# Cusps of caustics by reflection in ellipses

Gil Bor111 CIMAT, A.P. 402, Guanajuato, Gto. 36000, México<EMAIL_ADDRESS>Mark
Spivakovsky222CNRS UMR 5219, Institut de Mathḿatiques de Toulouse, 118, rte de
Narbonne, 31062 Toulouse cedex 9, France and Instituto de Matemáticas
(Cuernavaca) LaSol, UMI CNRS 2001, UNAM, Av. Universidad s/n. Col. Lomas de
Chamilpa, 62210, Cuernavaca, Morelos, México<EMAIL_ADDRESS>toulouse.fr Serge Tabachnikov333 Department of Mathematics, Penn State
University, University Park, PA 16802; USA<EMAIL_ADDRESS>
###### Abstract

This paper is concerned with the billiard version of Jacobi’s last geometric
statement and its generalizations. Given a non-focal point $O$ inside an
elliptic billiard table, one considers the family of rays emanating from $O$
and the caustic $\Gamma_{n}$ of the reflected family after $n$ reflections off
the ellipse, for each positive integer $n$. It is known that $\Gamma_{n}$ has
at least four cusps and it has been conjectured that it has exactly four
(ordinary) cusps. The present paper presents a proof of this conjecture in the
special case when the ellipse is a circle. In the case of an arbitrary
ellipse, we give an explicit description of the location of four of the cusps
of $\Gamma_{n}$, though we do not prove that these are the only cusps.

###### Contents

  1. 1 Introduction and statement of results
  2. 2 Preliminaries
    1. 2.1 Billiards in ellipses
    2. 2.2 Families of rays, envelopes, cusps
  3. 3 Proof of Theorem 1
  4. 4 Proof of Theorem 2
    1. 4.1 Two lemmas
    2. 4.2 Cusps by reflection in a circle
    3. 4.3 The four cusps are ordinary
  5. 5 Miscellanea
    1. 5.1 Liouville billiards
    2. 5.2 Cusps on axes
    3. 5.3 A light source outside an ellipse
    4. 5.4 The complexity of the caustics by reflection
    5. 5.5 Pseudo-integrable billiards
    6. 5.6 Caustics by refraction

##  1 Introduction and statement of results

The motivation for this work goes back to Jacobi’s 1842-3 “Lectures on
Dynamics” [13]. Recall that the conjugate locus of a point on a surface is the
locus of the first conjugate points on the geodesics that start at this point.
Jacobi considered the conjugate locus of a non-umbilic point on the surface of
a triaxial ellipsoid in 3-space. What is known as the Last Geometric Statement
of Jacobi is the claim that this conjugate locus has exactly four cusps, see
Figure 1. We refer to [17] for a detailed historical discussion.

Figure 1: Left: A sketch of the conjugate locus from [13]. Right: The first
(red) and second (blue) conjugate locus of a point on a triaxial ellipsoid.

The Last Geometric Statement of Jacobi was proved only recently [10]. In
contrast, it was known for a long time that the conjugate locus of a generic
point on a convex surface has at least four cusps; see [2] where this theorem
is attributed to C. Carathéodory and [22] for a recent proof.

The conjugate locus of a point is also called the first caustic. One considers
the loci of the second, third, etc., conjugate points on the geodesics
emanating from a point; these are the second, third, etc., caustics. These
curves are also the components of the envelope of the 1-parameter family of
geodesics that start at this point. Figure 1 (right) depicts the first and
second such caustics.

This article concerns the billiard versions of these problems. G. Birkhoff [1]
suggested to consider billiard trajectories in a convex plane domain as the
geodesics on a “pancake”, the surface obtained from the domain by
infinitesimally “thickening” it. This leads to the following set-up.

Consider an oval $C$, a smooth strictly convex closed curve in the plane, the
boundary of a billiard table. Let $O$ be a point inside $C$ and consider the
billiard trajectories that start at $O$. After $n$ reflections off $C$, we
obtain a 1-parameter family of lines whose envelope is a closed connected
curve in the real projective plane $\mathbb{RP}^{2}$, possibly with some cusps
and self intersections, called the $n$-th caustic by reflection from $O$. The
term caustic, meaning “capable of burning,” comes from optics, where $C$ is an
ideal mirror and $O$ is a light source.

We refer to [4, 5] and the literature cited therein for the study of the first
caustics by reflection, also known as catacaustics. In particular, A. Cayley
studied the first caustics by reflection and refraction in a circle in his
memoire [6] where he considered the cases when the source of light was inside
the circle, on the circle, and outside the circle, including at infinity.

We proved in [3] that, for every $n\geq 1$, if $O$ is a generic point inside
an oval, then the $n$-th caustic by reflection from $O$ has at least 4 cusps.
This is one of many variations on the classic 4-vertex theorem. Here are
refined versions of two conjectures made in [3].

######  Conjecture 1.

If $C$ is an ellipse and $O$ is an interior point which is not a focus of $C$
then, for all $n\geq 1$, the $n$-th caustic by reflection from $O$ has exactly
four cusps, and all four are ordinary ones.

See the Section 2.2 for a precise definition of “ordinary cusp”.

######  Remark 1.

The $n=1$ case of Conjecture 1 (without the “ordinary” part) can be thought of
as a “limiting case” of the Jacobi’s Last Geometric Statement, as one of the
axes of the ellipsoid tends to 0.

######  Conjecture 2.

If an oval $C$ is not an ellipse then there exists an $n\geq 1$ and an open
set $U$ inside $C$ such that for every $O\in U$ the number of cusps of the
$n$-th caustic by reflection from $O$ is greater than four.

An analogue of Conjecture 1 for the caustics of geodesics emanating from a
point on a tri-axial ellipsoid was experimentally studied in [17]. That paper
contains numerous computer generated images of first, second, third, and
fourth caustics, each having exactly four cusps.

This article is a step toward proving Conjecture 1. To state our first result,
we recall a well known property of billiards in an ellipse.

An ellipse $C$ defines two 1-parameter families of confocal conics, those
conics which share their foci with $C$. One family consists of ellipses, the
other of hyperbolas (including the major and minor axes of $C$). They form, in
the complement of the foci of $C$, a double foliation so that through each
point pass one confocal ellipse and one confocal hyperbola, intersecting
orthogonally at the point. A ray (directed line), incident to the interior of
$C$, is tangent to exactly one of these confocal conics (or incident to one of
the foci), and after reflection off $C$ it is tangent to the same conic. See
Figure 2 (left).

######  Theorem 1.

Let $O$ be a non-focal point inside an ellipse $C$, and let $E$ and $H$ be the
ellipse and hyperbola (respectively), passing through $O$ and confocal to $C$.
Consider the four rays emanating from $O$ and tangent to $E$ and $H$ (two
each). Then after $n$ reflections, the 4 rays are tangent to $E$ and $H$ at 4
points which are cusps of the $n$-th caustic by reflection from $O$. See
Figure 2 (right).

$C$
---

$O$
---
$H$
---
$H$
---
$E$
---
$C$
---

Figure 2: Left: a light ray reflected off an elliptical table $C$ stays
tangent to the same confocal conic, either an ellipse (blue) or a hyperbola
(red). Right: the first (red) and the second (blue) caustics by reflection
from $O$ each have cusps at the four tangency points with the confocal conics
through $O$ of the four reflected rays emanating from $O$ tangent to these
conics.

######  Remark 2.

  1. 1.

If $O$ lies on one of the axes of $C$, then the role of $H$ in the above
theorem is played by this axis. The location of the corresponding cusps along
this axis is then determined by the “mirror equation” of geometric optics. See
Section 5.2 below.

  2. 2.

The limiting case when $C$ is a circle is not excluded: in this case, the role
of the two confocal conics through $O$ are played by the concentric circle
through $O$ and the line through $O$ and the center. See Figure 3 (left).

