Euclidean and Non-Euclidean Geometries

This is a book to be used in undergraduate geometry courses at the junior-senior level. It develops a pelf-contained treatment of classical Euclidean geometry through both axiomatic and analytic methods. In addition, the text integrates the study of spherical and hyperbolic geometry. Euclidean and hyperbolic geometries are constructed upon a consistent set of axioms, as well as presenting the analytic aspects of their models and their isometrics. It also contains a study of the Euclideann-space. The text differs from the traditional textbooks on the foundations of geometry by taking a more natural route that leads to non-Euclidean geometries. The topics presented not only compare different parallel postulates, but place a certain emphasis on analytic aspects of some of the non-Euclidean geometries. I intend to show students how theories that underlie other fields of mathematics can be used to better understand the concrete models for the axiom systems of Euclidean, spherical, and hyperbolic geometries and to better visualize the abstract theorems. I use elementary calculus to compute lengths of curves in 3-space and on spheres, a topic usually found at the beginning of elementary books on differential geometry. Another feature of this book is the inclusion in the text of a few topics in linear algebra and complex variable. I treat those topics (and only those) which will be used in the book. They are introduced as needed to advance in the study of geometry. I do not assume any prior knowledge of complex variable and only a few basic facts about matrices. The topics of linear algebra are used to do a more advanced study of rigid motions of the n-dimensional Euclidean space, while complex variables are used to thoroughly study two models of the hyperbolic plane. This text also describes the connections between the study of geometric transformations and transformations groups -- for example, showing how a dihedral group is realized as the symmetry group of a regular polygon. The prerequisites for reading this book should be quite minimal. It has been written not presupposing any knowledge of Euclidean and, analytic geometry. However, basic elementary set theory is required for the axiomatic geometry part. The part containing the analytic methods is basically self-contained, assuming only that the students have had a basic single-variable calculus course. The book has been written assuming that a standard mathematical curriculum contains at most two semesters of geometry. Although some beautiful topics, such as projective geometry, have been left out, I believe that I have chosen crucial aspects of classical and modern geometry that provide an accessible introduction to advanced geometry. It is not unreasonable for the instructor to hope to cover the whole book in one year. Of course, it depends on the ability and experience of his/her students. But if some choices have to be made, the book is organized to permit a number of one- or two-semester course outlines so that instructors may follow their preferences. Moreover, I have attempted to discuss all topics in detail, so that the ones not covered could be undertaken as independent study. Since Chapter 1 sets the tone for the whole book and Chapter 2 contains the classical results of plane Euclidean geometry, they form the core of the book. It is possible to teach a course only on axiomatic geometry using this text. A one-semester course on Euclidean and hyperbolic geometries using only axiomatic methods would be Chapters 1, 2, and 8. If the instructor wishes to cover geometric transformations, without coordinatizing the plane, then Sections 3.1, 3.2, 3.3 should be included. Students will then be ready for the models of the hyperbolic plane and its isometries in Chapter 9. Another one-semester alternative is the one on 2and 3-dimensional Euclidean geometry. This would include Chapters 1 and 2, Sections 3.1, 3.2, 3.3, Sections 4Noronha, Helena is the author of 'Euclidean and Non-Euclidean Geometries', published 2002 under ISBN 9780130337177 and ISBN 013033717X.