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E-raamat: Collineations and Conic Sections: An Introduction to Projective Geometry in its History

  • Formaat: PDF+DRM
  • Ilmumisaeg: 01-Sep-2020
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030462871
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Collineations and Conic Sections: An Introduction to Projective Geometry in its HistorySuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 01-Sep-2020
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030462871
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This volume combines an introduction to central collineations with an introduction to projective geometry, set in its historical context and aiming to provide the reader with a general history through the middle of the nineteenth century. Topics covered include but are not limited to:
  • The Projective Plane and Central Collineations
  • The Geometry of Euclid's Elements
  • Conic Sections in Early Modern Europe
  • Applications of Conics in History
With rare exception, the only prior knowledge required is a background in high school geometry. As a proof-based treatment, this monograph will be of interest to those who enjoy logical thinking, and could also be used in a geometry course that emphasizes projective geometry.

Arvustused

The text is organised into fiffteen chapters and deals with a wealth of classical topics. The explicit exposition of all these topics is carried out in view of their history. The reader will find detailed information about Euclid's Elements and the treatment of conic sections by Greek mathematicians such as Apollonius of Perga and Archimedes. (Hans Havlicek, Mathematical Reviews, April, 2022)

This is a historically very well informed look at projective geometry, central collineations, and conics. The reader is exposed to both a wealth of results in projective geometry and to the motivations behind them . The author has rendered a genuine service to the reader interested in this topic, as there is no contemporary book one could turn to both learn the basic techniques of projective geometry and to find out about its historical intricacies. (Victor V. Pambuccian, zbMATH 1460.51001, 2021)

1 Introduction: The Projective Plane and Central Collineations
1(14)
1.1 The Projective Plane
1(1)
1.2 Homogeneous Coordinates and the Real Projective Plane
2(3)
1.3 Central Collineation: Definition and Elementary Properties
5(3)
1.4 Excursion: Finite Affine and Projective Planes of Minimum Size
8(1)
1.5 Looking Ahead
9(2)
1.6 Notes and Exercises
11(1)
1.7 Some Hints and Solutions to Exercises
12(3)
References
13(2)
2 Central Collineations: Properties
15(16)
2.1 Specifying a Central Collineation
15(3)
2.2 Central Collineations and Desargues' Theorem
18(1)
2.3 Composition of Central Collineations
19(1)
2.4 Group Properties
20(1)
2.5 Excursion: Two Commutative Groups of Central Collineations
20(1)
2.6 Notes and Exercises
21(6)
2.7 Some Hints and Solutions
27(4)
References
29(2)
3 The Geometry of Euclid's Elements
31(14)
3.1 Ancient Greek Mathematics Before Euclid
31(2)
3.2 The Geometry of Euclid's Elements: A. Preliminaries in Book 1
33(3)
3.3 The Geometry of Euclid's Elements: B. Straightedge/Compass Constructions in Book 1
36(1)
3.4 The Geometry of Euclid's Elements: C. Angles and Parallels
36(3)
3.5 The Geometry of Euclid's Elements: D. Triangle Similarity and Circles in Books 6 and 3
39(3)
3.6 Exercises
42(3)
References
43(2)
4 Conies in Greek Geometry: Apollonius, Harmonic Division, and Later Greek Geometry
45(14)
4.1 Conic Sections in Ancient Greece
45(1)
4.2 The Conks of Apollonius
46(4)
4.3 Harmonic Division of a Segment
50(1)
4.4 Conies and the Harmonic Relation
51(1)
4.5 Late Antiquity and Steps Toward Projective Geometry
52(3)
4.6 Notes and Exercises
55(1)
4.7 Some Solutions
56(3)
References
57(2)
5 Conic Sections in Early Modern Europe. First Part: Philippe de la Hire on Circles
59(12)
5.1 Philippe de la Hire
60(2)
5.2 On Circles: La Hire's First 17 Lemmas of 1673
62(6)
5.3 Notes and Exercises
68(3)
References
69(2)
6 Conic Sections in Early Modern Europe. Second Part: Philippe de la Hire on Conies
71(16)
6.1 Plani-Coniques
72(2)
6.2 Conic Properties Developed by La Hire, 1673
74(7)
6.3 Notes and Exercises
81(4)
6.4 Some Hints and Solutions
85(2)
References
85(2)
7 Central Collineations: Complete Quadrilateral, Involution, and Hexagon Theorems
87(12)
7.1 The Complete Quadrilateral
87(2)
7.2 Involution
89(1)
7.3 Collineations that Map a Circle to a Circle
90(3)
7.4 Theorems of Pascal and Brianchon
93(2)
7.5 Notes and Exercises
95(1)
7.6 Some Hints and Solutions
96(3)
References
97(2)
8 Nineteenth Century
99(18)
8.1 Monge and Carnot: Steps Toward Projective Geometry
99(1)
8.2 Jean-Victor Poncelet
100(1)
8.3 Dilations and the Inverse Homologue
101(4)
8.4 The Ideal Common Secant and Homology, 1813
105(4)
8.5 More Material in Poncelet's Cahiers of 1813-1814
109(1)
8.6 Poncelet's Traite of 1822
110(3)
8.7 Poncelet in 1822: Inverse Homologues, the Common Secant as Axis and Vanishing Line
113(2)
8.8 Notes and Exercises
115(2)
References
116(1)
9 Foci
117(10)
9.1 Foci Before Poncelet
117(6)
9.2 Foci in Poncelet and Chasles
123(4)
References
126(1)
10 Steiner: Cross-Ratio, Projective Forms, and Conies
127(18)
10.1 Cross-Ratio and Projective Forms
127(5)
10.2 Conies with Steiner, Chasles, and Cremona
132(3)
10.3 Constructions of Conies
135(1)
10.4 Excursion: Central Collineations and Perspectivities
136(2)
10.5 Notes and Exercises
138(4)
10.6 Some Hints and Solutions
142(3)
References
143(2)
11 Desargues and Involution
145(8)
11.1 Girard Desargues and Involution
145(5)
11.2 Foci in Desargues Work
150(1)
11.3 Notes and Exercises
151(2)
References
151(2)
12 Looking Ahead
153(4)
12.1 Projective Geometry After Steiner
153(2)
12.2 Notes
155(2)
References
156(1)
13 Matrices and Homogeneous Coordinates
157(10)
13.1 Matrices for Collineations
157(3)
13.2 Excursion: A 13-Point Projective Plane, and Yet Another Definition of a Conic
160(2)
13.3 Exercises
162(1)
13.4 Some Hints and Solutions
163(4)
References
165(2)
14 Some Applications of Conies and Collineations in History
167(8)
14.1 Archimedes' Quadrature of the Parabola
167(2)
14.2 An Islamic Sundial
169(3)
14.3 Central Collineations in Perspective Drawing: Brook Taylor and G. J.'s Gravesande
172(3)
References
174(1)
15 Vertical Stretch and Isaac Newton's Conies
175(10)
15.1 The Vertical Stretch
175(1)
15.2 Two Ellipse Properties
176(1)
15.3 Isaac Newton and the Principia of 1687
177(6)
15.4 A Conic Construction of Isaac Newton
183(1)
15.5 Notes
184(1)
References
184(1)
Index 185
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