Introduction to Classical Geometries

  • Ana Irene Ramírez Galarza
  • José Seade

Table of contents

  1. Front Matter
    Pages i-x
  2. Pages 1-73
  3. Pages 75-89
  4. Pages 91-147
  5. Pages 149-198
  6. Pages 199-210
  7. Back Matter
    Pages 211-219

About this book


This book follows Felix Klein’s proposal of studying geometry by looking at the symmetries (or rigid motions) of the space in question. In this way the classical geometries are studied: Euclidean, affine, elliptic, projective and hyperbolic. For simplicity the focus is on the two-dimensional case, which is already rich enough, though some aspects of the 3- or n-dimensional geometries are included. Once plane geometry is well understood, it is much easier to go into higher dimensions.

The fundamental ideas of the classical geometries are presented in a clear and elementary way, making them accessible to a wide audience, and relating them to more advanced topics in modern geometry, such as manifolds, Lie groups, the Gaussian curvature, group actions, and foliations.

The book appeals to, and develops, the geometric intuition of the reader. The only prerequisites are calculus, linear algebra and basic analytic geometry. After studying the material, the reader will have a good understanding of basic geometry as well as a clear picture of the relations of this beautiful subject to other branches of mathematics. This is supported by more than 100 carefully chosen illustrations and a large number of exercises. While mainly addressed to students at advanced undergraduate level, the text can be of interest to anyone wanting to learn classical geometry.


algebra calculus classical geometry euclidean geometry geometry hyperbolic geometry

Authors and affiliations

  • Ana Irene Ramírez Galarza
    • 1
  • José Seade
    • 2
  1. 1.Departamento de Matemáticas Facultad de CienciasUniversidad Nacional Autónoma de MéxicoMéxicoMéxico
  2. 2.Instituto de Matemáticas Unidad CuernavacaUniversidad Nacional Autónoma de MéxicoCuernavaca, MorelosMéxico

Bibliographic information