Format:
Online-Ressource
,
v.: digital
Edition:
Online-Ausg. Springer eBook Collection. Mathematics and Statistics Electronic reproduction; Available via World Wide Web
ISBN:
9783034800150
Series Statement:
Frontiers in Mathematics
Content:
The goal of the book is to present, in a complete and comprehensive way, a few areas of mathematics interlacing around the Poncelet porism: dynamics of integrable billiards, algebraic geometry of hyperelliptic Jacobians, and classical projective geometry of pencils of quadrics. Most important results and ideas connected to Poncelet theorem are presented: classical as well as modern ones, together with historical overview with analysis of classical ideas, their natural generalizations, and natural generalizations. Special attention is payed to realization of the Griffiths and Harris programme about Poncelet-type problems and addition theorems. This programme, formulated three decades ago, is aimed to understanding of higher-dimensional analogues of Poncelet problems and realization of synthetic approach of higher genus addition theorems. This problem is realized in a sequence of papers of the authors, published in last 20 years. Some of those results represent the key contribution to the development of the theory, while this book gives the complete image.
Note:
Includes bibliographical references and index
,
Poncelet Porisms and Beyond; Contents; Chapter 1 Introduction to Poncelet Porisms; Acknowledgement; Chapter 2 Billiards - First Examples; 2.1 Introduction to billiards; 2.2 Triangular billiards; 2.3 Billiards within an ellipse; 2.4 Periodic orbits of billiards and Birkhoff's theorem; 2.5 Bicentric polygons; Triangles; Quadrilaterals; 2.6 Poncelet theorem; Poncelet theorem; Mechanical interpretation of the Poncelet theorem; Chapter 3 Hyperelliptic Curves and Their Jacobians; 3.1 Riemann surfaces; Definition of Riemann surfaces; 3.2 Algebraic curves; First examples of algebraic curves
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Algebraic curves in the complex planeSingular points of algebraic curves; Algebraic curves in the complex projective plane; 3.3 Normalization theorem; Normalization Principle; Resolution of singularities of curves; 3.4 Further properties of Riemann surfaces; Divisors; Ramification index and Riemann-Hurwitz formula; Riemann-Roch theorem; Bézout theorem; 3.5 Complex tori and elliptic functions; Elliptic tori and cubic curves; Abelian group structure on cubic curve; Addition theorem for the Weierstrass function; Elliptic integrals and Jacobi functions; Cubic curves; 3.6 Hyperelliptic curves
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Definition of hyperelliptic curvesProperties of hyperelliptic curves; 3.7 Abel's theorem; Matrix of periods of a Riemann surface; Jacobian of a Riemann surface. The Abel mapping; Riemann theta-function; The Jacobi inversion problem and the Riemann theorem about zeroes of theta-function; The Baker-Akhiezer function; Riemann-Schottky problem and Novikov's conjecture; 3.8 Points of finite order on the Jacobian of a hyperelliptic curve; Chapter 4 Projective Geometry; 4.1 Preliminaries; 4.2 Conics and quadrics; Circles; 4.3 Projective structure on a conic; 4.4 Pencils of conics
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Examples of pencils of conics4.5 Quadrics and polarity; 4.6 Polarity and pencils of conics; General pencil of conics; 4.7 Invariants of pairs of conics; 4.8 Duality. Complete conics; Conics and duality; Complete conics; 4.9 Confocal conics; Confocal family of conics, elliptic coordinates; Tangential pencils; 4.10 Quadrics, their pencils and linear subsets; Pencils of quadrics; 4.11 Confocal quadrics; 4.12 (2-2)-correspondences; (1-1)and (2-1)-correspondences; (2-2)-correspondences, definition and properties; Symmetric correspondences
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Geometric interpretation of a symmetric (2-2)-correspondence. Poncelet theoremChapter 5 Poncelet Theorem and Cayley's Condition; 5.1 Full Poncelet theorem; Basic lemma; Circumscribed and Tangent Polygons; Proof of the Full Poncelet theorem; 5.2 Cayley's condition; Representation of conics of a pencil by points on a cubic curve; Condition for existence of a Poncelet triangle; Cayley's cubic; Sylvester's theory of residues; Condition for existence of a Poncelet polygon; Cayley's condition; Some applications of Lebesgue's results; 5.3 Another proof of Poncelet's theorem and Cayley's condition
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5.4 One generalization of the Poncelet theorem
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Electronic reproduction; Available via World Wide Web
Additional Edition:
ISBN 9783034800143
Additional Edition:
Erscheint auch als Druck-Ausgabe Dragović, Vladimir Poncelet porisms and beyond Basel : Birkhäuser Verlag, 2011 ISBN 3034800142
Additional Edition:
ISBN 9783034800143
Language:
English
Subjects:
Mathematics
Keywords:
Poncelet-Kurve
DOI:
10.1007/978-3-0348-0015-0
URL:
Volltext
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