On the Motion of Billiards in Ellipses

Already for two centuries, billiards in ellipses have attracted the attention of mathematicians, beginning with J.-V. Poncelet and A. Cayley. The assertion that one periodic billiard inscribed in an ellipse $e$ and tangent to a confocal ellipse $c$ implies a one-parameter family of such polygons, is known as the standard example of a Poncelet porism. It was Cayley who derived analytical conditions for such a pair $(e,c)$ of ellipses.

In 2005 S. Tabachnikov published a book on billiards from the viewpoint of integrable systems Tabach . The book DR_Buch and various papers by V. Dragović and M. Radnović addressed billiards in conics and quadrics within the framework of dynamical systems.

Computer animations carried out by D. Reznik, stimulated a new vivid interest on this well studied topic, where algebraic and analytic methods are meeting. Originally, Reznik’s experiments focused on billiard motions in ellipses, i.e., on the variation of billiards with a fixed circumscribed ellipse $e$ and a fixed caustic $c\,$. He published a list of more than 80 numerically detected invariants in 80 and contributed, together with his coauthors R. Garcia, J. Koiller and M. Helman, several proofs. Other authors like A. Akopyan, M. Bialy, A. Chavez-Caliz, R. Schwartz, and S. Tabachnikov published several proofs and found more invariants (e.g., in Ako-Tab ; Bialy-Tab ; Chavez ).

For a long time, at least since Jacobi’s proof of the Poncelet theorem on periodic billiards (see further references in (DR_russ, , p. 320)), it has been wellknown that there is a tight connection between billiards and elliptic functions (note also (Duistermaat, , Sect. 11.2) and Bobenko ). On the other hand, S. Tabachnikov proved in his book Tabach the existence of canonical parameters on ellipses with the property that the billiard transformation between consecutive vertices $P_{i}\mapsto P_{i+1}$ of a billiard acts like a shift on the parameters.

The goal of this paper is to prove that Jacobian elliptic functions with the numerical eccentricity of the caustic $c$ as modulus pave the way to canonical coordinates on the ellipse $e\,$. This is a consequence of a kinematic analysis of the billiard motion. It yields an infinitesimal transformation in the plane which preserves a family of confocal ellipses while it permutes the confocal hyperbolas as well as the tangents of the caustic. Integration results in a group of transformations with a canonical parameter. In terms of elliptic functions, we also obtain a mapping that sends a square grid together with the diagonals to a Poncelet grid.¹¹1 Recently, parametrizations of confocal conics in termins of elliptic functions were also presented in Bobenko , but not from the viewpoint of billiards. The paper concludes with applying the results of the velocity analysis to a few invariants of periodic billiards. These invariants deal mainly with the distances which occur on each side between the contact point with the caustic and the endpoints.

At the begin, we recall a few properties of confocal conics. A family of confocal central conics is given by

serves as a parameter in the family. All these conics share the focal points $F_{1,2}=(\pm d,0)$, where $d^{2}:=a^{2}-b^{2}$.

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

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

while conversely

Suppose that $(a,b)$ in (2.1) are the semiaxes $(a_{c},b_{c})$ 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

For the elliptic coordinates $(k_{e},k_{h})$ of $P$ follows from (2.2) that

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

we obtain

and $\|{\mathbf{t}}_{e}(t)\|^{2}=k_{e}-k_{h}(t)\,$. 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 1). Conversely, points of $e$ or $c$ on this hyperbola have a parameter out of $\{t,-t,\pi+t,\pi-t\}$.

Let $\theta_{i}/2$ denote the angle between the tangents drawn from any point $P_{i}\in e$ to $c$ and the tangent to $e$ at $P_{i}$ (Figures 2 or 3). Then we obtain for $P_{i}=(a_{e}\cos t_{i},\,b_{e}\sin t_{i})$ with elliptic coordinates $\left(k_{e},k_{h}(t_{i})\right)$

For a proof see Sta_I . We can assume a counter-clockwise order of the billiard. Hence, all exterior angles are positive.

From (2.6) follows

and furthermore

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 $b_{h}^{2}=-(b_{c}^{2}+k_{h})$.

