Poncelet's theorem:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI [u.a.]
American Math. Soc.
2009
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 240 S. Ill., graph. Darst. |
| ISBN: | 9780821843758 |
Internformat
MARC
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| 020 | |a 9780821843758 |9 978-0-8218-4375-8 | ||
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| 245 | 1 | 0 | |a Poncelet's theorem |c Leopold Flatto |
| 264 | 1 | |a Providence, RI [u.a.] |b American Math. Soc. |c 2009 | |
| 300 | |a XVI, 240 S. |b Ill., graph. Darst. | ||
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-022259469 | ||
Datensatz im Suchindex
| _version_ | 1804145331195084800 |
|---|---|
| adam_text | Contents
Preface xi
List of Commonly Used Symbols xv
Chapter 1. Introduction 1
1.1. The Theorems of Poncelet and Cay ley 1
1.2. The Poncelet and Steiner Theorems—
A Misleading Analogy 6
1.3. The Real Case of Poncelet s Theorem 9
1.4. Related Topics 10
Part I. Projective Geometry
Chapter 2. Basic Notions of Projective Geometry 15
2.1. Projective Plane 15
2.2. Projectivities 19
2.3. Projective Line 22
2.4. Algebraic Curves 24
vi _______________________ Contents
Chapter 3. Conies 31
3.1. Conies 31
3.2. Intersection of Line and Conic 34
3.3. Reduced Form 36
3.4. Projective Structure on a Smooth Conic 38
3.5. Parametric Equations of Smooth Conies 39
Chapter 4. Intersection of Two Conies 43
4.1. Intersection Numbers 43
4.2. Bezout s Theorem for Conies 51
4.3. Conic Pencils 53
4.4. Degenerate Conies in a Conic Pencil 55
Part II. Complex Analysis
Chapter 5. Riemann Surfaces 61
5.1. Definition of Riemann Surface 61
5.2. Examples of Riemann Surfaces 65
5.3. More Examples of Riemann Surfaces. Algebraic Curves 68
5.4. Examples of Conformal Maps 74
5.5. Covering Surfaces 76
5.6. Isomorphisms of Tori 79
Chapter 6. Elliptic Functions 83
6.1. Elliptic Functions 83
6.2. The Weierstrass p-Function 86
6.3. The Functions C and a 89
6.4. Differential Equation for p 92
6.5. The Elliptic Function w = sn(z) 94
Chapter 7. The Modular Function 97
7.1. The Functions 32, S3 97
7.2. The Modular Function J 98
7.3. Fundamental Region for T 100
Contents vii
7.4. Fourier Expansion of J 102
7.5. Values of J 104
7.6. Solution to the Inversion Problem 108
Chapter 8. Elliptic Curves 111
8.1. Elliptic Curves 111
8.2. Algebraic Models 113
8.3. Division Points of C/A 115
8.4. Division Points of S 117
Part III. Poncelet and Cayley Theorems
Chapter 9. Poncelet s Theorem 123
9.1. Poncelet Correspondence 123
9.2. Algebraic Equation for M 125
9.3. Complex Structure on M 128
9.4. M is an Elliptic Curve 130
9.5. The Automorphisms a, t, and r 131
9.6. Proof of Poncelet s Theorem 132
Chapter 10. Cayley s Theorem 135
10.1. Origin of M 135
10.2. Algebraic Equation for M 136
10.3. Proof of Cayley s Theorem 138
Chapter 11. Non-generic Cases 141
11.1. Fixed Points of T) 141
11.2. Equations for C, D, and M 142
11.3. The Riemann Surface Mo 144
11.4. Formulas for rj 147
11.5. Poncelet s Theorem 148
11.6. Existence of Circuminscribed n-Gons 150
viii Contents
Chapter 12. The Real Case of Poncelet s Theorem 153
12.1. Poncelet s Theorem for Two Circles 153
12.2. Poncelet s Theorem for Two Ellipses 155
12.3. Topological Conjugacy 157
Part IV. Related Topics
Chapter 13. Billiards in an Ellipse 165
13.1. Billiards in an Ellipse. Caustics 165
13.2. The Map r)R 167
13.3. Description of Mr 168
13.4. Invariant Measure. Rotation Number 170
13.5. Billiard Trajectories with the Same Caustic 172
13.6. Derivation of Invariant Measure 173
13.7. Proofs of Theorems 13.3 and 13.4 177
Chapter 14. Double Queues 179
14.1. The Two-Demands Model 180
14.2. Formulas 182
14.3. Riemann Surface 183
14.4. Automorphy Conditions 184
14.5. The Regions Vz and Vw 184
14.6. Analytic Continuation 186
Supplement
Chapter 15. Billiards and the Poncelet Theorem
S. Tabachnikov 191
15.1. Mathematical Billiards 191
15.2. Integrable Case 195
15.3. Poncelet Grid 198
15.4. Poncelet Theorem on Quadratic Surfaces 204
15.5. Outer Billiards in the Hyperbolic Plane 207
References 210
Contents ix
Appendices
Appendix A. Factorization of Homogeneous Polynomials 215
Appendix B. Degenerate Conies of a Conic Pencil. Proof of
Theorem 4.9 219
Appendix C. Lifting Theorems 223
C.I. Homotopy 223
C.2. Lifting Theorems 224
Appendix D. Proof of Theorem 11.5 229
Appendix E. Billiards in an Ellipse. Proof of Theorem 13.1 233
References 237
Index 239
|
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| ctrlnum | (OCoLC)637460298 (DE-599)HBZHT015814008 |
| dewey-full | 516.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.5 |
| dewey-search | 516.5 |
| dewey-sort | 3516.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV026718357 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T23:17:45Z |
| institution | BVB |
| isbn | 9780821843758 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022259469 |
| oclc_num | 637460298 |
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| owner | DE-188 DE-20 |
| owner_facet | DE-188 DE-20 |
| physical | XVI, 240 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | American Math. Soc. |
| record_format | marc |
| spelling | Flatto, Leopold 1929- Verfasser (DE-588)137202849 aut Poncelet's theorem Leopold Flatto Providence, RI [u.a.] American Math. Soc. 2009 XVI, 240 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Poncelet-Polygon (DE-588)4268395-6 gnd rswk-swf Poncelet-Polygon (DE-588)4268395-6 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1595-2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022259469&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Flatto, Leopold 1929- Poncelet's theorem Poncelet-Polygon (DE-588)4268395-6 gnd |
| subject_GND | (DE-588)4268395-6 |
| title | Poncelet's theorem |
| title_auth | Poncelet's theorem |
| title_exact_search | Poncelet's theorem |
| title_full | Poncelet's theorem Leopold Flatto |
| title_fullStr | Poncelet's theorem Leopold Flatto |
| title_full_unstemmed | Poncelet's theorem Leopold Flatto |
| title_short | Poncelet's theorem |
| title_sort | poncelet s theorem |
| topic | Poncelet-Polygon (DE-588)4268395-6 gnd |
| topic_facet | Poncelet-Polygon |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022259469&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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