Billiards in ellipses revisited
aa r X i v : . [ m a t h . M G ] J a n BILLIARDS IN ELLIPSES REVISITED
ARSENIY AKOPYAN, RICHARD SCHWARTZ, AND SERGE TABACHNIKOV
Abstract.
We prove some recent experimental observations of D. Reznik concerning periodicbilliard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiardpolygon remains constant in the one-parameter family of such polygons (that exist due to thePoncelet porism). In our proofs, we use geometric and complex analytic methods. Introduction
The billiard in an ellipse is a thoroughly studied completely integrable dynamical system,see, e.g., [16]. In particular, a billiard trajectory that is tangent to a confocal ellipse will remaintangent to it after each reflection. That is, the confocal ellipses are the caustics of this billiard.One of the properties of this system, a consequence of its complete integrability, is that aperiodic billiard trajectory tangent to a confocal ellipse includes into a 1-parameter family ofperiodic trajectories tangent to the same confocal ellipse and having the same period and thesame rotation number. This is the assertion of the Poncelet porism for confocal ellipses. ThePoncelet porism concerns the same kind of 1-parameter family of polygons that are simulta-neously inscribed and circumscribed in the same pair of conics, but in general the conics neednot be confocal.A classic result about a continuous 1-parameter family of billiard paths is that their perime-ters remain constant. (See [16], and also Lemma 2.3 below.) Recently Dan Reznik conducteda large series of computer experiments with periodic orbits in elliptic billiards and discoverednumerous new properties of these polygons that are similar in spirit to the constant-perimeterresult. See [9, 10, 11, 12, 6, 13]. In this paper we give proofs which verify some of these ob-servations. Essentially, we prove 3 main results. In the body of the paper, we will also prove anumber of variants and generalizations.We would also like to mention that another proof of the first two results is presented in [4];it is based on a non-standard generating functions for convex billiard discovered by Misha Bialy.
First Result:
Let α , ..., α n be the angles associated to a periodic billiard path. Figure 1shows the case n = 5. α α α α α p p p p p Figure 1.
Angles of a billiard trajectory.
Theorem 1.
The sum n X i =1 cos α i remains constant as P varies a 1-parameter family of periodic billiard paths on an ellipse. This result corresponds exactly to one of Reznik’s observations. We will give two proofs. Thefirst proof is based on the invariance of the perimeter, mentioned above, and on the invarianceof a quantity called the
Joachimsthal integral . The second proof is based on Liouville’s Theoremfrom complex analysis.
Second Result:
Let β , ..., β n be the angles of the polygon formed by tangents to the el-lipse at the vertices of a periodic billiard trajectory, as shown in Figure 2 for the case n = 5. β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β Figure 2.
The angles involved in Theorem 2.
Theorem 2.
The the product n Y i =1 cos β i remains constant as P varies in a 1-parameter family of periodic billiard paths on an ellipse. This result also corresponds exactly to one of Reznik’s observations. Our proof is the samekind of complex analysis argument that we use for Theorem 1. We don’t know a proof alongthe lines of our first proof of Theorem 1 but see [4].
Third Result:
Figure 3 shows how we construct a polygon Q starting from a Poncelet polygon P . QP Figure 3.
To Theorem 3.
ILLIARDS IN ELLIPSES REVISITED 3
Theorem 3.
Let P be a Poncelet n -gon with odd n , inscribed into an ellipse and circumscribedabout a concentric ellipse. Let Q be the polygon formed by the tangent lines to the outer ellipseat the vertices of P . The ratio of the areas of the two polygons remains constant within thePoncelet family containing P . This result is a mild generalization of an observation of Reznik. We will explain the exactrelationship after giving the proof. The proof itself is short. It follows readily from a resultcalled the Poncelet Grid Theorem [14, 8].2.
