This book provides a rigorous treatment of the fundamentals of plane geometry: Euclidean, spherical, elliptic, and hyperbolic. It is intended primarily for upper-level undergraduate mathematics students, since they will have acquired the ability to formulate mathematical propositions precisely and to construct and understand mathematical arguments.

The formal prerequisites are minimal, and all the necessary background material is included in the appendixes. However, it is difficult to imagine a student reaching the required level of mathematical maturity without a semester of linear algebra and some familiarity with the elementary transcendental functions. A previous course in group theory is not required. Group concepts used in the text can be developed as needed.

The book serves several purposes. The most obvious one is to acquaint the student with certain geometrical facts. These are basically the classical results of plane Euclidean and non-Euclidean geometry, congruence theorems, concurrence theorems, classification of isometries, angle addition, trigonometrical formulas, and the like. As such, it provides an appropriate background for teachers of high school geometry.

A second purpose is to provide concrete and interesting realizations of concepts students have encountered or will encounter in their other mathematics courses. All vector spaces are at most three-dimensional, so students do not get bogged down in summation signs and indices. The fundamental notions of linear dependence, basis, linear transformation, determinant, inverse, eigenvalue, and eigenvector all occur in simple concrete surroundings, as do many of the principal ideas of group theory.