$O$
---
$C$
---

Figure 3: Left: Theorem 1 for a circle. Right: The $n$-th caustic from a focus
of an ellipse is the other focus for odd $n$, the same focus for even $n$.

  3. 3.

If the point $O$ is a focus of the ellipse, then the $n$-th caustic by
reflection degenerates to one of the two foci, depending on the parity of $n$.
See Figure 3 (right).

  4. 4.

The stated location of the 4 cusps in Theorem 1 can be deduced from the
conjectures made in [17] about the location of the cusps of caustics of
envelopes of geodesics from a point on an ellipsoid.

  5. 5.

It is straightforward to extend Theorem 1 to an arbitrary non-degenerate conic
section $C$ (parabola and hyperbola). The complement of the closure of $C$ in
$\mathbb{RP}^{2}$ consists of two components, diffeomorphic to a disc and to a
Möbius band, respectively. The former can serve as a billiard table, and our
proof of Theorem 1 applies, mutatis mutandis, to it as well. See Figure 4.

$O$
---
$O$
---
Figure 4: Theorem 1 holds for any convex billiard table in the projective
plane, bounded by a conic; shown are a parabola (left) and a hyperbola
(right). Some cusps are out of sight.

  6. 6.

In Section 5.1 below Theorem 1 is further extended to “Liouville billiards”,
where the billiard table is formed by a coordinate line on a Liouville
surface.

Thus, after Theorem 1, proving Conjecture 1 amounts to showing that the 4
cusps described by Theorem 1 are the only cusps of the $n$-th caustic by
reflection from $O$ and that all 4 cusps are ordinary. We were able to show
this only in the case when $C$ is a circle, which is our next result.

######  Theorem 2.

Conjecture 1 holds if $C$ is a circle. Namely, if $O$ is an interior point of
a circle $C$, different from its center, then for every $n\geq 1$ there are
exactly 4 cusps on the $n$-th caustic by reflection from $O$; two of these
cusps lie on the line passing through $O$ and the center of the circle, the
other two on the circle through $O$ concentric with $C$. Furthermore, these 4
cusps are ordinary.

The content of this article is as follows. In Section 2 we recall relevant
facts about billiards in ellipses, envelopes of families of lines and their
cusps. In Section 3 we prove Theorem 1 and in Section 4 we prove Theorem 2.
Section 5 contains various additional results and suggested problems.

Acknowledgements. GB acknowledges hospitality of the Toulouse Mathematics
Institute during visits in 2023-4 and a CONAHCYT Grant A1-S-45886. ST
participated in the special program “Mathematical Billiards: at the Crossroads
of Dynamics, Geometry, Analysis, and Mathematical Physics” at Simons Center
for Geometry and Physics, where this work started; he is grateful to the
Center for its hospitality and the inspiring atmosphere. ST was supported by
NSF grants DMS-2005444 and DMS-2404535.

##  2 Preliminaries

###  2.1 Billiards in ellipses

Let us recall relevant facts concerning billiards in ellipses, in particular,
their complete integrability, see, e.g., [9, 11], or [20].

Consider a billiard table $C$ bounded by an ellipse

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,\quad\mbox{ where }0<b\leq a.$

Associated with the billiard table $C$ is a dynamical system whose phase space
$\mathcal{L}$ (topologically a cylinder) is the space of rays (oriented lines)
that intersect the interior of $C$. The billiard transformation $T$ is the
transformation of the phase space that sends an incoming ray to the outgoing
one upon reflection off $C$. See Figure 6 (left).

The phase cylinder $\mathcal{L}$ admits a $T$-invariant area form. If a ray is
characterized by its direction $\alpha$ and the signed distance from the
origin $p$ (see Figure 5), then the area form is $dp\wedge d\alpha$. This fact
is not specific to ellipses: this area form is invariant under the billiard
transformation in a billiard table of any shape.

$p>0$
---
$p<0$
---
$O$
---
$\alpha$
---

$alpha$
---

$\alpha$
---
Figure 5: The coordinates $(\alpha,p)$ on the space of oriented lines.

The ellipse $C$ is included in a confocal family of conics

$C_{\lambda}:\frac{x^{2}}{a^{2}-\lambda}+\frac{y^{2}}{b^{2}-\lambda}=1,\quad\lambda\in(-\infty,b^{2})\cup(b^{2},a^{2}).$

This is an ellipse for $\lambda<b^{2}$ and a hyperbola for
$b^{2}<\lambda<a^{2}$. For $0<\lambda<b^{2}$ the confocal ellipse
$C_{\lambda}$ is contained in the interior of $C$, for $\lambda<0$ it is
contained in the exterior of $C$. For $\lambda=0$ one has $C_{0}=C$.

As $\lambda$ tends to $b^{2}$ on the left, the confocal ellipse $C_{\lambda}$
tends to the line segment on the $x$-axis connecting the two foci of $C$; the
right limit is the closure of the complement of this segment in the $x$-axis.
As $\lambda$ tends to $a^{2}$ on the left, $C_{\lambda}$ tends to the
$y$-axis.

A ray $r\in\mathcal{L}$, not incident to one of the foci of $C$, is tangent to
a unique conic $C_{\lambda}$ from this confocal family, so $\lambda$ can be
considered as a function on $\mathcal{L}$. As is easy to show, it is given by

$\lambda=(a\sin\alpha)^{2}+(b\cos\alpha)^{2}-p^{2}.$

(This formula shows that $\lambda$ extends smoothly to all of $\mathcal{L}$,
including rays incident to the foci.)

After reflection, the ray $T(r)$ is tangent to the same conic [20, Theorem
4.4]. Thus the level curves of $\lambda$ define a (singular) $T$-invariant
foliation of the phase space ${\mathcal{L}}$, whose leaves consist of the rays
tangent to a fixed conic, see Figure 6 (right).

Note that the resulting foliation is non-singular away from the 4 marked
points on the $\alpha$-axis (the critical points of $\lambda$), corresponding
to the rays aligned with the major and minor axes of $C$. Note also that each
level curve of a regular value $\lambda\in(0,b^{2})\cup(b^{2},a^{2})$ has two
connected components. For $\lambda\in(0,b^{2})$ (rays tangent to a fixed
confocal ellipse), each of the two components is $T$-invariant. For
$\lambda\in(b^{2},a^{2})$ (rays tangent to a fixed confocal hyperbola,
including its asymptots), the two components are interchanged by $T$. The
figure $\infty$ (the level curve $\lambda=b^{2}$) corresponds to rays passing
through the foci. Orientation reversing acts on $\mathcal{L}$ by
$R:(\alpha,p)\mapsto(\alpha+\pi,-p)$, satisfying $R^{2}=(RT)^{2}=id$. The two
reflections about the major and minor axes of $C$ induce maps of $\mathcal{L}$
commuting with $T$.

$O$
---
$C$
---
$r$
---
$T(r)$
---
$C_{\lambda}$
---
$T^{2}(r)$
---

$p$
---
$\alpha$
---

Figure 6: Left: a billiards trajectory in an ellipse and the associated
confocal conic. Right: the phase space $\mathcal{L}$ of the billiard
transformation $T$ in an elliptical table (topologically a cylinder), and its
$T$-invariant foliation, which is regular away from the 4 marked points on the
$\alpha$ axis, corresponding to rays aligned with the major and minor axis of
the table. Reversing the orientation of a ray corresponds to the ‘glide-
reflection’ $(\alpha,p)\mapsto(\alpha+\pi,-p)$. The $\infty$-shaped curve
corresponds to rays incident to the foci of the ellipse, phase curves inside
it correspond to rays tangent to confocal hyperbolas (including their
asymptotes), phase curves outside it to rays tangent to confocal ellipses.

