# On cusps of caustics by reflection: a billiard variation on Jacobi’s Last
Geometric Statement

Gil Bor111 CIMAT, A.P. 402, Guanajuato, Gto. 36000, Mexico<EMAIL_ADDRESS>Serge
Tabachnikov222 Department of Mathematics, Penn State University, University
Park, PA 16802<EMAIL_ADDRESS>
###### Abstract

A point source of light is placed inside an oval. The $n$-th caustic by
reflection is the envelope of the light rays emanating from the light source
after $n$ reflections off the curve. We show that each of these caustics, for
a generic point light source, has at least 4 cusps. This is a billiard
variation on Jacobi’s Last Geometric Statement, concerning the number of cusps
of the conjugate locus of a point on a convex surface. We present various
proofs, using different ideas, including the curve shortening flow and
Legendrian knot theory.

##  1 Introduction

###  1.1 Motivation and background

The conjugate locus of a point on a surface is the locus of first conjugate
points along geodesics emanating from that point. In his “Lectures on
Dynamics” [11], published posthumously, Jacobi stated that the conjugate locus
of a generic point on an ellipsoid has exactly four cusps. This Last Geometric
Statement of Jacobi was proved only in this century, see [16]. Indeed, as
recently as the end of the 20th century, Marcel Berger wrote [4]:

> … this latter assumption depends on the scandalously unproved Jacobi
> “statement”: the conjugate locus of a non-umbilical point of an ellipsoid
> has exactly four cusps.

A related result is that the conjugate locus of a generic point on a convex
surface has at least four cusps, see [23] for a recent proof. This theorem was
attributed to C. Carathéodory (1912) by W. Blaschke (sect. 103 of [5]), who
presented a sketch of the proof. This theorem belongs to a long list of
results that stem from and are motivated by the celebrated 4-vertex theorem of
S. Mukhopadhyaya. See [2, 13] for surveys.

The conjugate locus can be equivalently described as the locus of the first
intersections of infinitesimally close geodesics emanating from a point. These
geodesics may intersect more than once, and the loci of their intersections
are known as second, third, etc., caustics of the point. Statements similar to
Jacobi’s statement and generalization to arbitrary convex surfaces about the
higher order caustics are still open. There is some experimental evidence that
if the surface is an ellipsoid then, for a non-umbilic point, each such
caustic has exactly four cusps, see [19] and Figure 1 from this paper
(presented with permission).

Figure 1: The first three caustics of an non-umbilic point on an ellipsoid.

In this article we consider a billiard version of this problem. Let $\gamma$
be an oval (a smooth strictly convex closed curve in ${\mathbb{R}}^{2}$), the
boundary of a billiard table or, equivalently, an ideal mirror. Let $O$ be a
point inside $\gamma$, a source of light. For $n=1,2.\ldots$, the 1-parameter
family of rays that have undergone $n$ optical reflections in $\gamma$
envelopes a curve $\Gamma_{n}$, the $n$-th caustic by reflection. See Figure
2.

$\gamma$$\Gamma_{1}$$\Gamma_{2}$$\Gamma_{3}$$O$ Figure 2: The $n$-th caustic
by reflection $\Gamma_{n}$ is the envelope of the family of rays emanating
from $O$ that have undergone $n$ reflections by $\gamma$.

These caustics may have singularities, generically, semi-cubical cusps. We
always assume that the caustics $\Gamma_{n}$ are in general position in this
sense. The singularities of caustics were thoroughly studied by Bruce, Giblin,
and Gibson; see [8] and the references therein.

Figure 3a shows that a caustic by reflection may extend beyond the interior of
$\gamma$, and furthermore, it can be disconnected in the Euclidean plane;
however, as the envelope of a 1-parameter family of lines, it is a connected
curve in the projective plane ${\mathbb{RP}}^{2}$ (possibly, with
singularities). Indeed, a 1-parameter family of lines is a curve in the space
of lines, and the respective envelope is projectively dual to this curve.

(a)

(b)

Figure 3: (a) The first three caustics by reflection in a circle, showing 4
cusps on each of them (red=1st caustic, blue=2nd, green=3rd). The small circle
is the light source. (b) The first caustic by reflection in a circle with an
external source of light (each ray optically reflects at both intersection
points with the circle). The caustic in this case has only 2 cusps. In this
paper we consider only an internal light source.

###  1.2 The main result and two conjectures

Our main result is as follows.

######  Theorem 1.

For every oval $\gamma\subset{\mathbb{R}}^{2}$, a generic light source inside
$\gamma$ and $n\geq 1$, the $n$-th caustic by reflection
$\Gamma_{n}\subset{\mathbb{RP}}^{2}$ has at least four cusps.

