PONCELET ’ S PORISM IN TRANSFORMATION GROUP FRAMEWORK

In this paper we introduce a transformation group connected with Poncelet’s porism. Several open questions following from our considerations end the paper. The aim of this paper is to give a new approach to find an algebraic Fuss-type formula for all natural n>2. Developed method may be applied to investigations of Poncelet’s porism.


INTRODUCTION
With each pair of circles we associate three positive numbers, namely their radii and the distance between their centers.If a pair of circles forms an annulus then their radii , and the distance between their centers satisfy the inequality .We consider a class of all annuli with a property of the Poncelet porism, see [3].This class is divided onto classes of annuli with a property of the Poncelet porism for a fixed natural number .In a case Poncelet's porism means that if we can circuminscribe in an annulus (inscribe in the outer circle and describe on the inner circle) one triangle then by each point of the annulus passes a circuminscribed triangle.For we have the following equality , see [2].Similarly, for we consider circuminscribed quadrangles and so on.For the following equality holds, see [2].A compact algebraic formula connecting , and for arbitrary does not exist.There exists an integral formula connecting , and for arbitrary given for example by Jacobi, see [6,2].Note, that simple formula involving elliptic integrals is given in [5].

A TRANSFORMATION GROUP
Let .We consider the multiplicative group and the set We define a function by the formula (2.2) Note that the function is well defined.It is easy to verify that for each and we have

PONCELET'S PORISM FOR n-GONS
It is proved in [5] that the annulus have a property of the Poncelet porism for -gons if and only if , (5.1) where . (5.2) We define a set as follows: if it satisfies the condition (5.1).For we have (5.3) Differentiating (5.3) with respect to and using (5.2) we obtain (5.4) where is some function.

FINAL REMARKS AND CONCLUSIONS
In this section we formulate a few open questions: 1. Taking into account the previous sections we conjecture that .
2. How to extend the group acting on in such a way that the left action is transitive?The solution of the 4 th problem gives Fuss formula of the form a = rh(t), where .
have Theorem 2.1.The group acts on from the left.PONCELET'S PORISM FOR TRIANGLES Let us fix two circles , satisfying the condition , where denotes the distance between their centers.If the annulus has the property of the Poncelet porism for then the Euler triangle formula is satisfied, namely (Let an element belongs to .Then it satisfies the equality (3.1).Hence we have (3.6)It is easy to see from the expression (3.6) that for .It follows from (3The transformation is a surjection.Proof.For an arbitrary fixed and we may write or where ρ is given by (3.6).The above equations lead us to .It is easy to verify that and .Let an element belongs to .Then it satisfies the equality given in (4.1).Hence, we have (4.5)Simple but long calculations show that .

3 .
Is it possible to repeat all considerations of the Sections 3 and 4 to all known algebraic Fuss formulas for Poncelet porism, see[10].

4 .
If the answer to the first question is positive then a is a function of two variables r, t of the form a = rh(t), where h is a monotonic function.Find a differential equation for the function h.