Proof of the First Result
Consider an ellipse in R given by the equation(1) x a + x a = 1 , or, in the matrix form, h Ax, x i = 1, where A = diag(1 /a , /a ).Recall the notion of polar duality: given a smooth convex closed planar curve γ , to a vector x ∈ γ there corresponds a unique covector x ∗ satisfying the conditionsKer x ∗ = T x γ and ( x, x ∗ ) = 1;here the parentheses denote the pairing of vectors and covectors. The points x ∗ comprise thedual curve γ ∗ in the dual plane.Identifying vectors and covectors via Euclidean metric, we see that, for the ellipse (1), x ∗ = Ax . The polar dual curve is the ellipse given by the matrix A − .The phase space of the billiard is 2-dimensional; it consists of the inward unit vectors u withthe foot point x on the ellipse. The billiard transformation is shown in Figure 4. y xv uu Figure 4.
The billiard transformation: ( x, u ) ( y, v ).Here is the key fact needed for our first result. The quantity J in this result is known as theJoachimsthal integral. Proposition 2.1.
The function J ( x, u ) := − ( u, x ∗ ) is invariant under the billiard transforma-tion: J ( x, u ) = J ( y, v ) .Proof. We claim that ( u, x ∗ ) = − ( u, y ∗ ) = ( v, y ∗ ). For the first equality, it is enough to showthat ( y − x, x ∗ ) = − ( y − x, y ∗ ), because y − x is parallel to u . Since A is self-adjoint operator onehas ( y, x ∗ ) = ( y, Ax ) = ( Ay, x ) = ( x, y ∗ ). Subtracting from it the equality ( x, x ∗ ) = ( y, y ∗ ) = 1we obtain the required one. Since u is collinear with y − x , one has h A ( x + y ) , u i = 0, asneeded. The second equality holds for billiards of every shape. By the law of billiard reflection,the vector u + v is tangent to the ellipse at point y , hence ( y ∗ , u + v ) = 0. This completes theproof. (cid:3) To each billiard trajectory P we can associate the quantity J ( P ) = J ( x, u ), where ( x, u ) isthe first point-vector pair associated to P . Corollary 2.2. J is constant as P varies in a -parameter family of periodic billiard paths inan ellipse. ARSENIY AKOPYAN, RICHARD SCHWARTZ, AND SERGE TABACHNIKOV
Proof.
Let E be a given ellipse. For the generic choice of caustic E ′ – i.e., a choice of ellipseconfocal with E – the billiard map is aperiodic and has dense orbits. By the preceding result, J is constant on a dense set of billiard paths tangent to E ′ . By continuity, J is constant onthe space of all billiard paths tangent to almost any caustic E ′ . By continuity, the same resultholds for every caustic. Finally, note that the polygons in a 1-parameter family of periodicbilliard paths are tangent to the same caustic. (cid:3) Here is the perimeter-invariance result mentioned in the introduction.
Lemma 2.3.
The perimiter of P is constant as P varies in a -parameter family of periodicbilliard paths in an ellipse.Proof. A n -periodic billiard path extremizes the perimeter among the n -gons inscribed in thebilliard table. The 1-parameter family of periodic polygons is a curve in the space of inscribedpolygons consisting of critical points of the perimeter function. A function has a constant valueon a curve of its critical points. See [16] for more details. (cid:3) The following identity, also noticed experimentally by Reznik, immediately implies Theorem1. The reason is that both the quantities J and L are constant for 1-parameter families ofperiodic billiard paths in ellipses. Theorem 4.
For an n -periodic billiard trajectory P in an ellipse, one has n X i =1 cos α i = J L − n, where L and J are the perimeter and the value of the Joachimsthal integral of P .Proof. We compute L = X i h p i +1 − p i , u i i = X i h p i , u i − i − h p i , u i i = X i h p i , u i − − u i i (this is a discrete version of integration by parts). By the law of billiard reflection, u i − − u i = 2 sin (cid:18) π − α i (cid:19) p ∗ i | p ∗ i | = 2 cos (cid:16) α i (cid:17) p ∗ i | p ∗ i | . Also one has J = − ( u i , p ∗ i ) = −| p ∗ i | cos (cid:16) π − α i (cid:17) = | p ∗ i | cos (cid:16) α i (cid:17) . Since ( p i , p ∗ i ) = 1, it follows that J L = X i (cid:16) α i (cid:17) = X i (1 + cos α i ) = n + X i cos α i , as needed. (cid:3) Remark . See [B] for a symplectic interpretation of these ideas.3.
