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Flatto L. — Poncelet's Theorem
Flatto L. — Poncelet's Theorem

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Название: Poncelet's Theorem

Автор: Flatto L.


Poncelet's theorem is a famous result in algebraic geometry, dating to the early part of the nineteenth century. It concerns closed polygons inscribed in one conic and circumscribed about another. The theorem is of great depth in that it relates to a large and diverse body of mathematics. There are several proofs of the theorem, none of which is elementary. A particularly attractive feature of the theorem, which is easily understood but difficult to prove, is that it serves as a prism through which one can learn and appreciate a lot of beautiful mathematics. The author's original research in queuing theory and dynamical systems figures prominently in the book. This book stresses the modern approach to the subject and contains much material not previously available in book form. It also discusses the relation between Poncelet's theorem and some aspects of queueing theory and mathematical billiards. The proof of Poncelet's theorem presented in this book relates it to the theory of elliptic curves and exploits the fact that such curves are endowed with a group structure. The book also treats the real and degenerate cases of Poncelet's theorem. These cases are interesting in themselves, and their proofs require some other considerations. The real case is handled by employing notions from dynamical systems. The material in this book should be understandable to anyone who has taken the standard courses in undergraduate mathematics. To achieve this, the author has included in the book preliminary chapters dealing with projective geometry, Riemann surfaces, elliptic functions, and elliptic curves. The book also contains numerous figures illustrating various geometric concepts.

Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2008

Количество страниц: 240

Добавлена в каталог: 09.03.2011

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Предметный указатель
$P_{2}$-version of Poncelet's theorem      3
1/2-periods      94
Affine coordinates      16
Affine plane      16
Algebraic curve      25
Algebraic model      113
Analytic atlas      62
Automorphism      63
Bezout's theorem      43
Billiard ball map      191
Branch point      76
Caustic      166
Cayley's theorem      4
Chart      61
Complete sets of zeros and poles      84
Complex structure      62
Conic pencil      53
Covering map      77
Covering surface      76
Degenerate conic      32
Division point      115
Double queues      11
Elliptic curve      111
Elliptic function      83
Elliptic function sn(z)      94
Fiber      77
Fundamental parallelogram      84
Fundamental region      100
Genus      73
Good measure      157
Homogeneous coordinate ratio      23
Homogeneous coordinates      16
Homotopy      223
Intersection number      45
Invariant measure      154
Inversion problem      97
Isomorphism      63
Lattice      66
Lift of a curve      79
Line at infinity      17
Local coordinate      61
Locally conformal      64
Modular function      98
Modular group      99
Moebius transformation      23
Multiplicity type      52
Natural projection      67
Order of a polynomial      25
Outer billiard transformation      207
Parametric disk      62
Pencil of lines      18
Point at infinity      16
Poncelet correspondence      5
Poncelet grid      198
Poncelet map      5
Poncelet's theorem      2
Projection map      76
Projective line      22
Projective plane      16
Projectivity      19
Quadratic parametrization      42
Regular point      76
Riemann surface      62
Riemann — Hurwitz Theorem      77
Rotation number      158
Smooth conic      32
Smooth cover      76
Steiner theorem      6
String construction      193
Tangent line      26
Topological conjugacy      158
Torus      67
Translation      82
Two-dimensional manifold      61
Weierstrass P-function      87
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