The following proposition is a special case of the Arnold-Liouville theorem on
completely integrable Hamiltonian systems [21]. We shall give a self-contained
proof in our case, following Chapter 4 of [20].

######  Proposition 1.

On each leaf $\gamma$ of the $T$-invariant foliation of $\mathcal{L}$ there is
a $T$-invariant non-vanishing 1-form, well defined up to multiplicative
constant. Consequently, there is a local coordinate $t$ on $\gamma$ in which
$T$ is given by $T(t)=t+c$ for some constant $c$.

###### Proof.

Choose a smooth function $f$ without critical points in a neighborhood of
$\gamma$, which is constant on each leaf of the $T$-invariant foliation (for
example, $f=\lambda$). Then $f\circ T=f$ implies $T^{*}df=df.$ Let $X_{f}$ be
the Hamiltonian vector field associated to $f,$ that is,
$\omega(X_{f},\,\cdot\,)=df,$ where $\omega=dp\wedge d\alpha$ is the
$T$-invariant area form on $\mathcal{L}$. Since both $df$ and $\omega$ are
$T$-invariant, the same holds for $X_{f}$. Since $X_{f}$ is non-vanishing and
tangent to $\gamma$, there is a unique 1-form $\alpha$ on $\gamma$ such that
$\alpha(X_{f})=1.$ Since $X_{f}$ is $T$-invariant, so is $\alpha$. In
neighborhoods of a point $r\in\gamma$ and its image $T(r)$, one can find
coordinates $t$ and $t_{1}$, respectively, such that $\alpha=dt$ near $r$ and
$\alpha=dt_{1}$ near $T(r)$. It follows that
$0=T^{*}\alpha-\alpha=d(t_{1}\circ T-t)$, thus $t_{1}\circ T-t=c$ for some
constant $c$; that is, $T$ is given near $r$ by $t_{1}=t+c.$

If one replaces $f$ by another function, say $g=\phi(f),$ then the
corresponding vector field changes to $X_{g}=(\phi^{\prime}\circ f)X_{f}$,
i.e., a non-zero constant multiple of $X_{f}$ along $\gamma$, so $\alpha$ is
also changed by a constant multiple. ∎

Once a choice of coordinate $t$ is made on each level curve of $\lambda$, one
can use $(t,\lambda)$ as coordinates on ${\mathcal{L}}$ (away from singular
leaves); the ray $r(t,\lambda)$ is tangent to the confocal conic corresponding
to the parameter $\lambda$, such that
$T(r(t,\lambda))=r(t+c(\lambda),\lambda).$

######  Remark 3.

It is important to note that the choice of the $t$ coordinate in the last
proposition depends only on the $T$-invariant foliation of $\mathcal{L}$,
which in turn depends on the family of conics confocal to the billiard table
$C$, and not on a particular choice of conic within this family as a billiard
table. That is, if one chooses, as a billiard table, any conic confocal to
$C$, say $C_{1}$, then the associated billiard map $T_{1}$ with respect to
$C_{1}$ admits the same invariant foliation of $\mathcal{L}$ as $T$, and is
thus given in each invariant leaf by the same formula,
$T_{1}(r(t,\lambda))=r(t+c_{1}(\lambda),\lambda).$

For example, consider an ellipse $E$ from a confocal family. One may think of
$t$ as a coordinate on $E$. Then the locus of the intersection points of the
tangent to $E$, whose $t$-coordinates differ by a constant, is a confocal
ellipse, and if the half-sum of the two $t$-coordinates is constant, then this
locus is a confocal hyperbola. We refer to [9, 11] and to a detailed
discussions in [18, 19].

######  Remark 4.

Note that the $T$-invariant leaf $\gamma$ in Proposition 1 need not be
connected for the proposition to hold. Indeed, each level curve of $\lambda$
has two components (each topologically a circle); in the elliptic case (level
curves above and below the $\infty$ shape in Figure 6 (right)), each of these
components is $T$-invariant, while in the hyperbolic case (level curves inside
the $\infty$ shape in Figure 6 (right)), the two components are interchanged
by $T$. By Proposition 1, even in this hyperbolic case, one can put a
coordinate on each of the two components, say $t$ on one component and $t_{1}$
on the other, such that $T$ is given by $T(t,\lambda)=(t_{1}+c,\lambda),$
$T(t_{1},\lambda)=(t+c,\lambda),$ for some constant $c$ (depending in
$\lambda$).

###  2.2 Families of rays, envelopes, cusps

The conjectures and theorems of the Introduction concern families of rays,
their envelopes (or caustics) and cusps. We briefly review here the pertinent
definitions. See, e.g., Section 8.4 of [8].

Define the “line” dual to a point in $\mathbb{R}^{2}$ as the curve in
$\mathcal{L}$ corresponding to the set of rays incident to the point (a
“pencil” of rays). See the dotted curves in Figure 7. We also include “lines”
dual to “points at infinity”, corresponding to pencils of parallel rays
sharing a common direction (vertical lines in the $(\alpha,p)$ coordinates on
$\mathcal{L}$). This defines a 2-parameter family of curves in $\mathcal{L}$,
a unique curve through each given point in a given tangent direction at this
point.

######  Definition 1.

Given a 1-parameter family of rays, that is, a smooth curve
$\gamma\subset\mathcal{L}$, an inflection point of $\gamma$ is a point where
the tangent “line” to $\gamma$ at this point has contact of order $m\geq 2$
with $\gamma$ (the tangent “line” to a curve has typically contact of order
1).

######  Definition 2.

The envelope (or caustic) of $\gamma$ is an oriented plane curve $\Gamma$
whose set of tangent lines is $\gamma$.

######  Definition 3.

An $m$-cusp of a plane curve $\Gamma$ is a point for which there is a
diffeomorphism taking a neighborhood of the point to a neighborhood of the
origin in the $(x,y)$-plane, taking the point to $(0,0)$ and $\Gamma$ to the
curve $y^{m}=x^{m+1}.$ An ordinary cusp is a 2-cusp (or a semi-cubical cusp).

A useful basic characterization of $m$-cusps is the following. Let $\gamma$ be
a smooth 1-parameter family of rays with envelope $\Gamma$. Then an $m$-cusp
of $\Gamma$ corresponds to an inflection point of $\gamma$ of order $m$. See
Example 8.2 of [8].

##  3 Proof of Theorem 1

We restate here Theorem 1 from the Introduction.

Theorem 1. Let $O$ be a non-focal point inside an ellipse $C$, and let $E$ and
$H$ be the confocal ellipse and hyperbola (respectively) passing through $O$.
Consider the four rays emanating from $O$ and tangent to $E$ and $H$ (two
each). Then after $n$ reflections, the 4 rays are tangent to $E$ and $H$ at 4
points which are cusps of the $n$-th caustic by reflection from $O$.

To prove Theorem 1, we first reformulate it as a statement about the
inflection points of a curve in the phase cyclinder $\mathcal{L}$, as
explained in Section 2.2.

Consider the pencil of rays incident to $O$ and let $\gamma$ be the
corresponding curve in $\mathcal{L}$ (a “line”). There are 4 points on
$\gamma$, corresponding to the 4 rays tangent to the confocal conics $E$ and
$H$ at $O$. The dual statement to Theorem 1 is then that $T^{n}$ maps these 4
points to inflection points of $T^{n}(\gamma)$.

We proceed as follows. Let $r_{0}\in\gamma$ be one of these 4 rays. We
separate the proof into 3 cases (see Figure 7):

  1. 1.

The ray $r_{0}$ is one of the two rays tangent to the confocal ellipse $E$
through $O$. In this case, $O$ may not lie on the line segment connecting to
two foci.

  2. 2.

The ray $r_{0}$ is one of the two rays tangent to the confocal hyperbola $H$
through $O$. In this case, $O$ may not lie on the minor axis of $C$, nor on
the major axis, on the complement of the line segment connecting the two foci.