Dual Minkowski billiards
Before we get to our complex analysis-based proofs, it is useful to discuss billiards in Minkowskimetrics. This is used to construct dual billiards, which offer new generalizations and help prov-ing the main results.A
Minkowski metric is a norm on a vector space. Let U be an n -dimensional vector spaceand V = U ∗ be its dual. Assume that U and V are equipped with Minkowski metrics, notnecessarily centrally-symmetric and dual to each other. Let M ⊂ U be the unit co-ball of themetric in V and N ⊂ V be he unit co-ball of the metric in U . One has two billiards: M innormed space U , and N in normed space V . ILLIARDS IN ELLIPSES REVISITED 5
Lemma 3.1.
The two billiards systems are isomorphic.Proof.
This is proved in [7], section 7. Here we sketch the ideas. Abusing notation, denote by D the polar duality that identifies the unit spheres and co-spheres of the metrics. The phasespace of the billiard in M consists of the pairs ( q, u ) where q ∈ ∂M and u is a (Minkowski)unit inward vector at q . Assign to it the pair ( q, p ) ∈ ∂M × ∂N where p = D ( u ). Likewise, thephase space of the billiard in N consists of the pairs ( p, v ) where p ∈ ∂N and v is a (Minkowski)unit inward vector at p . Assign to it the pair ( p, q ) ∈ ∂N × ∂M where q = D ( v ).The assertion is that a sequence ( . . . , q i , p i , q i +1 , . . . ) corresponds to a billiard orbit in M ifand only if the sequence ( . . . , p i , q i +1 , p i +1 , . . . ) corresponds to a billiard orbit in N . If M is anellipse with half-axes a and a in the Euclidean plane U , then N is the unit disc in the plane V whose unit ball is polar to M , i.e., is an ellipse with half-axes 1 /a and 1 /a . (cid:3) Remark . The affine map ( x , x ) → ( a x , a x ) isometrically maps the Minkowski plane V to the Euclidean plane, taking N back to the ellipse with half-axes a and a , that is, to M .One obtains a symmetry of the billiard in an ellipse, called the skew hodograph transformationand discovered by A. Veselov [17, 18]. Remark . It was noticed in [3] that minimal action of a billiard in Minkowski metric is relatedwith Ekeland–Hofer–Zehnder capacity, which made it possible to apply symplectic geometryto convex geometry problems. In particular it was shown that the famous Mahler conjecturefollowed from the Viterbo’s conjecture [2, 1].4.
Another Proof of the First Result
As before, let u , . . . , u n be the unit vectors along the sides of P . These vectors are pointsof the unit circle ω centered in the origin O , and they form an n -periodic trajectory of theMinkowski billiard in ω with the Minkowski metric defined by the ellipse dual to the originalone. Therefore u , . . . , u n are the vertices of a Poncelet n -gon, inscribed in ω and circumscribedabout some ellipse ξ centered at O . π − α u u u u u u u O Figure 5.
To Theorems 5 and 6.The angles of P satisfy α i = π − ∠ u i − Ou i , thus the next result is equivalent to Theorem 1. Theorem 5.
Let u , . . . , u n be a Poncelet polygon inscribed in a circle ω with center O andcircumscribed about an ellipse ξ with the same center. Then n X i =1 cos ∠ u i Ou i +1 is constant in the 1-parameter family of Poncelet n -gons.Proof. Following the approach of [15], we will complexify the situation, that is, extend thesetting to Poncelet polygons on the conics given by the same equations in the complex plane.
