# Euclidean and Non-Euclidean Geometry: An Analytic Approach

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 Opphavsrett

### Populære avsnitt

Side 2 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Side 1 - A point is that which has no part. 2 A line is breadthless length. 3 The extremities of a line are points. 4 A straight line is a line which lies evenly with the points on itself.
Side 1 - When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
Side 2 - Let the following be postulated: I. To draw a straight line from any point to any point. II. To produce a finite straight line continuously in a straight line. III. To describe a circle with any center and distance. IV. That all right angles are equal to one another. V. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two...
Side 1 - A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line; 9 And when the lines containing the angle are straight, the angle is called rectilineal.

### Referanser til denne boken

 Lie Sphere Geometry: With Applications to SubmanifoldsThomas E. CecilBegrenset visning - 2007
 Analytic Hyperbolic Geometry: Mathematical Foundations and ApplicationsAbraham A. UngarBegrenset visning - 2005
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