  3. 3.

The point $O$ lies on one of the axes of $C$ and $r_{0}$ is one of the two
rays aligned with this axis. In this case $O$ may be the center of $C$.

$p$
---
$\alpha$
---
1
---
1
---
2
---
2
---
$p$
---
3
---
3
---
1
---
1
---
$p$
---
3
---
1
---
3
---
1
---
Figure 7: The 3 types of inflection points of pencils. In each figure, the
pencil is the doted curve, with 4 points and their type marked on it. Left:
$O$ does not lie on an axis of $C$ (the generic case). Middle: $O$ lies on the
major axis, between a vertex and the nearby focus. Right: $O$ lies on the
minor axis.

#### Case 1.

Let $E=C_{\lambda_{0}}$, $b^{2}<\lambda_{0}<a^{2}$, be the confocal ellipse
passing through $O$ and $r_{0}\in\gamma$ one of the 2 rays tangent to
$C_{\lambda_{0}}$ at $O$. The $T$-invariant curve in $\mathcal{L}$ passing
through $r_{0}$ is given by $\lambda=\lambda_{0}$ in the $(t,\lambda)$
coordinates.

Note. We use $r_{0}$ to denote both a point in $\mathcal{L}$ and the
corresponding ray in $\mathbb{R}^{2}$.

######  Lemma 1.

$r_{0}$ is a tangency point of $\gamma$ with the $T$-invariant phase curve
$\lambda=\lambda_{0}$.

$C_{\lambda}$
---
$O$
---
$r_{0}$
---
$C_{\lambda_{0}}$
---

$\lambda$
---
$t$
---
$\lambda_{0}$
---
$\gamma$
---
$r_{0}$
---

Figure 8: Lemma 1.

###### Proof.

The rays of the pencil close to $r_{0}$ are tangent to confocal ellipses with
a greater value of the parameter $\lambda$, see Figure 8 (left). It follows
that $\gamma$, near $r_{0}$, drawn in the $(t,\lambda)$ plane, lies above the
horizontal line $\lambda=\lambda_{0}$ and is therefore tangent to it at
$r_{0}$. See Figure 8 (right). ∎

######  Lemma 2.

$T(r_{0})$ is an inflection point of $T(\gamma).$

###### Proof.

Let $r_{0}=(t_{0},\lambda_{0})$, $r_{1}=(t_{1},\lambda_{0})=T(r_{0})$, the
reflection of $r_{0}$ by $C$, where $t_{1}=t_{0}+c(\lambda_{0}).$ Then $r_{1}$
is tangent to $C_{\lambda_{0}}$ at some point, $O_{1}$. Let $\gamma_{1}$ be
the “line” dual to $O_{1}$, corresponding to the pencil of rays through
$O_{1}$. To show that $r_{1}$ is an inflection point of $T(\gamma)$ it is then
enough to show that the 2-jets at $r_{1}$ of $T(\gamma)$ and $\gamma_{1}$
coincide. See Figure 9.

$r_{1}$
---
$r_{0}$
---
$O$
---
$O_{1}$
---
$C_{\lambda_{0}}$
---
$C$
---
$\lambda$
---
$t$
---
$\lambda_{0}$
---
$\gamma$
---
$r_{0}$
---
$\gamma_{1}$
---
$r_{1}$
---
$t_{0}$
---
$t_{1}$
---
Figure 9: Proof of Lemma 2.

First, $r_{1}\in\gamma_{1}$, so $\gamma_{1}$ and $T(\gamma)$ intersect at
$r_{1}$ (their $0$-jets coincide). Second, $\gamma$ is tangent to the
$T$-invariant horizontal line $\lambda=\lambda_{0}$ at $r_{0}$ (Lemma 1) hence
$T(\gamma)$ is tangent to $\lambda=\lambda_{0}$ at $r_{1}=T(r_{0}).$ The same
holds for $\gamma_{1}$, by Lemma 1, hence $\gamma_{1}$ and $T(\gamma)$ are
tangent at $r_{1}$ (their $1$-jets coincide).

Next, the curve $\gamma$ intersects the horizontal line at a level
$\lambda>\lambda_{0}$ at two points, corresponding to the rays shown in Figure
8 (left). The billiard reflection in the ellipse with parameter $\lambda_{0}$
(the outer ellipse in Figure 8 (left)) takes one of these rays to the other
one. The difference of the $t$-coordinates of these two intersection points
depends only on $\lambda$ and $\lambda_{0}$, but not on $t_{0}$ (see Remark 3
of Section 2.1). It follows that the 2-jets of $\gamma$ and $\gamma_{1}$, at
$r_{0}$ and $r_{1}$ (respectively), are parametrized by

$\gamma:\varepsilon\mapsto(t_{0}+\varepsilon,\lambda_{0}+a\varepsilon^{2}),\quad\gamma_{1}:\delta\mapsto(t_{1}+\delta,\lambda_{0}+a\delta^{2}),$
(1)

where $a=a(\lambda_{0})$, $t_{1}=t_{0}+c(\lambda_{0}).$

Note. All calculations for the rest of the proof of this lemma are mod
$\varepsilon^{3}$ and $\delta^{3}$.

Now $T(t,\lambda)=(t+c(\lambda),\lambda)$, hence the 2-jet of $T(\gamma)$ at
$r_{1}$ is parametrized by

$\displaystyle T(\gamma):\varepsilon\mapsto$
$\displaystyle(t_{0}+\varepsilon+c(\lambda_{0}+a\varepsilon^{2}),\lambda_{0}+a\varepsilon^{2})$
$\displaystyle=(t_{0}+\varepsilon+c(\lambda_{0})+ac^{\prime}(\lambda_{0})\varepsilon^{2},\lambda_{0}+a\varepsilon^{2})$
$\displaystyle=(t_{1}+\varepsilon+ac^{\prime}(\lambda_{0})\varepsilon^{2},\lambda_{0}+a\varepsilon^{2}).$

Next we reparametrize this 2-jet by setting

$\delta=\varepsilon+ac^{\prime}(\lambda_{0})\varepsilon^{2},$

with inverse (mod $\delta^{3}$),

$\varepsilon=\delta-ac^{\prime}(\lambda_{0})\delta^{2}.$

It follows that the 2-jet of $T(\gamma)$ at $r_{1}$ can be reparametrized as

$T(\gamma):\delta\mapsto(t_{1}+\delta,\lambda_{0}+a\delta^{2}),$

coinciding with the expression (1) for the 2-jet of $\gamma_{1}$ at $r_{1}$,
as needed. ∎

Note that the last two lemmas are statements about the 2-jet of $\gamma$ at
$r_{0}$. That is, they remain valid if one replaces $\gamma$ with a curve
whose 2-jet at $r_{0}$ coincides with that of $\gamma$. We thus conclude: if
$r_{0}$ is an inflection point of a curve $\gamma\subset\mathcal{L}$, which is
also a point of tangency of $\gamma$ with the leaf of the $T$-invariant
foliation of $\mathcal{L}$ dual to an ellipse $E$ confocal to $C$, then the
same holds for $T(r_{0})\in T(\gamma).$ It follows by induction on $n$ that
the same holds for $T^{n}(r_{0})\in T^{n}(\gamma)$. This proves Case 1 of
Theorem 1.

#### Case 2.

This case is very similar to the previous one, so we omit the details. We only
note that in this case, like in case 1, the $T$-invariant leaf
$\lambda=\lambda_{0}$ consists of two components, but unlike case 1, $T^{n}$,
for $n$ odd, interchanges the two components; the argument however is
unaffected. See Remark 4 and Figure 10.

$O$
---
$C_{\lambda}$
---
$C_{\lambda_{0}}$
---
$r_{0}$
---
$\lambda$
---
$t$
---
$\lambda_{0}$
---
$\gamma$
---
$r_{0}$
---
Figure 10: Case 2 of Lemma 2. Compare to Figure 8.