We present three proof sketches.

Let $\mathcal{L}$ be the space of directed lines in ${\mathbb{R}}^{2}$. To
each caustic $\Gamma_{n}$ is associated its dual curve
$C_{n}\subset\mathcal{L}$, corresponding to the tangent lines along
$\Gamma_{n}$ (the rays of the $n$-th reflected beam). One can identify
$\mathcal{L}$ with the complement of the ‘north’ and ‘south’ pole of the unit
sphere $S^{2}\subset{\mathbb{R}}^{3}$, so that cusps of $\Gamma_{n}$
correspond to inflection points of $C_{n}$ (points with vanishing spherical
geodesic curvature). Using standard properties of convex billiards, we show
that $C_{n}$ is a closed simple smooth curve in $S^{2}$, intersecting every
great circle. A theorem of B. Segre from 1968 [18, 24] states that such a
curve has at least four spherical inflection points, thus completing the proof
of Theorem 1.

Another approach, starts with a realization of $\mathcal{L}$ as the vertical
cylinder circumscribing $S^{2}$ and the curve $C_{n}\subset\mathcal{L}$
representing the tangent lines of $\Gamma_{n}$. Following S. Angenent [1],
apply the curve shortening flow with respect to the flat metric on the
cylinder to the curve $C_{n}$ to deform it to the graph of a function
$F:S^{1}\to{\mathbb{R}}$ with zero mean value. Spherical inflection points of
$C_{n}$ correspond to the zeroes of $F^{\prime\prime}+F$, a function with
vanishing constant and first order Fourier terms. By the Sturm-Hurwitz
theorem, it has at least four zeros.

Yet another approach is to use the relation between the cusps of the caustic
$\Gamma_{n}$ and the vertices (critical points of the curvature) of its normal
front $\Delta_{n}$, a closed planar curve whose normal lines, parametrized by
$C_{n}$, are the lines tangent to $\Gamma_{n}$, see Figure 4. The relation
bewteen $\Gamma_{n}$ and $\Delta_{n}$ is the familiar relation between
evolutes and involutes, see, e.g., [13].

$\Gamma_{2}$$\Delta_{2}$$\gamma$ Figure 4: The 2nd caustics $\Gamma_{2}$ (red)
with an involute $\Delta_{2}$ (green) for an elliptical billiard table
$\gamma$ (black). The rays (gray) correspond to $C_{2}$, are normal to the
wave front $\Delta_{2}$ and are tangent to $\Gamma_{2}$.

We show that $\Delta_{n}$ exists as a closed curve, possibly with cusps (in
fact, there is a 1-parameter equidistant family of such curves). To show that
$\Delta_{n}$ has at least four vertices, we use a theorem of Chekhanov and
Pushkar [9], stating that a planar cooriented closed wave front has at least
four vertices, provided its Legendrian lift to the space of cooriented contact
element in the plane is a ‘Legendrian unknot’, that is, is Legendrian isotopic
to the Legendrian lift of a circle.

The rest of the article provides background information and details of these
arguments. In the last section we present some generalizations of Theorem 1 to
spherical and hyperbolic geometry, as well as “projective billiards.”

We present two conjectures; the first one is supported by experimental
evidence, the second one might be over-optimistic.

######  Conjecture 1.

If $\gamma$ is an ellipse, then the caustic by reflection $\Gamma_{n}$ for a
light source inside $\gamma$ and different from a focus has exactly four cusps
for every $n\geq 1$ (see Figure 5).

This conjecture is only known to hold in the case of $n=1$ (the ‘catacaustic’,
see next section).

Figure 5: The 2nd, 5th and 8th caustics by reflection in an ellipse, each with
4 cusps (marked by gray disks).

######  Conjecture 2.

If $\gamma$ is not an ellipse then, for some choice of light source inside
$\gamma$ and some $n\geq 1$, the caustic by reflection $\Gamma_{n}$ has more
than four cusps.

Figure 6 shows caustics with $>4$ cusps for non-elliptical $\gamma$ .

Figure 6: First caustic by reflection with more than four cusps of non-
elliptical ovals. Left: $x^{4}+y^{4}=1$, $O=(.6,.4)$. Right:
$.5x^{2}+(1+.25x)y^{2}=1$, $O=(.5,.3)$

###  1.3 Catacaustics

The first caustics by reflections, called catacaustics, are well studied. We
give a brief summary of what is known about them, referring to [7, 8, 15] and
the literature cited in these articles.