Let $\dots P_{1}P_{2}P_{3}\dots$ be a billiard in the ellipse $e$. Then the extended sides intersect at points

where $i=\dots,1,2,3,\dots$ and $j=1,2,\dots$. These points are distributed on different confocal conics: For fixed $j$, there are ellipses $e^{(j)}$ passing through the points $S_{i}^{(j)}$. On the other hand, the points $S_{i}^{(2)}$, $S_{i}^{(4)},\dots$ are located on the confocal hyperbola through $P_{i}$, while $S_{i}^{(1)}$, $S_{i}^{(3)},\dots$ belong to the confocal hyperbola through the contact point $Q_{i}$ between the side $P_{i}P_{i+1}$ and the caustic $c\,$. This configuration is called the associated Poncelet grid (Figure 1). For periodic billiards the sets of points $S_{i}^{(j)}$ and associated conics are finite. The turning number $\tau$ of a periodic billiard in $e$ with an ellipse as caustic counts how often one period of the billiard surrounds the center $O$ of $e$ (note Figure 1).

For each billiard $P_{1}P_{2}\dots$ in $e$ with caustic $c$ there exists a conjugate billiard $P_{1}^{\prime}P_{2}^{\prime}\dots$ in $e$ with the same caustic. The axial scaling $c\to e$ maps the contact point $Q_{i}\in c$ of $P_{i}P_{i+1}$ to $P_{i}^{\prime}$ while the inverse scaling brings $P_{i}$ to the contact point $Q_{i-1}^{\prime}$ of $P_{i-1}^{\prime}P_{i}^{\prime}$ with the caustic. The relation between these billiards is symmetric. For further details see (Sta_I, , Sect. 3.2).

Let the first vertex of a billiard $P_{1}P_{2}\dots$ move smoothly along the circumscribed ellipse $e$. Then this induces a continuous variation of all other vertices along $e$ and also of the intersection points $S_{i}^{(j)}$ along $e^{(j)}$. We call this a billiard motion, though it neither preserves angles or distances nor is an affine or projective motion.

According to Graves’s construction (Conics, , p. 47), we can conceive the periodic billiard $P_{1}P_{2}\dots P_{N}$ as a flexible chain of fixed total length $L_{e}$ and the caustic $c$ as a fixed non-circular chain wheel. The vertices $P_{1},P_{2},\dots$ move along $e$ and relative to the chain such that they keep the chain strengthened, while the chain contacts $c$ only at the single points $Q_{1},Q_{2},\dots,Q_{N}$.

Let us pick out a single vertex $P_{2}$ (see Figure 2). In the language of kinematics, the line spanned by the straight segment $Q_{1}P_{2}$ rolls at $Q_{1}$ on $c$ (= fixed polode) while point $P_{2}$ moves along the line (= moving polode) with the velocity vector ${\mathbf{v}}_{t_{1}}$ . The instantaneous rotation about $Q_{1}$ with the angular velocity $\omega_{1}$ assigns to $P_{2}$ a velocity vector ${\mathbf{v}}_{n_{1}}$ orthogonal to $Q_{1}P_{2}$ in order to keep the vector of absolute velocity of $P_{2}\,$, namely ${\mathbf{v}}_{2}={\mathbf{v}}_{t_{1}}+{\mathbf{v}}_{n_{1}}$, tangent to the ellipse $e$.

Similarly, we have a second decomposition ${\mathbf{v}}_{2}={\mathbf{v}}_{t_{2}}+{\mathbf{v}}_{n_{2}}$, since at the same time the line $[Q_{2},P_{2}]$ rotates about $Q_{2}$ with the angular velocity $\omega_{2}$, while $P_{2}$ moves relative to this line. Due to the constant length of the chain, the tangential components in these two decompositions must be of equal lengths $\|{\mathbf{v}}_{t_{2}}\|=\|{\mathbf{v}}_{t_{1}}\|$. Since the tangent $t_{P}$ to $e$ at $P_{2}$ bisects the exterior angle of $Q_{1}P_{2}Q_{2}$, the second decomposition is symmetric w.r.t. $t_{P}$ to the first one. From $\|{\mathbf{v}}_{n_{2}}\|=\|{\mathbf{v}}_{n_{1}}\|$ follows for the distances $r_{2}:=\overline{P_{2}Q_{1}}$ and $l_{2}:=\overline{P_{2}Q_{2}}$

and similarly for all other vertices. If the billiard is $N$-periodic, then the product of all ratios $l_{i}/r_{i}$ for $i=1,\dots,N$ yields

which results in the equation

listed as k116 in (80, , Table 2).