#### Case 3.

This case is simpler then the previous two. First, a lemma.

######  Lemma 3.

Let $\rho$ denote the involution of $\mathcal{L}$ induced by the reflection
about one of the axes of $C$, major or minor. Let $r_{0}$ be one of the two
fixed points of $\rho$ (a ray aligned with the axis of reflection) and
$\gamma\subset\mathcal{L}$ a $\rho$-invariant curve containing $r_{0}$. Then
$r_{0}$ is an inflection point of $\gamma$.

###### Proof.

Assume that $\rho$ is given by reflection about the major axis of $C$ (the
$x$-axis) and $r_{0}$ is the ray along this axis, oriented eastwards. We use
the coordinates $(\alpha,p)$ on $\mathcal{L}$, see Figure 5. Then
$\rho(\alpha,p)=(-\alpha,-p)$ and $r_{0}=(0,0).$ Assume the tangent to
$\gamma$ at $r_{0}$ is not vertical. Then the 2-jet of $\gamma$ at $r_{0}$ can
be parametrized by

$\varepsilon\mapsto(\varepsilon,a\varepsilon+b\varepsilon^{2}),$ (2)

for some $a,b\in\mathbb{R}.$ This is mapped by $\rho$ to

$\varepsilon\mapsto(-\varepsilon,-a\varepsilon-b\varepsilon^{2}).$

Renaming $-\varepsilon$ by $\varepsilon$, this 2-jet of $\rho(\gamma)$ at
$r_{0}$ can be reparametrized as

$\varepsilon\mapsto(\varepsilon,a\varepsilon-b\varepsilon^{2}).$ (3)

Since $\rho(\gamma)=\gamma$ and $r_{0}$ is a fixed point of $\rho$, the 2-jets
(2) and (3) must coincide. It follows that $b=0$, hence the 2-jet of $\gamma$
at $r_{0}$ is parametrized by

$\varepsilon\mapsto(\varepsilon,a\varepsilon).$ (4)

On the other hand, the tangent “line” to $\gamma$ at $r_{0}$ is the graph of
$p=a\sin\alpha$ (see Lemma 4 below). Its 2-jet at $r_{0}$ is given by formula
(4). This shows that $r_{0}$ is an inflection point of $\gamma$.

If the tangent to $\gamma$ at $r_{0}$ is vertical then the tangent “line” at
$r_{0}$ is $\alpha=0$ and we can parametrize the 2-jet of $\gamma$ at $r_{0}$
by $\varepsilon\mapsto(a\varepsilon^{2},\varepsilon)$. As before,
$\rho$-invariance of $\gamma$ implies that $a=0,$ hence the 2-jets of $\gamma$
and the $\alpha=0$ at $r_{0}$ coincide. Thus in this case $r_{0}$ is an
inflection point of $\gamma$ as well.

The other 3 cases, where $r_{0}=(\pi,0)$ and $\rho$ is the reflection about
the $x$-axis, or $\rho$ is the reflection about the $y$-axis and
$r_{0}=(\pm\pi/2,0)$, are treated similarly and their proof is omitted. ∎

We can now complete the proof of Case 3 of Theorem 1. Let $O$ be a point on
one of the axes of $C$ (major or minor, or both, when $O$ is the center of
$C$, if $C$ is not a circle). Let $\gamma\subset\mathcal{L}$ be the dual
“line” (the curve corresponding to the pencil of rays through $O$). Let
$r_{0}\in\gamma$ be one of the two rays aligned with the axis through $O$.
Then $\gamma$ is $\rho$-invariant and $r_{0}$ is a fixed point of $\rho$.
Clearly, $\rho$ and $T$ commute, hence $T^{n}(\gamma)$ is $\rho$-invariant and
$T^{n}(r_{0})\in T^{n}(\gamma)$ is a fixed point of $\rho$. Lemma 3 implies
that $T^{n}(r_{0})$ is an inflection point of $T^{n}(\gamma),$ as needed. ∎

##  4 Proof of Theorem 2

###  4.1 Two lemmas

The billiard table $C$ here is the unit circle $x^{2}+y^{2}=1$. We use the
same coordinates $(\alpha,p)$ in the space of oriented lines in
$\mathbb{R}^{2}$ that were introduced in Section 2.1, Figure 5.

######  Lemma 4.

The pencil of rays through a point $(a,b)\in\mathbb{R}^{2}$, the “line” dual
to $(a,b)$, is given by the equation

$p(\alpha)=a\sin\alpha-b\cos\alpha.$ (5)

See Figure 11.

$p$
---
$\alpha$
---

$\alpha$
---
$(a,b)$
---
$p$
---
$\theta$
---
$r$
---
$\alpha^{\prime}$
---

Figure 11:  Left: the solid curve represents the pencil of rays through a
point inside a circular table $C$. The dotted curve is its image under the
billiard map $T$. The 4 marked points on it are its inflection points. The
horizontal lines are the $T$ invariant foliation of the phase cylinder
$\mathcal{L}$. Right: the proof of Lemma 4.

###### Proof.

Let $(a,b)=r(\cos\theta,\sin\theta)$ and $\alpha^{\prime}=\pi/2-\alpha.$ Then

$\displaystyle p$
$\displaystyle=r\cos(\theta+\alpha^{\prime})=r(\cos\theta\cos\alpha^{\prime}-\sin\theta\sin\alpha^{\prime})$
$\displaystyle=r(\cos\theta\sin\alpha-\sin\theta\cos\alpha)=a\sin\alpha-b\cos\alpha.\qed$
††margin:

Let $\gamma$ be a curve in the phase space ${\mathcal{L}}.$ Using the same
terminology as in Section 3, an inflection point of $\gamma$ is a second order
tangency with the “line” tangent to $\gamma$ at the point. If $\gamma$ is the
graph of a function $p(\alpha)$, the tangent “line” is a graph of a function
given by (5), i.e., a solution to the ODE $f^{\prime\prime}+f=0$, hence the
inflection points of $\gamma$ are given by the zeros of the function
$p^{\prime\prime}(\alpha)+p(\alpha)$.

If a line tangent to $\gamma$ is vertical, i.e., $\gamma$ is tangent at
$r_{0}=(\alpha_{0},p_{0})$ to the vertical line $\alpha=\alpha_{0}$, then
$\gamma$ is the graph of a function $\alpha(p)$ near $r_{0}$, and $r_{0}$ is
an inflection point if and only if $\alpha(p)=\alpha_{0}+O(|p-p_{0}|^{3}),$
degenerate if $\alpha(p)=\alpha_{0}+O(|p-p_{0}|^{4}).$

Next consider a map $T:\mathcal{L}\to\mathcal{L}$ given by

$T(\alpha,p)=(\tilde{\alpha},p),\ \tilde{\alpha}=\alpha+\phi(p)\ ({\rm mod}\
2\pi),$

where $\phi(p)$ is some function. Let $(\alpha_{0},p_{0})$ be the coordinates
of a point $r_{0}$ on a curve $\gamma\subset\mathcal{L},$ the graph of a
function $p(\alpha)$. We ask: what is the condition on the 2nd order jets of
$p(\alpha)$ and $\phi(p)$ at $\alpha_{0}$ and $p_{0}$ (respectively) so that
$T(\gamma)$ has an inflection point at $T(r_{0})$? The answer is given by the
following lemma.