A version of the string construction that recovers a billiard curve from a
billiard caustic (see, e.g., [22]) makes it possible to reconstruct the curve
$\gamma$ from its first caustic by reflection $\Gamma_{1}$. This construction
involves a parameter, the length of the string. See Figure 7 (left).

$A$
---
$\Delta_{1}$
---
$Z$
---
$\gamma$
---
$X$
---
$O$
---
$B$
---
$\Gamma_{1}$
---
(a)
---

$O$
---
$\Gamma_{1}$
---
$B$
---
$X$
---
$\gamma$
---
(b)
---

Figure 7: (a) The curve $\gamma$ is the locus of points $X$ such that
$|OX|+|XB|+\stackrel{{\scriptstyle\smile}}{{|BA|}}=const$ (point $A$ is fixed
on $\Gamma_{1}$). The point $Z$ is the reflection of $O$ in the tangency line
to $\gamma$ at $X$. The locus of points $Z$ is the orthotomic curve
$\Delta_{1}$, orthogonal to $BZ$ and whose evolute is $\Gamma_{1}$. (b) The
catacaustic $\Gamma_{1}$ (red) is the locus of 2nd foci $B$ of the osculating
Kepler conic (gray) to the curve $\gamma$ (blue) with 1st foci at $O$.

The orthotomic curve $\Delta_{1}$ is an involute of the catacaustic
$\Gamma_{1}$, see Figure 7 (left), and $\Gamma_{1}$ is the evolute of
$\Delta_{1}$, that is, the envelope of its normals. The cusps of $\Gamma_{1}$
correspond to the vertices of $\Delta_{1}$. It is known that when $\gamma$ is
an ellipse and $O$ is not one of its foci then $\Gamma_{1}$ has 4 vertices
[8].

A Kepler conic is a conic with one focus fixed at the origin $O$. The curve
$\gamma$ has a Kepler conic that has 3-point contact with it at every point,
see [6]. The locus of the second foci of these osculating Kepler conics is the
first caustic $\Gamma_{1}$ – this follows from the optical properties of
conics (a ray from one focus reflects to another focus). It follows that the
cusps of the catacaustics $\Gamma_{1}$ correspond to the points where the
Kepler conics hyperosculate the curve $\gamma$.

#### Computer graphics and animations.

Most figures in this article were made using the computer program Mathematica.
They are complemented with some animations on the web page
https://www.cimat.mx/~gil/caustics/.

#### Acknowledgements:

GB acknowledges support from the Shapiro visiting program in Penn State and
CONACYT Grant A1-S-4588. ST was supported by NSF grant DMS-2005444.

##  2 Background material

###  2.1 The phase cylinder and the billiard ball map

This section contains some standard material on mathematical billiards, see,
e.g., [22].

Denote by $\mathcal{L}$ the space of oriented lines in ${\mathbb{R}}^{2}$. We
use the ‘cylinder model’ of $\mathcal{L}$, with coordinates $(\alpha,p)$
defined as follows: $\alpha\in S^{1}={\mathbb{R}}/2\pi{\mathbb{Z}}$ is the
direction of the line and $p\in{\mathbb{R}}$ is the signed distance from the
oriented line to the origin $O$ (which we choose to be the center of the
initial beam of light). The sign of $p$ is defined by the right-hand rule, see
Figure 8. Thus $\mathcal{L}$ is an infinite cylinder.

$p<0$
---
$p>0$
---
$\alpha$
---
$\alpha$
---
$O$
---
Figure 8: The coordinates $(\alpha,p)$ of an oriented line in
${\mathbb{R}}^{2}$.

The space $\mathcal{L}$ of oriented lines in ${\mathbb{R}}^{2}$ admits an area
form, unique up to scale, invariant under the Euclidean group action. In
coordinates, this area form is $\omega=d\alpha\wedge dp.$

The phase cylinder of the billiard system inside an oval
$\gamma\subset{\mathbb{R}}^{2}$ is the set $M\subset\mathcal{L}$ of oriented
lines intersecting $\gamma$. It is a bounded cylinder whose two boundary
components correspond to the lines tangent to $\gamma$, one component for each
orientation. The “equator” $p=0$ corresponds to the lines through $O$, see
Figure 9.

$C_{0}$$M$$\alpha$$p=0$$p$$\mathcal{L}$

$r$$T(r)$$\gamma$

Figure 9: Left: the phase cylinder $M\subset\mathcal{L}$. Right: the billiard
ball map $T:M\to M$.