ARSENIY AKOPYAN, RICHARD SCHWARTZ, AND SERGE TABACHNIKOV
We show that the function in question is bounded, and then the Liouville theorem implies thatthis function is constant. To extend our function to complex plane, we need to represent thefunction cos ∠ u i Ou i +1 in a more convenient way. Since | Ou i | = 1 for all i , we have:cos ∠ u i Ou i +1 = h u i , u i +1 i . In other words, for the proof of the first statement, we need to show that the sum h u i , u i +1 i is constant. Let us emphasize that here we consider the usual dot product, not the Hermitianone.Consider the standard rational parametrization of the circle ω : p ( t ) = (cid:18) tt + 1 , t − t + 1 (cid:19) . The points at infinity correspond to the value of the parameter t = ± I . Here I = √−
1. Itis clear that the only possibility for the Poncelet polygon to have an infinite P h u i , u i +1 i is tohave one of its vertex at infinity.Let us show that when a vertex goes to infinity, the inner products in which this vertexparticipates cancel each other.Consider a point at infinity, say, p ( I ). We claim that its two neighboring vertices of thePoncelet polygon, denoted by a ( I ) and b ( I ), are opposite points of ω . Indeed, the lines p ( I ) a ( I )and p ( I ) b ( I ) are tangent to ξ and are parallel, therefore the tangency points of these lines with ξ are symmetric with respect to O , and hence their intersection points with ω are also symmetric(the point p ( I ) is invariant under the reflection in O , given by t
7→ − /t ). Thus, for any finitepoint q on ω , we have h q, a ( I ) i + h q, b ( I ) i = 0 . Now, consider point p ( t + I ) with t tending to zero and its neighboring vertices a ( I + t ) and b ( I + t ) of the Poncelet polygon. Notice that p ( t + I ) tends to infinity as O (1 /t ), while a ( t + I ), b ( t + I ) tend to their limit a ( I ), b ( I ) linearly. Furthermore, due to the symmetry, as t goes tozero, the linear in t terms are vectors with the same absolute value and the opposite directions: a ( t + I ) = a ( I ) + ~k · t + O ( t ) , b ( t + I ) = − a ( I ) − ~k · t + O ( t ) . Now we can bound above the sum for small t : h p ( t + I ) , a ( t + I ) i + h p ( t + I ) , b ( t + I ) i = h p ( t + I ) , a ( t + I ) + b ( t + I ) i = h O (1 /t ) , O ( t ) i = O ( t ) . That is, the sum tends to zero as t goes to 0, and therefore it is bounded. (cid:3) p ( I ) b ( I ) a ( I ) O~k
Figure 6.
The behavior of the polygon at infinity.
ILLIARDS IN ELLIPSES REVISITED 7 Proof of the Second Result
Referring to the construction in the previous section, the angles β i in Figure 2 are given bythe formula β i = π − ∠ u i − Ou i +1 ∠ u i − u i u i +1 . Theorem 6.
Let u , . . . , u n be a Poncelet polygon inscribed in a circle ω with center O andcircumscribed about an ellipse ξ with the same center. Then n Y i =1 cos ∠ u i − u i u i +1 is constant in the 1-parameter family of Poncelet n -gons.Proof. Since the product of cosines changes continuously, if the absolute value of this productis constant in the family, then its sign is fixed as well. Therefore, instead of the product ofcosines, we may consider the product of their squares:cos ∠ u i − u i u i +1 = cos ∠ u i − Ou i +1 ∠ u i − Ou i +1 h u i − , u i +1 i . Thus we need to prove that the product Q (1 + h u i − , u i +1 i ) is bounded. Again the onlypossibility for this product to be infinite is when one of the vertices goes to infinity.Similarly to the previous proof, and using the same parameterization of the circle, we assumethat p ( t + I ) is the vertex that goes to infinity, a ( t + I ) and b ( t + I ) are its neighboring vertices,and a ′ ( t + I ) and b ′ ( t + I ) are its second removed neighbors (which are also centrally symmetric,but it plays no role here).Let us show that the corresponding product is bounded:(1 + h p ( t + I ) , a ′ ( t + I ) i )(1 + h p ( t + I ) , b ′ ( t + I ) i )(1 + h a ( t + I ) , b ( t + I ) i ) = O ( t − ) · O ( t − ) · (1 + h a ( I ) + ~k · t + O ( t ) , − a ( I ) − ~k · t + O ( t ) i ) = O ( t − ) · (1 + h a ( I ) , − a ( I ) i − h a ( I ) , ~k · t i + O ( t )) = O ( t − ) · ( − h a ( I ) , ~k · t i + O ( t )) = O ( t − ) · ( − h a ( I ) , ~k · t i ) + O (1) . It is left to notice that ~k is tangent to ω at a ( I ), therefore h a ( I ) , ~k · t i = 0 . Thus the productis bounded.If n is odd, then only one vertex can go to infinity: if p ( ± I ) is a vertex of the Ponceletpolygon, then p ( ∓ I ) is not its vertex.For even n , a vertex u i goes to infinity simultaneously with u i + n/ . A simple combinatorialanalysis of the configurations shows that the only case when the above considered factorscoincide is when n = 4. In that case the polygon is always a rectangle, and the statement isobvious. (cid:3) Variants and Generalizations
Here we list some variants and generalizations of the results we have proved so far.
