The Poncelet annuli

Cieślak, Waldemar

doi:10.1007/s13366-013-0142-x

Cited by 6 publications

(14 citation statements)

References 4 publications

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“…This is exactly the formula (5.6) from [5], and we note that it implies Poncelet's porism property for n-gons. Introducing…”

Section: From (12) and (14) It Follows Immediately That (15) B (2π) =supporting

confidence: 71%

Rotation indices related to Poncelet’s closure theorem

Cieślak¹,

Martini²,

Mozgawa³

2014

Annales UMCS, Mathematica

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Abstract. 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. , 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.,

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“…This is exactly the formula (5.6) from [5], and we note that it implies Poncelet's porism property for n-gons. Introducing…”

Section: From (12) and (14) It Follows Immediately That (15) B (2π) =supporting

confidence: 71%

Rotation indices related to Poncelet’s closure theorem

Cieślak¹,

Martini²,

Mozgawa³

2014

Annales UMCS, Mathematica

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“…Next, from w(t) we draw the tangent line to C r and it is tangent to C r at a point denoted by z(ϕ(t)). It was shown in [3] that the function ϕ satisfies the following differential…”

Section: Geometric Interpretation Of the Function Bmentioning

confidence: 99%

“…We denote by L(t) the perimeter of a circuminscribed quadrangle in C r C a,R passing through a point z(t). The perimeter L is given by the formula 1 2 L = λ + λ • ϕ + λ • ϕ [2] + λ • ϕ [3] . Applying (3.7), L = 8λ − 4a(3 sin ϕ + 2 sin ϕ [2] + sin ϕ [3] ).…”

Section: Critical Points Of the Perimeter Lmentioning

confidence: 99%

“…He expressed the area of the triangle in terms of the lengths of parts of Poncelet transversals and then he determined the extremal triangle. In this paper we solve the extremal problem connected with perimeters of quadrangles determined by the Poncelet closure theorem, using the differential equation given in [3]. The essential difficulty in the approach used by Radić and our approach in this paper is the necessity to use n-porism formulas in explicit form (see [2,8]).…”

Section: Introductionmentioning

confidence: 99%

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Extremal Perimeters of Quadrangles in The poncelet Porism

Cieślak¹,

Mozgawa²

2015

Bull. Aust. Math. Soc.

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“…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 .…”

Section: Introductionmentioning

confidence: 99%