The billiard ball map $T:M\to M$, sending an incoming ray to the reflected
one, is an area preserving transformation, that is, $T^{*}\omega=\omega$.
Since $\omega=-d(pd\alpha)$, the differential 1-form
$T^{*}(pd\alpha)-pd\alpha$ is closed. In fact, more is true: as we will now
show, it is exact, that is, $T^{*}(pd\alpha)-pd\alpha=dF$ for some function
$F:M\to{\mathbb{R}}$. An example of an area preserving, but non-exact, map is
$(\alpha,p)\mapsto(\alpha,p+1).$

######  Proposition 1.

The billiard ball map $T:M\to M$ is exact.

In order to prove Proposition 1, consider another description of the phase
cylinder, as the set of unit vectors with a foot point on $\gamma$, pointing
inwards, the initial position and velocity of the billiard ball. These unit
vectors are in one-to-one correspondence with the oriented lines that they
generate. Let $\gamma(t)$ be an arc length counterclockwise parameterization
and $\varphi$ be the angle between the tangent $\gamma^{\prime}(t)$ and the
unit vector. See Figure 10a.

’$\small\gamma(t)$$\small\gamma^{\prime}(t)$$p$$O$$\alpha$$\small\varphi$(a)

$\gamma(t)$$\gamma^{\prime}(t)$$\alpha$$\varphi$$\varphi_{1}$$\varphi_{1}$$\gamma(t_{1})$(b)$L$

Figure 10: (a) The coordinates $(t,\varphi)$ on $M$ and their relation to
$(\alpha,p).$ (b) The generating function $L$ of the billiard ball map $T$.

Consider the differential 1-form $\cos\varphi\,dt$. Let
$L=|\gamma(t_{1})-\gamma(t)|$ be the distance between the intersection points
of a line with $\gamma$. See Figure 10b.

######  Lemma 2.

$T^{*}(\cos\varphi\ dt)-\cos\varphi\ dt=dL$.

###### Proof.

One has: $T(t,\varphi)=(t_{1},\varphi_{1})$ and

$\frac{\partial L(t,t_{1})}{\partial t}=-\cos\varphi,\ \frac{\partial
L(t,t_{1})}{\partial t_{1}}=\cos\varphi_{1},$

that is, $dL=\cos\varphi_{1}\ dt_{1}-\cos\varphi\ dt,$ as needed. ∎

Two differential 1-forms are cohomologous if their difference is the
differential of a function.

######  Lemma 3.

The 1-form $p\,d\alpha$ is cohomologous to $\cos\varphi\ dt$.

###### Proof.

Using complex notation, see Figure 10a, one has

$e^{i(\alpha-\varphi)}=\gamma^{\prime}(t),\ p=\det(\gamma(t),e^{i\alpha}).$

Differentiating these equations, we get

$d\alpha\equiv d\varphi\ (\mbox{mod }dt),\ dp\equiv\sin\varphi\,dt\ (\mbox{mod
}d\alpha),$

so

$d\alpha\wedge dp=\sin\varphi\,d\varphi\wedge dt=-d(\cos\varphi\ dt),$

hence $pd\alpha-\cos\varphi\ dt$ is closed.

To show that $p\,d\alpha-\cos\varphi\,dt$ is exact it suffices to show that
its integral along a non-contractible closed curve in $M$ vanishes. As such a
curve take $C$, the boundary component of the phase cylinder $M$ given by
$\varphi=0$. Clearly, $\int_{C}\cos\varphi\,dt$ equals the perimeter of
$\gamma$.

On the other hand, by the Cauchy-Crofton formula, this perimeter equals $\pi$
times the average length of the orthogonal projection of $\gamma$ on a line,
that is, $\int_{C}pd\alpha$. This implies the result. ∎

Proof of Proposition 1. Lemma 2 shows that $T$ preserves the integral
$\int_{C}\cos\varphi\ dt$, and Lemma 3 shows that $\int_{C}\cos\varphi\
dt=\int_{C}p\,d\alpha$, hence $T$ preserves the integral $\int_{C}p\,d\alpha$,
i.e., it is exact. ∎

The signed area enclosed by an oriented closed curve $C$ in $\mathcal{L}$ is
the line integral $\int_{C}pd\alpha$. Clearly, the curve $C_{0}$ representing
the initial beam of light encloses zero area. Since $T$ is an exact map, so
does $C_{n}=T^{n}(C_{0})$. See Figure 11.