######  Lemma 5.

Let $\gamma$ be the graph of $p(\alpha)$, $r_{0}=(\alpha_{0},p_{0})\in\gamma$,
with

$\displaystyle p(\alpha_{0}+\varepsilon)=p_{0}+p_{1}\varepsilon+{p_{2}\over
2}\varepsilon^{2}+O(\varepsilon^{3}),$
$\displaystyle\phi(p_{0}+\delta)=\phi_{0}+\phi_{1}\delta+{\phi_{2}\over
2}\delta^{2}+O(\delta^{3}).$

Then $T(r_{0})$ is an inflection point of $T(\gamma)$ if and only if

$p_{2}+p_{0}(1+p_{1}\phi_{1})^{3}=p_{1}^{3}\phi_{2}.$ (6)

###### Proof.

Calculating mod $\varepsilon^{3},\delta^{3}$ throughout, set

$\delta=p_{1}\varepsilon+{p_{2}\over 2}\varepsilon^{2},$

then

$\displaystyle\phi(p(\alpha_{0}+\varepsilon))$
$\displaystyle=\phi(p_{0}+\delta)=\phi_{0}+\phi_{1}\delta+{\phi_{2}\over
2}\delta^{2}$
$\displaystyle=\phi_{0}+\phi_{1}p_{1}\varepsilon+{p_{1}^{2}\phi_{2}+p_{2}\phi_{1}\over
2}\varepsilon^{2}.$

The 2-jet of $\gamma$ at $r_{0}=(\alpha_{0},p_{0})$ is parametrized by

$\varepsilon\mapsto\left(\alpha_{0}+\varepsilon,\
p_{0}+p_{1}\varepsilon+{p_{2}\over 2}\varepsilon^{2}\right),$

hence the 2-jet of $T(\gamma)$ at $T(r_{0})=(\alpha_{0}+\phi_{0},p_{0})$ is
parametrized by

$\varepsilon\mapsto\left(\alpha_{0}+\phi_{0}+(1+p_{1}\phi_{1})\varepsilon+{p_{1}^{2}\phi_{2}+p_{2}\phi_{1}\over
2}\varepsilon^{2},\ p_{0}+p_{1}\varepsilon+{p_{2}\over
2}\varepsilon^{2}\right).$

Let

$\tilde{\varepsilon}:=(1+p_{1}\phi_{1})\varepsilon+{p_{1}^{2}\phi_{2}+p_{2}\phi_{1}\over
2}\varepsilon^{2},$

then, assuming $1+p_{1}\phi_{1}\neq 0$, one can invert this (mod
$\tilde{\varepsilon}^{3}$),

$\varepsilon=\frac{\tilde{\varepsilon}}{1+p_{1}\phi_{1}}-\frac{p_{1}^{2}\phi_{2}+p_{2}\phi_{1}}{2(1+p_{1}\phi_{1})^{3}}\tilde{\varepsilon}^{2}.$

Thus the 2-jet of $T(\gamma)$ at $T(r_{0})$ is parametrized by

$\tilde{\varepsilon}\mapsto\left(\alpha_{0}+\phi_{0}+\tilde{\varepsilon},\
p_{0}+\tilde{p}_{1}\tilde{\varepsilon}+{\tilde{p}_{2}\over
2}\tilde{\varepsilon}^{2}\right),$

where

$\tilde{p}_{1}=\frac{p_{1}}{1+p_{1}\phi_{1}},\
\tilde{p}_{2}=\frac{p_{2}-p_{1}^{3}\phi_{2}}{(1+p_{1}\phi_{1})^{3}}.$

The inflection condition at $r_{1}$ is then $\tilde{p}_{2}+p_{0}=0,$ which
reduces to the stated formula (6).

If $1+p_{1}\phi_{1}=0$ then $p_{1}=p^{\prime}(\alpha_{0})\neq 0$ so one can
invert $p(\alpha)$ near $\alpha_{0}$,

$\alpha(p_{0}+\delta)=\alpha_{0}+\alpha_{1}\delta+{\alpha_{2}\over
2}\delta^{2},$

where

$\alpha_{1}={1\over p_{1}},\ \alpha_{2}=-\frac{p_{2}}{p_{1}^{3}}$ (7)

and

$p_{1}={1\over\alpha_{1}},\ p_{2}=-\frac{\alpha_{2}}{\alpha_{1}^{3}}.$ (8)

The inflection condition for $p(\alpha)$ at $\alpha_{0}$ is $p_{2}+p_{0}=0.$
Substituting for $p_{2}$ from equation (8), this is

$\alpha_{2}=p_{0}(\alpha_{1})^{3}.$ (9)

Now $T(\gamma)$ is the graph of $\alpha(p)+\phi(p),$ hence the inflection
condition at $T(r_{0})$ is

$\alpha_{2}+\phi_{2}=p_{0}(\alpha_{1}+\phi_{1})^{3}.$

Substituting for $\alpha_{1},\alpha_{2}$ from equation (7), one obtains
equation (6). ∎

###  4.2 Cusps by reflection in a circle

The billiard ball map inside the unit circle $C$ is given by
$T(\alpha,p)=(\alpha+2\arccos p,p)$. Fix a point $O=(a,b)$ inside $C$ and let
$\gamma$ be the dual “line” (5). This takes us to the setting of Lemma 2 with

$p(\alpha)=a\sin\alpha-b\cos\alpha,\ \phi(p)=2n\arccos(p),\ -1<p<1.$

We are looking for points $r_{0}=(\alpha_{0},p_{0})\in\gamma$ such that
$T^{n}(r_{0})$ is an inflection point of $T(\gamma)$. Using circular symmetry,
we may assume, without loss of generality, that $\alpha_{0}=0$, $0\leq b<1$
and $0<a^{2}+b^{2}<1.$ We substitute in formula (6)

$p_{1}=a,\ p_{2}=-p_{0}=b,\ \phi_{1}={-2n\over\sqrt{1-b^{2}}},\
\phi_{2}={2bn\over(1-b^{2})^{3/2}},$

obtaining the inflection condition at $T^{n}(r_{0})$:

$b-b\left[1-{2an\over\sqrt{1-b^{2}}}\right]^{3}={2a^{3}bn\over(1-b^{2})^{3/2}}.$
(10)

This is satisfied if $a=0$ or $b=0$, corresponding to four inflection points
of the curve $T^{n}(\gamma)$, as described by Theorem 2.

We claim that there are no other solutions to equation (10) with $n\geq 1$ and
$0<a^{2}+b^{2}<1$. Set $x=a/\sqrt{1-b^{2}}$. Assuming $a,b\neq 0$, equation
(10) becomes

$(4n^{2}-1)x^{2}-6nx+3=0.$ (11)

The discriminant of this quadratic equation in $x$ is a positive multiple of
$1-n^{2}$. Thus equation (11) has a solution with $n\geq 1$ only for $n=1$.
But in this case the solution is $x=1$, that is, $a^{2}+b^{2}=1,$ which is out
of range.∎

###  4.3 The four cusps are ordinary

Dually, this amounts to proving the non-degeneracy of the 4 inflection points
of $T^{n}(\gamma)$. Suppose, without loss of generality, that $O=(a,0)$,
$a>0$, hence $\gamma$ is given by $p=a\cos\alpha$, and the inflection points
of $T^{n}(\gamma)$ are $T^{n}(r_{0})$, where $r_{0}=(0,0),(\pi,0)$ or
$\pm(\pi/2,a).$

Begin with $r_{0}=(0,0).$ The 3-jet of $\gamma$ at this point is parametrized
by

$\varepsilon\mapsto(\varepsilon,a\varepsilon-{a\over 6}\varepsilon^{3}).$

Then $r_{n}:=T^{n}(r_{0})=(n\pi,0)$. We calculate mod $\varepsilon^{4}$:

$\arccos(a\varepsilon-{a\over
6}\varepsilon^{3})=\frac{\pi}{2}-a\varepsilon+\frac{a(1-a^{2})}{6}\varepsilon^{3},$

hence the 3-jet of $T^{n}(\gamma)$ at $r_{n}$ is parametrized by

$\varepsilon\mapsto\left(n\pi+(1-2na)\varepsilon+\frac{na\left(1-a^{2}\right)}{3}\varepsilon^{3},a\varepsilon-{a\over
6}\varepsilon^{3}\right).$ (12)

