This book offers a concrete and accessible treatment of Euclidean, projective and hyperbolic geometry, with more stress on topological aspects than is found in most textbooks. The author's purpose is to introduce students to geometry on the basis of elementary concepts in linear algebra, group theory, and metric spaces, and to deepen their understanding of these topics in the process. A large number of exercises and problems is included, some of which introduce new topics.
Author(s): Elmer G. Rees
Series: Universitext
Publisher: Springer
Year: 2005
Language: English
Pages: 117
Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 7
Contents......Page 8
Introduction......Page 10
The Linear Groups......Page 12
The Relationship Between 0(n) and GL(n,R)......Page 14
Atfine Subspaces and Affine Independence......Page 16
The Parallel Axiom Z......Page 18
Isometries of R^2......Page 22
Isometnes of R^3......Page 24
Some Subsets of R^3......Page 26
Finite Groups of Isometries......Page 30
The Platonic Solids......Page 32
Duality......Page 36
The Symmetry Groups of the Platonic Solids......Page 37
Finite Groups of Rotations of R^3......Page 44
Crystals......Page 46
Rotations and Ouaternions......Page 51
Problems......Page 54
Part II Projective Geometry......Page 60
Homogeneous Co-ordinates......Page 61
The Topology of P' and P2......Page 62
Duality......Page 66
Projective Groups......Page 68
The Cross-Ratio......Page 70
The Elliptic Plane......Page 71
Conics......Page 74
Diagonalization of Quadratic Forms......Page 76
Polarity......Page 78
Problems......Page 84
The Beltrami (or projective) Model......Page 88
Stereographic Projection......Page 91
The Poincart Model......Page 93
The Local Metnc......Page 65
Areas......Page 98
Trigonometry......Page 100
Isometries......Page 101
Lines and Polanty......Page 103
Elliptic Trigonometry......Page 108
Problems......Page 111
Further Reading......Page 114
List of Symbols......Page 115
Index......Page 117
Back Cover......Page 124