$p$$\alpha$$C_{1}$$C_{2}$$C_{3}$ Figure 11: The succesive iterates
$C_{n}=T^{n}(C_{0})$, for $n=1,2,3$, drawn on the phase cylinder $\mathcal{L}$
(spread flat). Each is a simple closed smooth curve of 0 signed area (blue
area = green area). The billiard table is the ellipse $4x^{2}/5+y^{2}=1$ and
the source is $(.6,.2)$.

###  2.2 Contact elements, Legendrian knots, and wave fronts

A contact element in the plane is a pair consisting of a point and a line
through it. A coorientation of a contact element is a choice of one of the
sides of the line. More conceptually, the space of cooriented contact elements
is the spherization of the cotangent bundle $ST^{*}{\mathbb{R}}^{2}$: assign
to a covector $\eta$ its kernel, a tangent line, and define coorientation by
choosing the side on which $\eta$ is positive.

The space of contact elements carries a contact structure, a 2-dimensional
distribution defined by the “skating condition”: the foot point may move along
the line, and the line may rotate about the foot point. Let $(x,y)$ be the
standard coordinates in ${\mathbb{R}}^{2}$ and $\theta$ the angle between the
positive $x$-axis and the direction of the line; then the contact distribution
is the kernel of the 1-form $\sin\theta\,dx-\cos\theta\,dy.$

A smooth curve in $ST^{*}{\mathbb{R}}^{2}$ that is tangent to the contact
distribution is called Legendrian. Its projection to the plane is a wave
front, a curve that may have singularities, generically semicubical cusps, but
that has a tangent line at every point. Conversely, such a curve has a unique
lift to the space of contact elements as a Legendrian curve.

We introduce coordinates $(\alpha,p,z)$ in $ST^{*}{\mathbb{R}}^{2}$, where
$(\alpha,p)\in T^{*}S^{1}$ are the coordinates of the orthogonal line at the
foot point $A$, and $z$ is the (signed) distance of the line to the origin
$O$. See Figure 12.

$A$$p$$\ell$$z$$\theta$$O$$\alpha$ Figure 12: Coordinates $(\alpha,p,z)$ on
the space $ST^{*}{\mathbb{R}}^{2}$ of cooriented contact elements in
${\mathbb{R}}^{2}$. With a cooriented line $\ell$ through $A$ we associate the
(signed) distance $z$ to $O$ and the coordinates $(\alpha,p)$ of the
perpendicular oriented line at $A$, pointing to the positive side of $\ell$.

This defines an identification of $ST^{*}{\mathbb{R}}^{2}$ with $J^{1}S^{1}$,
the space of 1-jets of functions $f:S^{1}\to{\mathbb{R}}$, where $z=f(\alpha)$
and $p=f^{\prime}(\alpha).$ On $J^{1}S^{1}$ there is a standard contact form
$dz-pd\alpha$ (the 1-jets of functions $z=f(\alpha)$ are Legendrian curves).

######  Lemma 4.

The identification $ST^{*}{\mathbb{R}}^{2}=J^{1}S^{1}$ is a contactomorphism.

###### Proof.

Using the notation of Figure 12,

$\alpha=\theta+\pi/2,\ p=\det(A,e^{i\alpha}),\ z=\det(A,e^{i\theta}),$

so

$dz-pd\alpha=\det(dA,e^{i\theta})=\sin\theta dx-\cos\theta dy,$

as claimed. ∎

We have two projections

${ST^{*}{\mathbb{R}}^{2}}$${{\mathbb{R}}^{2}}$${\mathcal{L}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$

where $\pi_{1}$ maps a cooriented contact element $(A,\ell)$ to $A$ and
$\pi_{2}$ maps it to the line through $A$, orthogonal to $\ell$, oriented
towards its positive side.

In coordinates, $\pi_{2}:(\alpha,p,z)\mapsto(p,\alpha)$ (“forgetting $z$”).
The fibers of $\pi_{1}$ are Legendrian, spanned by the vector field
$\partial_{\alpha}-z\partial_{p}+p\partial_{z}$, while the fibers of
$\pi_{2}$, spanned by the vector field $\partial_{z}$, are transverse to the
contact distribution. Hence $\pi_{2}$ projects a smooth Legendrian curve in
$ST^{*}{\mathbb{R}}^{2}$ to a smooth curve in $\mathcal{L}$.

Conversely, let $C$ be a closed curve in $\mathcal{L}$. We want to lift it via
$\pi_{2}:ST^{*}{\mathbb{R}}^{2}\to\mathcal{L}$ to a Legendrian curve
$\widetilde{C}\subset ST^{*}{\mathbb{R}}^{2}$. Since the contact distribution
on $ST^{*}{\mathbb{R}}^{2}$ is transverse to the fibers of $\pi_{2}$, once the
initial point of the lifting is chosen, the lifting is uniquely determined,
but it may fail to close up.

