The principle aim of this unique text is to illuminate the beauty of the subject both with abstractions like proofs and mathematical text, and with visuals, such as abundant illustrations and diagrams. With few mathematical prerequisites, geometry is presented through the lens of linear fractional transformations. The exposition is motivational and the well-placed examples and exercises give students ample opportunity to pause and digest the material. The subject builds from the fundamentals of Euclidean geometry, to inversive geometry, and, finally, to hyperbolic geometry at the end. Throughout, the author aims to express the underlying philosophy behind the definitions and mathematical reasoning.
This text may be used as primary for an undergraduate geometry course or a freshman seminar in geometry, or as supplemental to instructors in their undergraduate courses in complex analysis, algebra, and number theory. There are elective courses that bring together seemingly disparate topics and this text would be a welcome accompaniment.
Author(s): Arseniy Sheydvasser
Series: Undergraduate Texts in Mathematics
Publisher: Springer
Year: 2023
Language: English
Pages: 241
City: Cham
978-3-031-25002-6
1
Preface
Acknowledgements
Contents
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1 Euclidean Isometries and Similarities
1.1 The Complex Plane and Affine Maps
1.2 Isometries
1.3 Similarities
1.4 Classifying Similarities
1.5 Applications
1.6 A Little Bit of Group Theory
Problems
1.1 COMPUTATIONAL EXERCISES
1.2 PROOFS
1.3 PROOFS (Calculus)
1.4 PROOFS (Group Theory)
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2 Inversive Geometry
2.1 The Extended Euclidean Plane
2.2 A Little Bit More Group Theory
2.3 Circle Inversions
2.4 Generalized Circles
2.5 Oriented Circles
2.6 Angles, Revisited
2.7 The Cross-Ratio
2.8 The Group of Möbius Transformations
Problems
2.1 COMPUTATIONAL EXERCISES
2.2 PROOFS
2.3 PROOFS (Calculus)
2.4 PROOFS (Group Theory)
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3 Applications of Inversive Geometry
3.1 Steiner's Porism
3.2 Apollonian Gaskets
3.3 Inversive Coordinates
3.4 The Special Linear Group
3.5 Inversive Distance
3.6 Steiner's Porism Revisited
3.7 Apollonian Gaskets Revisited
3.8 Descartes' Theorem
Problems
3.1 COMPUTATIONAL EXERCISES
3.2 PROOFS
3.3 PROOFS (Group Theory)
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4 Construction of Hyperbolic Geometry
4.1 Metric Geometry
4.2 The Real Special Linear Group
4.3 The Poincaré Half-Plane
4.4 Circles and Lines
4.5 Isometries and Geometric Notions
4.6 The Poincaré Disk Model
4.7 Quaternions
4.8 Hyperbolic 3-Space
4.9 Hyperbolic Spheres, Planes, and Isometries
Problems
4.1 COMPUTATIONAL EXERCISES
4.2 PROOFS
4.3 PROOFS (Calculus)
4.4 PROOFS (Linear Algebra)
4.5 PROOFS (Metric Geometry)
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5 Properties of Hyperbolic Geometry
5.1 Peculiarities of Hyperbolic Geometry
5.2 Decomposing via the Trace
5.3 Elliptic Elements
5.4 Hyperbolic Elements
5.5 Parabolic Elements
5.6 Loxodromic Elements
5.7 Other Decomposition Theorems
Problems
5.1 COMPUTATIONAL EXERCISES
5.2 PROOFS
5.3 PROOFS (Calculus)
5.4 PROOFS (Group Theory)
1 (1)
Appendix A Set Theory
A.1 Basic Constructions
A.2 Some Common Sets
A.3 Ordered Pairs and Relations
A.4 Injections, Surjections, and Bijections
References
Index