Let

$\tilde{\varepsilon}:=(1-2na)\varepsilon+\frac{na\left(1-a^{2}\right)}{3}\varepsilon^{3}.$

If $1-2na\neq 0$ this can be inverted,

$\varepsilon=\frac{\tilde{\varepsilon}}{1-2na}-\frac{na\left(1-a^{2}\right)\tilde{\varepsilon}^{3}}{3(1-2na)^{4}},$

so that

$a\varepsilon-{a\over
6}\varepsilon^{3}=\frac{a}{1-2na}\tilde{\varepsilon}-\frac{a\left(1-2na^{3}\right)}{6(1-2na)^{4}}\tilde{\varepsilon}^{3}.$

The 3-jet of $T^{n}(\gamma)$ at $r_{n}$ can thus be reparametrized as

$\tilde{\varepsilon}\mapsto\left(n\pi+\tilde{\varepsilon},\frac{a}{1-2na}\tilde{\varepsilon}-\frac{a\left(1-2na^{3}\right)}{6(1-2na)^{4}}\tilde{\varepsilon}^{3}\right).$

The tangent “line” at $r_{n}=(n\pi,0)$ is the graph of

$p(\alpha)=\frac{a}{1-2na}\sin(\alpha-n\pi),$

with 3-jet at $r_{n}$ parametrized by

$\varepsilon\mapsto\left(n\pi+\varepsilon,\frac{a}{1-2na}\varepsilon-\frac{a}{6(1-2na)}\varepsilon^{3}\right).$

This coincides with the 3-jet of $T^{n}(\gamma)$ at $r_{n}$ if and only if

$\frac{a\left(1-2na^{3}\right)}{(1-2na)^{4}}=\frac{a}{1-2na},$

which simplifies to

$(4n^{2}-1)a^{2}-6na+3=0.$ (13)

The only solution is $a=n=1$, which is excluded.

######  Remark 5.

We notice a mysterious coincidence between Equations (11) and (13). We could
not find an explanation.

If $1-2na=0$ then the parametrized 3-jet (12) becomes

$\varepsilon\mapsto\left(n\pi+\frac{1-a^{2}}{6}\varepsilon^{3},a\varepsilon-{a\over
6}\varepsilon^{3}\right).$ (14)

Let

$\tilde{\varepsilon}:=a\varepsilon-{a\over 6}\varepsilon^{3},$

with inverse

$\varepsilon=\frac{\tilde{\varepsilon}}{a}+\frac{\tilde{\varepsilon}^{3}}{6a^{3}}.$

Then (14) can be reparametrized as

$\tilde{\varepsilon}\mapsto\left(n\pi+\frac{\left(1-a^{2}\right)}{6a^{3}}\tilde{\varepsilon}^{3},\tilde{\varepsilon}\right).$

This is vertical at $r_{n}=(n\pi,0)$, so the tangent “line” at $r_{n}$ is the
vertical line $\alpha=n\pi$. It coincides with the 2-jet of the above, but not
with the 3-jet, as claimed. The argument for $r_{0}=(\pi,0)$ is similar and is
omitted.

For $r_{0}=(\pi/2,a)$ we proceed in a similar way. The “line” $\gamma$ is the
graph of $p=a\sin\alpha$, whose 3-jet at $r_{0}$ is parametrized by

$\varepsilon\mapsto\left({\pi\over 2}+\varepsilon,a-{a\over
2}\varepsilon^{2}\right).$

The image of this 3-jet under $T^{n}$ is the 3-jet at $T^{n}(r_{0})$
parametrized by

$\varepsilon\mapsto\left(\alpha_{n}+\varepsilon+\frac{an}{\sqrt{1-a^{2}}}\varepsilon^{2},a-{a\over
2}\varepsilon^{2}\right),\quad\alpha_{n}={\pi\over 2}+2n\arccos a.$

Let

$\tilde{\varepsilon}:=\varepsilon+\frac{an}{\sqrt{1-a^{2}}}\varepsilon^{2},$

with inverse

$\varepsilon=\tilde{\varepsilon}-\frac{an}{\sqrt{1-a^{2}}}\tilde{\varepsilon}^{2}+\frac{2a^{2}n^{2}}{1-a^{2}}\tilde{\varepsilon}^{3}.$

We get the parametrization of the 3-jet of $T^{n}(\gamma)$ at $T^{n}(r_{0}),$

$\tilde{\varepsilon}\mapsto\left(\alpha_{n}+\tilde{\varepsilon},a-{a\over
2}\tilde{\varepsilon}^{2}+\frac{a^{2}n}{\sqrt{1-a^{2}}}\tilde{\varepsilon}^{3}\right).$
(15)

The “line” tangent to $T^{n}(\gamma)$ at $T^{n}(r_{0})$ is given by
$p=a\cos(\alpha-\alpha_{n})$, with 3-jet at $T^{n}(r_{0})$ parametrized by

$\varepsilon\mapsto\left(\alpha_{n}+\varepsilon,a-{a\over
2}\varepsilon^{2}\right).$

This coincides with the 2-jet of (15), but not the 3rd, as claimed. The case
$r_{0}=(-\pi/2,-a)$ is similar and is omitted. ∎

##  5 Miscellanea

We present here briefly some results and conjectures, inspired by the previous
sections.

###  5.1 Liouville billiards

Recall that a Riemannian metric in a 2-dimensional domain is called a
_Liouville metric_ if there exist coordinates $(x,y)$ in which it is given by
the formula

$(f(x)+g(y))(dx^{2}+dy^{2}),$

where $f$ and $g$ are smooth functions of one variable, such that
$f(x)+g(y)>0$ for all $x,y$. The coordinate lines form a _Liouville net_ ,
consisting of two families of mutually orthogonal curves.

The Euclidean metric in the plane admits a Liouville net consisting of
confocal conics, corresponding to the respective elliptic coordinates. The
degenerations of this net include the net of confocal parabolas and the net
consisting of concentric circles and the radial lines (as well as the trivial
net consisting of the horizontal and vertical lines).

The elliptic coordinates in 3-space, restricted to a triaxial ellipsoid which
is a level surface of one of the coordinates, define a Liouville metric whose
Liouville net consists of the lines of curvature, see Figure 12.

Figure 12:  The lines of curvature on an ellipsoid form a Liouville net,
associated with the elliptic coordinates in $\mathbb{R}^{3}$. The billiard
bounded by such a curve is a completely integrable.

One considers a billiard system in a geodesically convex domain with a smooth
closed boundary on a Riemannian surface: the trajectories are made of geodesic
segments, and the law of reflection is the same as in the Euclidean case (the
angle of incidence equals the angle of reflection). Similar to the case of
billiards in an ellipse in the plane, the billiard system on a Liouville
surface whose billiard table is bounded by a coordinate line from the
Liouville net is integrable: a generic trajectory has all its segments tangent
to a fixed curve of the Liouville net. See [9, 11, 14, 15, 16] for details.

The main ingredient in the proof of Theorem 1 was the complete integrability
of the billiard ball map in ellipses and its consequences, such as a version
of the Arnold-Liouville theorem (Proposition 1). For this reason, Theorem 1
and its proof extend, with appropriate adjustments, to Liouville billiards as
well.

We note that this set-up includes billiards bounded by conics in the
hyperbolic and spherical geometries, the closest “relatives” of the Euclidean
billiard inside an ellipse. Concerning spherical and hyperbolic conics, see,
e.g., [12].

###  5.2 Cusps on axes

As noted in Remark 21, when a light source $O$ is placed on one of the axes of
an ellipse, two of the cusps on the $n$-th caustic by reflection from $O$ will
be located on this axis, but Theorem 1 does not give their location. Here we
fill this gap, using the classical “mirror equation” of geometric optics
(Equation (5.9) of [20]).

