Rotation indices related to Poncelet ’ s closure theorem

Let CRCr denote an annulus formed by two non-concentric circles CR, Cr in the Euclidean plane. We prove that if Poncelet’s closure theorem holds for k-gons circuminscribed to CRCr, then there exist circles inside this annulus which satisfy Poncelet’s closure theorem together with Cr, with ngons for any n > k.


Introduction.
Poncelet's closure theorem, going back to the 19th century, has various interesting forms and applications; cf.[2], [7], [4], [9], and the excellent survey [3] as well as [4].The rich history of this theorem is presented in [1,Ch. 16], [8, § 2.4], and [7], and our paper refers to circular versions of it.Let C R , C r be two circles with radii R > r > 0 and C r lying inside C R .From any point on C R , draw a tangent to C r and extend it to C R again, using the obtained new intersection point with C R for starting with a new tangent to C r , etc.; the system of tangential segments obtained in this way inside C R is called a Poncelet transverse (or bar billiard).We say that the annulus C R C r has Poncelet's porism property if there is a starting point on C R for which a Poncelet traverse is a closed polygon.Poncelet's closure theorem (for circles) says that then the transverse will also close for any other starting point from C R .It is known that such closing polygons (with or without self-intersections) correspond to rational rotations; e.g., the rotation number or index 1  3 is related to a triangle "between" C R and C r , and the index 2  5 to a (self-intersecting) pentagram.In [6] it was proved that "close" to a pair of circles, which have Poncelet's porism property for index 1  3 , there exist unique pairs of circles having this property with respect to indices 1  4 and 1 6 , and it was conjectured there that this holds true for arbitrary indices.
In the present paper we show that this conjecture is true in the following sense: for a pair of circles having Poncelet's porism property for index 1  k , with k ≥ 3 as natural number, we prove that there exists a circle lying between the starting circles such that this circle together with the smaller given circle has Poncelet's porism property for any given index 1  n , where n is an arbitrary natural number with n > k.

Basic notions and tools.
Let us consider a circular annulus C r C a,R formed by two circles C r and C a,R .The circles C r and C a,R are given by the equations x 2 + y 2 = r 2 and (x − a) 2 + y 2 = R 2 , respectively, with Recall the following form of Poncelet's closure theorem which is suitable for our purpose; see [1].
If there exists a one circuminscribed (i.e., simultaneously inscribed in the outer circle and circumscribed about the inner circle) n-gon in a circular annulus, then any point of the outer circle is the vertex of some circuminscribed n-gon.
If Poncelet's closure theorem holds for n = 3, then Euler's condition (2) is satisfied.We will denote this condition by Pct (C r C a,R , 3).There is no elementary formula for the analogously defined condition respectively; see [3].
It is amazing that for particular natural numbers we have elementary conditions involving also radicals, while for an arbitrary natural number n ≥ 3 only the Jacobi formula (cf.formula (7) in [10]), using elliptic functions, is involved.
For further use we introduce a convenient parametrization of the annulus C r C a,R .Namely, we take the parametrization z (t) = re it for C r , and for C a,R we use ( 5) where The line which is tangent to the circle C r at a point z (t) intersects the circle C R at a point w (t) = z (t) + λ(t)ie it .Let us draw a second tangent line to C r , passing at w (t).It intersects C r at a point z (ϕ (t)), where ϕ (t) satisfies the condition (6) tan In [5] it is proved that where ( 8) It is routine to check that the solution of this differential equation with initial condition ϕ (0) = m is given by the formula ( 9) .

Theorem 2. Assume that Poncelet's closure theorem holds in an annulus
C r C a,R for k-gons, k ≥ 3. Then for any n > k there exists γ ∈ (0, 1) such that Poncelet's closure theorem holds in the annulus C r C γa,(1−γ)r+γR for n-gons.
Proof.Using the equality (20) from the proof of Theorem 1, we introduce the function First we have From now on we assume that the starting annulus C r C a,R has Poncelet's porism property for a natural number k ≥ 3, and we consider n > k.Then by (20) we have (29) k Using this condition, we get In order to evaluate F n (0), we first calculate the value F n (ε) for ε ∈ (0, 1).We have ds.
First we prove that (30) lim for some positive constant C. We calculate Next, we claim that (33) lim