Figure 3 shows a graphical velocity analysis for the billiard motion of a 5-sided periodic billiard in $e$. We can begin this analysis by choosing an arbitrary length for the arrow representing the velocity vector ${\mathbf{v}}_{2}$ of $P_{2}$. This defines the two components ${\mathbf{v}}_{t_{2}}$ and ${\mathbf{v}}_{n_{2}}$, where the latter determines the angular velocity $\omega_{2}$ of the side $P_{2}P_{3}$ and furtheron the absolute velocity ${\mathbf{v}}_{3}$ of $P_{3}$. This can be continued. From now on, we denote the norms $\|{\mathbf{v}}_{t_{1}}\|=\|{\mathbf{v}}_{t_{2}}\|$ and $\|{\mathbf{v}}_{n_{1}}\|=\|{\mathbf{v}}_{n_{2}}\|$ of the respective components of the velocity vector ${\mathbf{v}}_{i}$ of $P_{i}$ with $v_{t|i}$ and $v_{n|i}$.

In terms of the exterior angles $\theta_{1},\dots,\theta_{N}$ of the billiard, we obtain from (3.1)

Let $R_{i}$ denote the pole of the line $[P_{i},P_{i+1}]$ with respect to (w.r.t. in brief) $e\,$. Since the poles of a line $\ell$ w.r.t. confocal conics lie on a line orthogonal to $\ell$, the side $P_{1}P_{2}$ is orthogonal to $[Q_{1},R_{1}]$ (Figure 3), which means

From (3.4) and (3.3) follows

This shows that, by virtue of (2.7), the products

for $i=1,2,\dots,N$ are equal along the billiard. We denote this quantity with $C$. Instead of a free choice of $v_{2}\,$, it means no restriction of generality to set $C=k_{e}\,$. Then we obtain for the point $P_{i}=(a_{e}\cos t_{i},\,b_{e}\sin t_{i})$ of the ellipse $e$, by virtue of (2.7),

Our specification of the quantity $C$ assigns to the vertex $P_{i}\in e$ with parameter $t_{i}$ a non-vanishing velocity vector ${\mathbf{v}}_{i}=\|{\mathbf{t}}_{c}(t_{i})\|\,{\mathbf{t}}_{e}(t_{i})\,$. This assignment can immediately be extended to all points of $e\,$. There exists a parameter $u$ on $e$ such that the differentiation by $u$ gives the said velocity vector. Let a dot indicate this differentiation. Then

We can even extend this to all confocal ellipses of $c\,$. The assignment of a velocity vector

to each point $P=(a_{e}\cos t,\,b_{e}\sin t)$ with $a_{e}^{2}-b_{e}^{2}=a_{c}^{2}-b_{c}^{2}$ defines an ‘instant motion’ of the plane, where

We prove below, that this ‘instant motion’ is compatible with the billiard and the associated Poncelet grid. This means in particular, that the velocity (3.6) is also valid for all points $S_{i}^{(j)}\in e^{(j)}$.

Figure 4 shows a portion of the Poncelet grid and the velocity vectors of a couple of points, each represented by a scaled arrow. As indicated, for all points on the tangent $t_{Q}$ to $c$ at any point $Q\in c$, the normal components $\|{\mathbf{v}}_{n}\|$ of the respective velocity vectors ${\mathbf{v}}$ are proportional to the distance to $Q$. On the other hand, all points on any confocal hyperbola share the tangential component $\|{\mathbf{v}}_{t}\|$.

The vector field defines a canonical parameter $u$ for the one-parameter Lie group $\Gamma$, i.e., for transformations $\gamma(u)\in\Gamma$ holds $\gamma(u_{2})\circ\gamma(u_{1})=\gamma(u_{1}+u_{2})$. At the same time, $u$ provides canonical coordinates²²2 Of course, we still obtain canonical coordinates on the ellipses, when all velocity vectors are multiplied with any constant $\lambda\in\mathbb{R}\setminus{0}$. on each confocal ellipse with the property

In order to express the action of the transformation $\gamma(u)\in\Gamma$ on an initial point $(a_{e}\cos t,\,b_{e}\sin t)$, we integrate the differential equation (4.2)

with $m:=d/a_{c}<1$ as numerical eccentricity of the caustic $c$. The substitution

results in

The initial condition $\varphi=0$ for $u=0$ yields the unique solution

with $F(\varphi,m)$ as the elliptic integral of the first kind with the modulus $m$. The equation (4.7) shows the canonical coordinate $u$ in terms of $\varphi$ with the quarter period

For the sake of simplicity, we introduce

as a new canonical coordinate.