######  Proposition 2.

Let $O=(x_{0},0),$ $|x_{0}|<a,$ and let $O_{n}$ (resp. $O_{n}^{\prime}$) be
the cusp of the $n$-th caustic by reflection from $O$ along the trajectory
leaving $O$ in the positive (resp. negative) direction of the $x$-axis. Then

$O_{n}=(-1)^{n}f^{n}(O),\quad O^{\prime}_{n}=(-1)^{n+1}f^{n}(-O),$

where $f$ is a hyperbolic Möbius transformation of the $x$-axis with fixed
points at the foci $\pm F=(\pm c,0)$, $c=\sqrt{a^{2}-b^{2}}$. Furthermore, $F$
is an unstable fixed point of $f$ and $-F$ is stable. Thus, as $n\to\infty$,

$O_{2n}\to-F,\quad O_{2n+1}\to F,\quad O^{\prime}_{2n}\to F,\quad
O^{\prime}_{2n+1}\to-F.$

Explicitly,

$f(x)=\frac{(a^{2}+c^{2})x-2ac^{2}}{-2ax+a^{2}+c^{2}}.$ (16)

Exception: if $C$ is a circle then $f$ is parabolic, with a single fixed point
at $(0,0)$. Thus, $\lim O_{n}=\lim O_{n}^{\prime}=(0,0)$, as $n\to\infty.$

###### Proof.

Let $(R(x),0)$ be the image of $(x,0)$ after reflection off $C$ at $(a,0)$ and
$(L(x),0)$ the image after reflection at $(-a,0)$. The $x$-coordinate of the
successive images of $(x_{0},0)$, starting with a reflection at $(a,0)$, are
then

$R(x_{0}),LR(x_{0}),RLR(x_{0})\ldots.$

Note that $L(x)=-R(-x),$ hence the $n$-th term in the above sequence is

$x_{n}=(-1)^{n}(-R)^{n}(x_{0}).$

It remains to find an explicit formula for $f(x):=-R(x)$.

The “mirror equation” states that if an object is placed on the line normal to
a convex mirror, where the curvature of the mirror is $k$, at a distance $d$
from the mirror, then a reflected image of the object will form at a distance
$d^{\prime}$ from the mirror, given by

${1\over d}+{1\over d^{\prime}}={2k}.$ (17)

The curvature of $C$ at $(a,0)$ is $a/b^{2}$, so setting
$d=a-x,d^{\prime}=a+f(x)$ in formula (17), we obtain

${1\over a-x}+{1\over a+f(x)}={2a\over b^{2}}.$ (18)

Formula (16) for $f(x)$ follows. From formula (16) follows that

$f^{\prime}(c)=\frac{(a+c)^{2}}{(a-c)^{2}},\quad
f^{\prime}(-c)=\frac{(a-c)^{2}}{(a+c)^{2}}.$

Thus $f^{\prime}(c)>1$ and $0<f^{\prime}(-c)<1.$ It follows that $c$ is an
unstable fixed point of $f$ and $-c$ is stable.

The formula for $O^{\prime}_{n}$ is obtained is a similar manner by
considering the sequence $L(x_{0}),RL(x_{0}),LRL(x_{0}),\ldots$. ∎

Next we study the case when $O$ is on the minor axis.

######  Proposition 3.

Let $O=(0,y_{0}),$ $|y_{0}|<b,$ and let $O_{n}$ (resp. $O_{n}^{\prime}$) be
the cusp of the $n$-th caustic by reflection from $O$ along the trajectory
leaving $O$ in the positive (resp. negative) direction of the $y$-axis. Then

$O_{n}=(-1)^{n}g^{n}(O),\quad O^{\prime}_{n}=(-1)^{n+1}g^{n}(-O),$

where $g$ is an elliptic Möbius transformation of the $y$-axis, conjugate to a
rotation by $4\theta$, where $c+ib=ae^{i\theta}$ (that is, $\theta$ is the
angle between the $x$-axis and line through $(0,b)$ and $-F=(-c,0)$).

Explicitly,

$g(y)=\frac{y\left(c^{2}-b^{2}\right)-2bc^{2}}{2by+c^{2}-b^{2}}.$ (19)

###### Proof.

The proof of formula (19) is very similar to the above proof of formula (16)
and is omitted. Using formula (19), one finds that $g(y)$ has no fixed points,
hence it is elliptic, i.e., conjugate to a rotation. The angle of rotation is
given by the derivative at the complex fixed points. The complex fixed points
of (19) are $\pm ic$, with

$g^{\prime}(ic)=\left({c-ib\over c+ib}\right)^{2},\
g^{\prime}(-ic)=\left({c+ib\over c-ib}\right)^{2}.$

Let $c+ib=ae^{i\theta}.$ Then
$g^{\prime}(ic)=e^{-4i\theta},g^{\prime}(-ic)=e^{4i\theta},$ from which
follows the statement about the angle of rotation. ∎

###  5.3 A light source outside an ellipse

Let us place a light source $O$ outside an ellipse $C$. For each line through
$O$ intersecting the interior of $C$ we consider the two billiard trajectories
in the interior of $C$, whose initial rays are aligned with the line. One then
finds analogues of the 2 conjectures and 2 theorems of this article, with “4”
replaced by “2” throughout: the $n$-th caustic by reflection of these rays is
tangent to $C$ at the contacts points with $C$ of the two tangents to $C$
through $O$, and has 2 cusps, located on the hyperbola confocal with $C$ and
passing through $O$. See Figure 13.

$O$
---
$C$
---
$O$
---
$C$
---
$O$
---
(a)
---
(b)
---
(c)
---
(d)
---
Figure 13: Caustics by reflection from an external light source $O$: (a) each
line through $O$, incident to the interior of $C$, produces 2 billiards
trajectories, tangent to the same conic confocal to $C$; (b) the 1st caustic
by reflection off an ellipse, showing 2 cusps, lying on the confocal hyperbola
through $O$. (c) The 1st two caustics by reflection off a circle, from an
exterior light source $O$, showing two cusps for each caustic, lying on the
line through $O$ and the center of the cicrcle. (d) A coffee cup “half-
caustic”, showing a single cusp.

###  5.4 The complexity of the caustics by reflection

Figure 14 illustrates the observation that the complexity of the $n$-th
caustic by reflection in an ellipse increases with $n$. There are many ways to
measure “complexity”; for example, one may consider the number of times that
the caustic goes to infinity (these points correspond to the vertical tangents
of the curve $T^{n}(\gamma)\subset\mathcal{L}$). It would be interesting to
make conjectures in this direction.

Figure 14: The 2nd, 5th, and 8th caustics by reflection in an ellipse. The
cusps are marked by gray circles.

###  5.5 Pseudo-integrable billiards

One may consider billiard tables bounded by arcs of confocal conics; such
billiards were introduced in [7]. Since confocal conics intersect at right
angles, these billiard tables have angles that are multiples of $\pi/2$.
Figure 15 shows caustics by reflection in a table bounded by two confocal
parabolas. Although four cusps still lie on the confocal parabolas that pass
through the source of light, there are additional cusps, and their number
increases with $n$.

Figure 15: The first three caustics by reflection in a table bounded by two
confocal parabolas.

###  5.6 Caustics by refraction

One could extend the experimental study and make conjectures about caustics by
refraction in ellipses. Cayley considered the first such caustic in the case
of a circle in [6]. See Figure 16, taken from p. 286 of Cayley’s text.

Figure 16: Caustics by refraction of a parallel beam in a circle. The figures
show 3 distinct values of the index of refraction $\mu$ (from left to right):
$1<\mu<2$, $\mu=2$, $\mu>2.$ The 4 cusps occur on the concentric circle of
radius $1/\mu.$

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