Multi-dimensional version:
The billiard inside an ellipsoid in R n +1 is also completely inte-grable: the phase space is 2 n -dimensional, the trajectories are confined to n -dimensional tori,and the motion on these tori is quasi-periodic, see [5]. In particular, if a point is periodic,then all points of the torus are periodic with the same period, and the respective polygonshave the same perimeters. The billiard map still has the Joachimsthal integral, constant on the ARSENIY AKOPYAN, RICHARD SCHWARTZ, AND SERGE TABACHNIKOV orbits confined to an invariant torus, and the above arguments go through, proving a multi-dimensional version of Theorem 1.
Sizes of the Angles:
Concerning Theorem 2, it was pointed out to us by M. Bialy thatthis theorem implies that the sign of the quantity β i − π/ Q lie on a ellipse polar dualto the inner ellipse (the caustic) with respect to the outer one. If we fix this outer ellipse andvary the caustic, then these polar dual ellipses, into which the polygons Q are inscribed, forma pencil of conics. This pencil contains the orthoptic circle , the locus of points from which anellipse is seen under the right angle. This orthoptic circle separates the two cases: when allangles β i are obtuse and when they are all acute. Additional Invariants:
The following theorem is a generalization of Theorem 1, which isits k = 1 case. We explain how to deduce this theorem from Theorem 1. Theorem 7.
For each k = 1 , ..., n , the quantity C k = n X i =1 cos( α i + α i +1 + . . . + α i + k − ) remains constant as P varies a 1-parameter family of periodic billiard paths on an ellipse. Figure 7.
Four 13-periodic billiard orbits in confocal ellipses tangent to thesame caustic.
Proof.
To line up our proof with a previously published result that we use, we state things interms of Poncelet polygons and the Poncelet porism. Label the lines containing the sides of aPoncelet n -gon cyclically. Fix k and consider the intersections of i th and ( i + k )th lines, where i = 1 , , . . . , n . This set of points lies on a confocal ellipse and comprises several polygons, eacha periodic billiard trajectory (the number of polygons equals gcd ( n, k )). This statement is apart of the Poncelet Grid theorem [14, 8]. Figure 7 illustrates the case of n = 13. The anglesof these new polygons are expressed via the angles of the original one. Namely, the new anglesare equal to α i + α i +1 + . . . + α i + k − − ( k − π. ILLIARDS IN ELLIPSES REVISITED 9
Therefore Theorem 1, applied to the new polygons, implies that C k remains constant in theirPoncelet family. (cid:3) Theorem 7 has a reformulation, generalizing Theorem 5, which is the k = 1 case. Theorem 8.
Let u , . . . , u n be a Poncelet polygon inscribed in a circle ω with center O andcircumscribed about an ellipse ξ with the same center. Then, for each k , n X i =1 cos ∠ u i Ou i + k are constant in the 1-parameter family of Poncelet n -gons. Sums of Squared Lengths:
Here is a corollary of Theorem 8. Note that | u i + k − u i | = 2 − ∠ u i Ou i + k . This implies
Corollary 6.1.
Consider a Poncelet polygon inscribed in a circle and circumscribed about aconcentric ellipse. Then the sum of the squared lengths of its k -diagonals remains constant inthe 1-parameter family of Poncelet polygons. Product of sines of half-angles:
Notice that the angles of polygons formed by the tangentscan be represented through angles of the billiard trajectory: β i = α i + α i +1 . Therefore, the statement of Theorem 2 can be formulated in terms of the angles of the billiardtrajectory and then can be extended for the angles of lines in the corresponded Poncelet gridas in Theorem 7. Doing this for odd n and the polygon formed by sides of our trajectory withstep k = ( n − /
2, we find that the corresponding angle of the tangential polygon β ′ i equals β ′ i = P j = i α j π ( n − − α i . Since n is odd we getcos β ′ i = cos π ( n − − α i ± cos π − α i ± sin α i . This gives us the following result, also noticed by D. Reznik:
Corollary 6.2.