######  Lemma 5.

The lifted curve $\widetilde{C}$ is closed if and only if
$\int_{C}pd\alpha=0$: the curve $C$ encloses zero signed area.

###### Proof.

The curve $\widetilde{C}$ is closed if and only if the value of the third
coordinate $z$ is the same at the endpoints. Since $dz=pd\alpha$ along a
Legendrian curve, the values of $z$ at the endpoints are equal if and only if
$\int_{C}pd\alpha=0$. ∎

The curve $C\subset\mathcal{L}$ defines a 1-parameter family of oriented
lines. The projection of the lifted Legendrian curve $\widetilde{C}\subset
ST^{*}{\mathbb{R}}^{2}$ to ${\mathbb{R}}^{2}$ is a wave front $\Delta$ that is
orthogonal to this family of lines and is cooriented by their directions. If a
closed front $\Delta$ exists, i.e., $C$ encloses zero signed area, then there
exists a whole 1-parameter family of fronts that are equidistant from each
other. This non-uniqueness corresponds to the choice of the initial point of
the lifted curve $\widetilde{C}$.

The situation is the same as in the familiar relation between evolutes and
involutes: for an involute of a closed curve to close up it is necessary and
sufficient for the curve to have zero signed length (the sign changes after
each cusp), and the equidistant family of curves share their normals, and
hence their evolutes.

###  2.3 Vertices of wave fronts and Legendrian isotopies

A vertex of a plane curve is an extremum of its curvature or, equivalently, a
cusp of the evolute, the envelope of its normals. The notion of vertex extends
to cooriented wave fronts: the curvature at cusps is infinite, changing from
$-\infty$ to $\infty$ (so cusps are not vertices).

The classical 4-vertex theorem asserts that a simple closed convex curve has
at least four vertices. Let $\Delta$ be a cooriented wave front whose
Legendrian lift to $ST^{*}{\mathbb{R}}^{2}$ is embedded, i.e., is a Legendrian
knot. V. Arnold conjectured [2, 3] that if this Legendrian knot is homotopic
as a Legendrian knot to the Legendrian lift of a circle, then $\Delta$ has at
least four vertices. This conjecture was proved by Chekanov and Pushkar [9]
using Legendrian knot theory.

A generic regular homotopy of a cooriented wave front is a composition of a
number of moves, similar to the Reidemeister moves in knot theory, see Figure
13, borrowed from [9]. The first five moves are isotopies of the respective
Legendrian knot, but the “dangerous” self-tangency with coinciding
coorientations correspond to self-intersection of the Legendrian lifted curve
and changes the Legendrian knot type.

Figure 13: Generic “perestroikas” of cooriented wave fronts.

For example, the curve on the left of Figure 14 has only two vertices, but the
curve on the right is Legendrian isotopic to a circle, therefore it has at
least four vertices no matter how one draws it. Thus these curves are not
Legendrian isotopic. On the other hand, the Whitney winding number of both
curves is one, hence they are regularly isotopic.

Figure 14: Left: only two vertices; right: at least four vertices.

###  2.4 Summary

With each “beam” of light rays (a 1-parameter family of oriented lines in
${\mathbb{R}}^{2}$) we have associated four curves,

$C\subset\mathcal{L},\qquad\widetilde{C}\subset
ST^{*}{\mathbb{R}}^{2},\qquad\Delta\subset{\mathbb{R}}^{2},\qquad\Gamma\subset{\mathbb{RP}}^{2}.$

The correspondences between the cusps, vertices and inflection points on these
curves is depicted in Figure 15.

$C$$\Delta$$\Gamma$inflection ptscuspsverticesprojectiveduality

normals

wave front

involute

evvolute

$\widetilde{C}$$\pi_{2}$$\pi_{1}$ Figure 15:

##  3 Proofs of Theorem 1

#### First proof.

Cusps of $\Gamma_{n}$ correspond, by projective duality, to inflection points
of $C_{n}$. These inflections points are 3-point contacts of $C_{n}$ with the
curves in $\mathcal{L}$ corresponding to the 1-parameter families of lines
passing through a fixed point. If the point is $(a,b)\in{\mathbb{R}}^{2}$,
then the respective curve in $\mathcal{L}$ is the graph of the first harmonic

$p=a\sin\alpha-b\cos\alpha,$

that is, it is an ellipse obtained as the intersection of the cylinder
$\mathcal{L}$ with a plane through the origin. Note that this graph encloses
zero signed area.

