Elementary geometry provides the foundation of modern geometry. For the most part, the standard introductions end at the formal Euclidean geometry of high school. Agricola and Friedrich revisit geometry, but from the higher viewpoint of university mathematics. Plane geometry is developed from its basic objects and their properties and then moves to conics and basic solids, including the Platonic solids and a proof of Euler's polytope formula. Particular care is taken to explain symmetry groups, including the description of ornaments and the classification of isometries by their number of fixed points. Complex numbers are introduced to provide an alternative, very elegant approach to plane geometry. The authors then treat spherical and hyperbolic geometries, with special emphasis on their basic geometric properties.

This largely self-contained book provides a much deeper understanding of familiar topics, as well as an introduction to new topics that complete the picture of two-dimensional geometries. For undergraduate mathematics students the book will be an excellent introduction to an advanced point of view on geometry. For mathematics teachers it will be a valuable reference and a source book for topics for projects.

The book contains over 100 figures and scores of exercises. It is suitable for a one-semester course in geometry for undergraduates, particularly for mathematics majors and future secondary school teachers.

**Ilka Agricola**: Humboldt-Universität zu Berlin, Berlin, Germany,

**Thomas Friedrich**: Humboldt-Universität zu Berlin, Berlin, Germany

• Preface to the German Edition 8

• Chapter 1. Introduction: Euclidean space 14

o Exercises 19

• Chapter 2. Elementary geometrical figures and their properties 22

o §2.1. The line 22

o §2.2. The triangle 32

o §2.3. The circle 58

o §2.4. The conic sections 76

o §2.5. Surfaces and bodies 90

o Exercises 102

• Chapter 3. Symmetries of the plane and of space 112

o §3.1. Affine mappings and centroids 112

o §3.2. Projections and their properties 118

o §3.3. Central dilations and translations 121

o §3.4. Plane isometries and similarity transforms 127

o §3.5. Complex description of plane transformations 140

o §3.6. Elementary transformations of the space E[sup(3)] 144

o §3.7. Discrete subgroups of the plane transformation group 152

o §3.8. Finite subgroups of the spatial transformation group

o Exercises 169

• Chapter 4. Hyperbolic geometry 180

o §4.1. The axiomatic development of elementary geometry 180

o §4.2. The Poincaré model 187

o §4.3. The disc model 196

o §4.4. Selected properties of the hyperbolic plane 198

o §4.5. Three types of hyperbolic isometries 202

o §4.6. Fuchsian groups 207

o Exercises 217

• Chapter 5. Spherical geometry 222

o §5.1. The space S[sup(2)] 222

o §5.2. Great circles in S[sup(2)] 224

o §5.3. The isometry group of [sup(2)] 228

o §5.4. The Möbius group of S[sup(2)] 229

o §5.5. Selected topics in spherical geometry 231

o Exercises 239

• Bibliography 242

• List of Symbols 248

• Index