Euclidean and Non-Euclidean Geometries

M. Helena Noronha

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This book develops a self-contained treatment of classical Euclidean geometry through both axiomatic and analytic methods. Concise and well organized, it prompts readers to prove a theorem yet provides them with a framework for doing so. Chapter topics cover neutral geometry, Euclidean plane geometry, geometric transformations, Euclidean 3-space, Euclidean n-space; perimeter, area and volume; spherical geometry; hyperbolic geometry; models for plane geometries; and the hyperbolic metric.

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Review:

What's so sacred about parallel lines? Students and general readers who want a solid grounding in the fundamentals of space would do well to let M. Helena Noronha's Euclidean and Non-Euclidean Geometries be their guide. Noronha, professor of mathematics at California State University, Northridge, breaks geometry down to its essentials and shows students how Riemann, Lobachevsky, and the rest built their own by re-evaluating the parallel postulate. Each chapter devotes itself to rigorous study of one topic: neutral geometry, Euclidean 3-space, hyperbolic geometry, and more reveal themselves to the reader through the author's clear analyses and proofs. Problem sets help the student become comfortable with techniques and reach the conclusions through their own work, gaining a visceral understanding impossible through passive reading. Little mathematical background is needed beyond a bit of set theory, calculus, and a willingness to persevere. --Rob Lightner

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 Euclidean n-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 4.1, 4.2, 4.3, and Chapter 6.

The instructor who wants to study hyperbolic geometry mainly through the Poincaré models can move to Chapters 9 and 10 after having covered only Sections 8.1, 8.2 and 8.3. Likewise, in a one-semester course, spherical geometry can be briefly studied in Section 9.1, which is independent of Chapter 7 where this topic is presented with some thoroughness. The diagram on page xv shows the dependencies among the various chapters, and the instructor can customize the coverage by choosing the desired topics.

This text fits very well the needs of an undergraduate in the secondary teaching option. It contains the classical Euclidean geometry required for obtaining a teaching credential, and it is an introduction to non-Euclidean geometry. The majority of mathematics majors in the secondary teaching option are not required to take courses on advanced calculus and complex variables. It is for such majors that I wrote a self-contained text. The book could also be used in an introductory graduate course for secondary or community college teachers. This text should also appeal to undergraduate mathematics majors interested in geometry. The analytic methods used in the text will complement and deepen the knowledge of certain areas of mathematics involved in such methods, including Euclidean geometry itself.

Throughout the text I follow the same guiding principles, precisely stating definitions and theorems. Proofs are presented in detail, and when some steps are missing the reader is told precisely from which exercises they will follow. A difficulty usually found at the beginning of courses that build a geometry upon a set of postulates is to have the students understand that results derived from these postulates hold in some of the non-Euclidean geometries as well, and therefore their proofs cannot rely on facts obtained from their drawings. To clarify this point, in all proofs in the first sections of Chapter 1 I precisely indicate which axioms have been used. I also clearly point out assertions that follow from results that have already been proved. Likewise, after introducing the Euclidean parallel postulate in Chapter 2, I repeatedly point out in subsequent proofs where such a postulate or an equivalent result is used.

The degree of difficulty of exercises varies. Some have the purpose of only fixing the concepts or practicing a method, while others complete the results proved in the text or extend some of the theory, sometimes proving a result that will be needed later in the book. All challenging problems contain hints that should get the students started on the solution. Moreover, bearing in mind that to write logical and coherent proofs for theorems is a difficult task for beginners, in several exercises I sketch the proofs, and students are asked to justify the steps or explain the contradictions.

Acknowledgments

This text evolved from geometry courses that I taught at California State University, Northridge. I acknowledge my debt and my gratitude to all students who attended these courses in 1998 and 1999. Their reaction to my classes was the best encouragement I received to write this book. I would especially like to thank my colleagues Yuriko Baldin, Elena Marchisotto, Eliane Quelho, Patrick Shanahan, and Joel Zeitlin for having read the preliminary notes and making helpful suggestions. Particular thanks go to Adonai Seixas and Dennis Kletzing for their help with LaTeX, and to George Lobell and all Prentice Hall staff for helping me to refine the text. In addition, I wish to thank the reviewers Roger Cooke, David Ewing, Loren Johnsen, Sandy Norman, and Alvin Tinsley for numerous useful suggestions.

Finally, I want to thank my husband for spending many hours drawing the figures, for his encouragement, and tolerance.

M. Helena Noronha
maria.noronha@csun.edu

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