Euclidean and Non-Euclidean Geometry: An Analytic Approach

Forside
Cambridge University Press, 27. jun. 1986 - 215 sider
This book gives a rigorous treatment of the fundamentals of plane geometry: Euclidean, spherical, elliptical and hyperbolic. The primary purpose is to acquaint the reader with the classical results of plane Euclidean and nonEuclidean geometry, congruence theorems, concurrence theorems, classification of isometries, angle addition and trigonometrical formulae. However, the book not only provides students with facts about and an understanding of the structure of the classical geometries, but also with an arsenal of computational techniques for geometrical investigations. The aim is to link classical and modern geometry to prepare students for further study and research in group theory, Lie groups, differential geometry, topology, and mathematical physics. The book is intended primarily for undergraduate mathematics students who have acquired the ability to formulate mathematical propositions precisely and to construct and understand mathematical arguments. Some familiarity with linear algebra and basic mathematical functions is assumed, though all the necessary background material is included in the appendices.

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Innhold

Plane Euclidean geometry
8
The innerproduct space R²
9
The Euclidean plane E² Lines
11
Orthonormal pairs
14
Perpendicular lines
16
Parallel and intersecting lines
17
Reflections
19
Congruence and isometries
20
Fixed points and fixed lines of isometries
101
Segments
103
Rays angles and triangles
107
Spherical trigonometry
108
Rectilinear figures
109
Congruence theorems
111
Symmetries of a segment
114
Right triangles
115

Symmetry groups
21
Plane Euclidean geometry Translations
22
Rotations
26
Glide reflections
29
Structure of the isometry group
31
Fixed points and fixed lines of isometries
33
Affine transformations in the Euclidean plane
39
Fixed lines
40
The affine group AF2
42
Fundamental theorem of affine geometry
43
Affine Reflections
44
Shears
46
Dilatations
47
Similarities
48
Affine symmetries
49
Rays and angles
50
Rectilinear figures
52
The centroid
54
Symmetries of a segment
55
Symmetries of an angle
56
Barycentric coordinates
58
Addition of angles
60
Triangles
61
Symmetries of a triangle
62
Congruence of angles
64
Congruence theorems for triangles
65
Angle sums for triangles
66
Finite groups of isometries of E²
71
Conjugate subgroups
72
Orbits and stabilizers
74
Leonardos theorem
75
Regular polygons
76
Similarity of regular polygons
78
Symmetry of regular polygons
80
Figures with no vertices
81
Geometry on the sphere
84
The cross product
85
Orthonormal bases
86
Planes
87
Incidence geometry of the sphere
88
Distance and the triangle inequality
90
Parametric representation of lines
91
Perpendicular lines
92
Orthogonal transformations of E³
96
Eulers theorem
98
Isometries
100
Concurrence theorems
116
Congruence theorems for triangles
117
Finite groups of isometries of S²
120
The projective plane P²
124
Incidence properties of P²
125
Two famous theorems
126
Applications to E²
127
Desargues theorem in E²
129
The fundamental theorem of projective geometry
130
A survey of projective collineations
131
Polarities
134
Cross products
136
Distance geometry on P²
141
Isometries
143
Motions
145
Elliptic geometry
146
The hyperbolic plane
150
Incidence geometry of H²
154
Perpendicular lines
155
Pencils
156
Distance in H2 Distance in H²
157
Isometries of H² Reflections
159
Motions
160
H² as a subset of P²
162
Parallel displacements
163
Translations
165
Glide reflections
166
Products of more than three reflections
167
Fixed points of isometries
169
Segments rays angles and triangles
171
Addition of angles
172
Triangles and hyperbolic trigonometry
173
Asymptotic triangles
174
Quadrilaterals
175
Regular polygons
176
Circles horocycles and equidistant curves
178
The axiomatic approach
184
Sets and functions
186
Groups
189
Linear algebra
193
Proof of Theorem 22
203
Trigonometric and hyperbolic functions
206
References
210
Index
213
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