## Euclidean and Non-Euclidean Geometry: An Analytic ApproachThis 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 |

210 | |

213 | |

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Euclidean and Non-Euclidean Geometry International Student Edition: An ... Patrick J. Ryan Begrenset visning - 2009 |

### Vanlige uttrykk og setninger

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