Consider the central projection of this cylinder to the unit sphere. This
projection sends the graphs of the first harmonics to great circles.

Since $C_{n}$ encloses zero signed area, it intersects every graph of the
first harmonics. It follows that $\bar{C}_{n}$, the image of the curve
$C_{n}$, is a smooth spherical curve that is not contained in any hemisphere.
In particular, the convex hull of $\bar{C}_{n}$ contains the origin.

The (geodesic) inflections of $\bar{C}_{n}$ in the standard metric of the
sphere are its 3-point contacts with great circles. By the Segre theorem
mentioned earlier, $\bar{C}_{n}$ has at least four inflections. Therefore so
does $C_{n}$. ∎

#### Second proof.

Following [1], one can use the curve shortening flow to prove that the curve
$C_{n}$ has at least four inflections. Recall that under the curve shortening
flow, each point of the curve moves in the normal direction with the speed
equal to the curvature; see [12, 14] and the book [10].

Equip $\mathcal{L}$ with the flat Riemannian metric $d\alpha^{2}+dp^{2}$ and
apply the curve shortening flow to $C_{n}$. Let $C_{n}(s)$ be arclength
parametrization, then the flow is given by the partial differential equation
$C_{t}=C_{ss}$.

A variation of the standard proof shows that the evolution is defined for all
$t\geq 0$, deforming $C_{n}$ through embedded curves, shrinking it to a
horizontal curve $p=const$, which is a closed geodesic. A version of the
maximum principle implies that the number of inflections does not increase
during this evolution, see [1]. This is illustrated in Figure 16.

Figure 16: In the curve shortening flow, two nearby inflection points may
cancel each other, but they cannot appear on a convex arc.

Next, a version of Lemma 3.1.7 in [12] or Lemma 1.10 of [14] shows that the
evolving curves enclose zero signed areas.

######  Lemma 6.

The curve shortening flow $C_{t}=C_{ss}$ preserves the signed area
$\int_{C}pd\alpha.$

###### Proof.

Let $(\alpha(s),p(s))$ be arc length parameterization of a non-contractible
closed curve in $\mathcal{L}$, so that $\alpha_{s}^{2}+p_{s}^{2}=1$. Then its
time evolution under the curve-shortening flow is given by
$(\alpha_{t},p_{t})=kN=k(-p_{s},\alpha_{s}),$ where the subscript denotes the
derivative, $k$ is the curvature, and $N$ is the unit normal. It follows that

$\frac{d\int
pd\alpha}{dt}=\int[k\alpha_{s}^{2}-p(kp_{s})_{s}]ds=\int[k\alpha_{s}^{2}+kp_{s}^{2}]ds=\int
k(s)ds,$

where the second equality is due to integration by parts.

It remains to note that the total curvature of a closed curve that goes around
the cylinder equals zero. ∎

As $C_{n}$ approaches $C_{0}$, it is given by the graph of a function
$p=F(\alpha)$. The inflection points of this graph are the points where it is
tangent to 2nd order to the graphs of functions of the form
$h(\alpha)=a\cos(\alpha)+b\sin(\alpha).$

For each $\alpha\in S^{1}$, one can find unique $a,b$ such that
$F(\alpha)=h(\alpha)$, $F^{\prime}(\alpha)=h^{\prime}(\alpha).$ Since
$h^{\prime\prime}+h=0,$ the equation
$F^{\prime\prime}(\alpha)=h^{\prime\prime}(\alpha)$ holds if and only if
$F^{\prime\prime}(\alpha)+F(\alpha)=0.$ So the inflection points of $F$ are
the zeros of the function $G:=F^{\prime\prime}+F.$

To conclude that $G$ has no less than four zeros, apply the Sturm-Hurwitz
theorem that states that the number of zeros of a $2\pi$-periodic function is
not less than the number of zeros of its first non-trivial harmonic, see,
e.g., [2, 3].

Since the differential operator $d^{2}+1$ preserves the order of Fourier terms
and kills 1st order terms, $G$ has no first harmonics. Since the curve
encloses zero signed area, $F$ has zero constant term, and so does $G$, as
needed. ∎

######  Remark 7.

The Sturm-Hurwitz theorem has many proofs, see Section 8.1 of [17].
Interestingly, one of them, due to G. Polya, makes use of the heat equation, a
close relative of the curve shortening flow.

#### Third proof.

This argument relies on the correspondence between the cusps of $\Gamma_{n}$
and the vertices of $\Delta_{n}$, its normal front.