For odd n , the quantity n Y i =1 sin α i remains constant as P varies in a 1-parameter family of periodic billiard paths on an ellipse. Hyperbolic Interpretation:
One can interpret Theorems 7 and 6 in terms of hyperbolicgeometry. Consider ω as the absolute of the Klein model of the hyperbolic plane, and ξ asan ellipse in it. The hyperbolic and the Euclidean measures of the angles u i Ou k coincide. Weobtain the following corollary. Corollary 6.3.
Let u , . . . , u n be an ideal n -gon in the hyperbolic plane whose sides are tangentto an ellipse with center O . Then, for each k , n X i =1 cos ∠ u i Ou i + k and n Y i =1 cos (cid:18) ∠ u i − Ou i +1 (cid:19) are constant in the 1-parameter family of ideal Poncelet n -gons. A Dual Version:
Under the duality transform with respect to the circle ω , the inner ellipse ξ goes to a concentric ellipse ξ ∗ , and we again obtain a Poncelet polygon, this time inscribedin ξ ∗ and circumscribed about ω . The angles between the unit vectors u i become the anglesbetween the sides of the Poncelet polygon, and we obtain the following corollary of Theorems5 and 6. v v v v v v v O α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α Figure 8.
To Corollary 6.4.
Corollary 6.4 (Ellipse-Circle version) . Let v , . . . , v n be a Poncelet polygon circumscribed abouta circle and inscribed in a concentric ellipse. Denote its angles by α i . Then n X i =1 cos α i and n Y i =1 cos ∠ v i Ov i +1 are constant in the 1-parameter family of Poncelet n -gons. Proof of the Third Result
Theorem 3 is closely related to the experimental observation of D. Reznik illustrated inFigure 9. Consider an n -periodic billiard trajectory in an ellipse with odd n (the pentagon Q in Figure 9). The tangent lines to the ellipse at these n points form a new n -gon (the pentagon P in Figure 9). The observation is that the ratio of the areas of these two polygons remainsconstant as the n -periodic billiard trajectory varies in its 1-parameter family. p p p p p q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 9.
The ratio of the areas of the polygons P and Q remains constant inthe 1-parameter family of Poncelet polygons.There exists an affine transformation that makes the two ellipses confocal: first turn one ofthe ellipses into a circle by an affine transformation, and then stretch along the axes of the otherellipse to make the two confocal. Since an affine transformation does not affect the ratio of theareas, it remains to prove the claim for a Poncelet polygon on confocal ellipses, see Figure 9.This is the case observed by Reznik. ILLIARDS IN ELLIPSES REVISITED 11
Now we deal with the confocal case. The Poncelet grid theorem [8] implies that the affinetransformation that takes the inner ellipse to the outer one by scaling its main axes and reflectingin the origin takes the inner polygon to the outer one as well (it maps each vertex to the“opposite one”, see the labelling in Figure 9). Since the ratio of the areas is invariant under anaffine transformation, the result follows.
Acknowledgements . This paper would not be written if not for Dan Reznik’s curiosity andpersistence; we are very grateful to him. We also thank R. Garcia and J. Koiller for interestingdiscussions. It is a pleasure to thank the Mathematical Institute of the University of Heidelbergfor its stimulating atmosphere. ST thanks M. Bialy for interesting discussions and the Tel AvivUniversity for its invariable hospitality.AA was supported by European Research Council (ERC) under the European Union’s Hori-zon 2020 research and innovation programme (grant agreement No 78818 Alpha). RS is sup-ported by NSF Grant DMS-1807320. ST was supported by NSF grant DMS-1510055 andSFB/TRR 191.
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E-mail address : [email protected] Richard Schwartz, Department of Mathematics, Brown University, Providence, RI 02912,USA
E-mail address : [email protected] Serge Tabachnikov, Department of Mathematics, Penn State University, University Park,PA 16802, USA
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