This book, taking a holistic view of geometry, introduces the reader to axiomatic, algebraic, analytic and differential geometry.

Starting with an informal introduction to non-Euclidean plane geometries, the book develops the theory to put them on a rigorous footing. It may be considered as an explication of the Kleinian view of geometry a la Erlangen Programme. The treatment in the book, however, goes beyond the Kleinian view of geometry.

Some noteworthy topics presented include:

• various results about triangles (including results on areas of geodesic triangles) in Euclidean, hyperbolic, and spherical planes
• affine and projective classification of conics
• twopoint homogeneity of the three planes and the fact that the set of distance preserving maps (isometries) are essentially the same as the set of lengths preserving maps of these planes.

Geometric intuition is emphasized throughout the book. Figures are included wherever needed. The book has several exercises varying from computational problems to investigative or explorative open questions.