Since $C_{n}$ encloses zero signed area, Lemma 5 implies that it admits a
Legendrian lift $\widetilde{C}_{n}\subset ST^{*}{\mathbb{R}}^{2}$, and its
projection to ${\mathbb{R}}^{2}$ is a closed curve, possibly with cusps, which
is normal to the rays of $C_{n}$.

A homotopy of the curve $C_{n}$ to $C_{0}$ in the class of smooth closed
embedded curves that enclose zero signed area induces a Legendrian isotopy
between the Legendrian knots $\widetilde{C}_{n}$ and $\widetilde{C}_{0}$.
(Such a homotopy is provided by the curve shortening flow but, unlike the
second proof, one can use any other homotopy for this purpose).

Now our “black box”, the Pushkar-Chekanov theorem [9], implies that
$\Delta_{n}$ has at least four vertices. ∎

##  4 Variations

In this last section we briefly mention other variants of our result.

#### Spherical and hyperbolic geometry.

One can extend Theorem 1 to geodesically convex billiards in spherical and
hyperbolic geometries (the former lie in one hemisphere).

The space of oriented geodesics (great circles) in $S^{2}$ is identified with
$S^{2}$ itself via the usual equator/pole correspondence. The phase cylinder
$M\subset S^{2}$ parametrizes oriented geodesics intersecting $\gamma$. The
billiard ball map preserves the standard area form on $S^{2}$, and the curve
$C_{n}=T^{n}(C_{0})$ bisects the area of the sphere.

According to Arnold’s “tennis ball theorem” [2] (which also follows from the
theorem of Segre), a smooth closed embedded spherical curve that bisects the
area has at least four inflections. As before, the curve $C_{n}$ is dual to
the caustic $\Gamma_{n}$, hence the latter has at least four cusps.

In the hyperbolic case, consider the hyperboloid model
$H^{2}=\\{x^{2}+y^{2}-z^{2}=-1,\ z>0\\}$, with the Minkowski metric
$dx^{2}+dy^{2}-dz^{2}$ restricted to $H^{2}$. An oriented geodesic is given by
intersecting $H^{2}$ with an oriented plane through the origin, the orthogonal
complement (in the Minkowski sense) of a space-like unit vector.

These vectors comprise a hyperboloid of one sheet
$H^{1,1}=\\{x^{2}+y^{2}-z^{2}=1\\}$, equipped with the area form induced from
the ambient Minkowsky space (this area form is invariant under the group of
motions $SO(2,1)$). The phase cylinder $M\subset H^{1,1}$ corresponds to
geodesics intersecting $\gamma$, and the billiard ball map is exact area
preserving.

To show that the curve $C_{n}$ has at least four inflections, we use the same
argument as in our first proof of Theorem 1: centrally project the hyperboloid
to the sphere. This projection takes inflections to inflections, and the image
of $C_{n}$ contains the origin in its convex hull. Then the Segre theorem
implies the result.

#### Projective billiards.

For any convex curve $\gamma\subset{\mathbb{R}}^{2}$ with a transverse vector
field $v$ along it one can define the projective billiard map $T:M\to M$ [20,
21]. The reflection law is as follows. Consider an incoming ray at a point
$x\in\gamma$ in the direction $u$, decompose $u=u_{1}+u_{2}$, where $u_{1}$ is
tangent to $\gamma$ at $x$ and $u_{2}$ is a multiple of $v(x)$. Then the
outgoing ray passes through $x$ in the direction $u_{1}-u_{2}$. Equivalently,
the tangent line, the transverse line, the incoming, and the outgoing ones,
form a harmonic quadruple of lines.

If the transverse field consists of the normals, one has the usual law “the
angle of incidence equals the angle of reflection”.

If $\gamma$ is an origin-centered ellipse and the transverse field $v$ is
given by the gradient of a homogeneous function of two variables, then the
projective billiard ball map is again exact area preserving. The area form on
the phase space $M$ is the same as the one on the space of oriented geodesics
in the hyperbolic plane, but this time one considers the projective, or
Cayley-Klein, model of hyperbolic geometry in the interior of the ellipse
$\gamma$. The total area of $M$ is infinite in this case.

As before, Theorem 1 holds: see Figure 17 for an illustration.

Figure 17: The 1st caustic by reflection, showing 8 cusps, in a projective
billiard system with a circular table and the exact transverse field
$v=\nabla(x^{4}+y^{4})$.

#### Other initial beams.

Finally, we note that Theorem 1, along with its proofs, extends to some other
initial beams of light. For example, one may consider a 1-dimensional source,
an oval that lies inside $\gamma$ and that emanate rays of light in the
outward normal directions.

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