The inverse function of $\tilde{u}=F(\varphi,m)$, namely the Jacobian amplitude $\varphi=\mathrm{am}\mskip 1.0mu(\tilde{u})$ leads to the Jacobian elliptic functions, the elliptic sine

with $\mskip 2.0mu\mathrm{sn}(-\tilde{u})=-\mskip 2.0mu\mathrm{sn}\mskip 2.0mu\tilde{u}\,$, the elliptic cosine

with $\mskip 2.0mu\mathrm{cn}(-\tilde{u})=\mskip 2.0mu\mathrm{cn}\mskip 2.0mu\tilde{u}\,$, and the delta amplitude

with $\mskip 2.0mu\mathrm{dn}(-\tilde{u})=\mskip 2.0mu\mathrm{dn}\mskip 2.0mu\tilde{u}\,$ as the third elliptic base function Hoppe . Moreover, for $k\in\mathbb{Z}$ holds

This gives rise to the canonical parametrization of the ellipse $e$ with semiaxes $(a_{e},b_{e})$ as

Corollary 3 reveals that $\varDelta\tilde{u}$ serves as a canonical coordinate for confocal ellipses in the exterior of $c$. If $\varDelta\tilde{u}$ corresponds by (4.10) to the ellipse $e$ with semiaxes $(a_{e},b_{e})$, then $2\mskip 1.0mu\varDelta\tilde{u}$ is the shift for the billiards in $e$ with caustic $c\,$. If these billiards have the turning number 1, then increasing the shift by $\varDelta\tilde{u}$ means to increase the turning number of the billiard in a confocal ellipse by 1, while the caustic $c$ remains fixed (Figure 6). The billiard $P_{1}P_{2}\dots$ and its conjugate $P_{1}^{\prime}P_{2}^{\prime}\dots$ in $e$ (cf. (Sta_I, , Sect. 3.2)) intersect each other along the ellipse with the canonical coordinate $\varDelta\tilde{u}/2\,$. Note that the ellipses $e^{(j)}$ and $e^{(N-2-j)}$ coincide while the corresponding $\varDelta\tilde{u}$’s differ in their signs. For even $N$, the points $S_{i}^{(N/2-1)}$ are at infinity, and the line at infinity as a limit of a confocal ellipse corresponds to $\varDelta\tilde{u}=K$.

The following formulas express the elliptic coordinates $(k_{e},k_{h})$ of the point $P=(-a_{e}\mskip 2.0mu\mathrm{sn}\mskip 2.0mu\tilde{u},\,b_{e}\mskip 2.0mu\mathrm{cn}\mskip 2.0mu\tilde{u})$ of $e$ in terms of the canonical coordinate $\tilde{u}$ on $e$ and the shift $\varDelta\tilde{u}$ corresponding to $e\,$.

This follows from

and

Note that $k_{h}=k_{h}(\tilde{u})$ is a solution of (4.9).

In Izmestiev , an unordered pair of coordinates $(r,s)$ is proposed for each point $P$ in the exterior of $c$, namely with $r$ and $s$ as canonical coordinates of the tangency points for the tangent lines from $P$ to $c$ (see also (MonGeom, , p. 358)). This means for $P=(-a_{e}\mskip 2.0mu\mathrm{sn}\mskip 2.0mu\tilde{u},\,b_{e}\mskip 2.0mu\mathrm{cn}\mskip 2.0mu\tilde{u})$ that

where $\varDelta\tilde{u}$ corresponds to $e$ according to (4.10).

If we keep the sum $\tilde{u}+\varDelta\tilde{u}$ or the difference $\tilde{u}-\varDelta\tilde{u}$ constant, the corresponding point $P$ runs along a tangent of the caustic $c$ (compare with (Bobenko, , Prop. 8.3)). For the sake of simplicity, we replace in the summary below $\varDelta\tilde{u}$ by $\tilde{v}\,$.