ot_{P}}=\frac{a_{e}b_{e}}{\|{\mathbf{t}}_{e}\|}\,.$ (4.6)

From (4.5) follows for the vertices of a billiard a result of [3, Cor. 3.2].
The invariant k119, first proved by P. Roitmann, deals with the curvature of
$e\,$, namely

$\kappa_{e}(t):=\frac{a_{e}b_{e}}{\|{\mathbf{t}}_{e}(t)\|^{3}}$

by [15, p. 79]. The result referring to this and given below is again a
consequence of (4.5).

######  Corollary 4.9.

The squared distances from the center $O$ to the tangents $t_{P_{i}}$ at the
vertices $P_{i}$ of the periodic $N$-sided elliptic billiard in the ellipse
$e$ have a constant sum, independent of the initial vertex, namely

$\sum_{i=1}^{N}\overline{Ot_{P}}^{\,2}=\frac{a_{e}b_{e}}{2\sqrt{k_{e}}}\,L_{e}\,.$

The curvatures $\kappa_{i}$ of $e$ at the vertices $P_{i}$ give rise to an
invariant sum

$\sum_{i=1}^{N}\kappa_{i}^{2/3}=\frac{L_{e}}{2\sqrt{k_{e}}}\,(a_{e}b_{e})^{-1/3}\,.$

We conclude this section with a comment on (4.1) in connection with (3.8) and
(3.10). For example, we obtain $k_{e|1}=k_{e}$ for $N=3$, $k_{e}=a_{c}\,b_{c}$
for $N=4$, $k_{e|2}=k_{e|1}$ for $N=5$, and $k_{e|2}=\infty$ for $N=6$. This
yields algebraic conditions for the dimension of the ellipse $e$ with an
inscribed $N$-periodic billiard when the ellipse $c$ is given as caustic.
However, we need to recall that A. Cayley gave already an explicit solution
for this problem in a projective setting ([16] or [15, Theorem 9.5.4]). Other
approaches are provided in [4, Sect. VI] and [14, Sect. 11.2.3.9]. Equivalent
conditions in terms of elliptic functions can be deduced from [24, Corollary
3].

### Acknowledgment

The author is grateful to Dan Reznik and Ronaldo Garcia for inspirations and
interesting discussions.

## References

  * [1] A.W. Akopyan, A.I. Bobenko: Incircular nets and confocal conics. Trans. Amer. Math. Soc. 370 (2018), 2825–2854.
  * [2] A. Akopyan, R. Schwartz, S. Tabachnikov: Billiards in ellipses revisited. Eur. J. Math. 2020. doi:10.1007/s40879-020-00426-9
  * [3] M. Bialy, S. Tabachnikov: Dan Reznik’s identities and more. Eur. J. Math. 2020, doi:10.1007/s40879-020-00428-7
  * [4] G.D. Birkhoff: Dynamical Systems. American Mathematical Society, Colloquium Publications, Vol. 9, Providence/Rhode Island, 1966
  * [5] A.I. Bobenko, W.K. Schief, Y.B. Suris, J. Techter: On a discretization of confocal quadrics. A geometric approach to general parametrizations. Int. Math. Res. Not. IMRN 2020, no. 24, 10180–10230.
  * [6] A.I. Bobenko, A.Y. Fairley: Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics. Discrete Comput. Geom. 2021. https://doi.org/10.1007/s00454-021-00277-5
  * [7] W. Böhm: Die Fadenkonstruktionen der Flächen zweiter Ordnung. Math. Nachr. 13 (1955), 151–156.
  * [8] W. Böhm: Ein Analogon zum Satz von Ivory. Ann. Mat. Pura Appl. (4) 54 (1961), 221–225.
  * [9] M. Chasles: Propriétés générales des arcs d’une section conique, dont la difference est rectifiable. Comptes Rendus hebdomadaires de séances de l’Académie des sciences 17 (1843), 838–844.
  * [10] M. Chasles: Résumé d’une théorie des coniques sphériques homofocales. Comptes Rendus des séances de l’Académie des Sciences 50 (1860), 623–633.
  * [11] A. Chavez-Caliz: More about areas and centers of Poncelet polygons. Arnold Math. J. 7 (2021), 91–106.
  * [12] V. Dragović, M. Radnović: Integrable billiards and quadrics. Russian Math. Surveys 65/2 (2010), 319–379.
  * [13] V. Dragović, M. Radnović: Poncelet porisms and beyond. Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics. Birkhäuser/Springer Basel AG, Basel 2011.
  * [14] J.J. Duistermaat, Discrete Integrable Systems. QRT Maps and Elliptic Surfaces. Springer Monographs in Mathematics 304, Springer, New York (2010)
  * [15] G. Glaeser, H. Stachel, B. Odehnal: The Universe of Conics. From the ancient Greeks to 21st century developments. Springer Spectrum, Berlin, Heidelberg 2016.
  * [16] Ph. Griffiths, J. Harris: On Cayley’s explicit solution to Poncelet’s porism. L’Enseignement Mathématique 24/1-2 (1978), 31–40.
  * [17] L. Halbeisen, N. Hungerbühler: A Simple Proof of Poncelet’s Theorem (on the occasion of its bicentennial). Amer. Math. Monthly 122/6 (2015), 537–551.
  * [18] I. Izmestiev, S. Tabachnikov: Ivory’s Theorem revisited. J. Integrable Syst. 2/1 (2017), xyx006, https://doi.org/10.1093/integr/xyx006
  * [19] D. Reznik, R. Garcia, J. Koiller: Can the elliptic billiard still surprise us ? Math. Intelligencer 42 (2020), 6–17.
  * [20] D. Reznik, R. Garcia: The Circumbilliard: Any Triangle can be a 3-Periodic. arXiv:2004.06776[math.DS], 2020.
  * [21] D. Reznik, R. Garcia, J. Koiller: Eighty New Invariants of N-Periodics in the Elliptic Billiard. arXiv:2004.12497v11[math.DS], 2020.
  * [22] R.E. Schwartz: The Poncelet grid. Adv. Geom. 7 (2007), 157–175.
  * [23] H. Stachel: Recalling Ivory’s Theorem. FME Transactions 47, No 2 (2019), 355–359.
  * [24] H. Stachel: On the Motion of Billiards in Ellipses. arXiv:2105.03624v1[math.DG] (2021).
  * [25] H. Stachel: Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids. arXiv:2105.05295v1[nlin.CD] (2021).
  * [26] S. Tabachnikov: Geometry and Billiards. American Mathematical Society, Providence/Rhode Island 2005.