Smith J.T. Methods of Geometry

John Wiley, 2000, -507 p.This book is a text for an upper-division general geometry course offered by many universities. It builds on knowledge gained in school and lower-division university courses, to introduce advanced geometry, emphasizing transformations and symmetry groups in two and three dimensions. These theories underly many fields of advanced mathematics, and are essential for application in science, engineering, and graphics. The book analyzes the content of the lower-level courses from an advanced standpoint, to lend prospective and in-service teachers insight into the structure of their subject as well as excitement about its beautiful methods and where they lead.
Mathematical preparation for this course should include vector and matrix algebra and the solution of square and nonsquare linear systems. Appendix C provides a streamlined summary of that material, with most of the nonobvious proofs. The book often uses simple concepts related to equivalence relations, described briefly in appendix A. Concepts related to transformations, their compositions and inverses, and transformation groups are developed in detail in section 6.1, but this book applies them only to isometries and similarities. The least upper bound principle is discussed in Appendix B and applied once in section 3.14, which is about π.
This should probably not be a student's first upper-division course with an algebraic flavor. All topics mentioned in the previous paragraph are indeed covered here, but if this is the first exposure to too many of them, there won't be enough time nor energy for geometry.
Chapter 4 consists entirely of nonroutine exercises related to chapter 3, on elementary Euclidean geometry. These are not the sort that you'd find in a school text. Their goal is to show what you can do with school mathematics beyond what's covered there routinely. You'll also find large exercise sets in the other chapters, to enrich and extend the theory presented in the text, and provide experience with the related computations. It's intended that results stated in some exercises be used in solving later ones. Often one aim of an exercise is to make students ask questions. They may need help to determine just what the problem is, what constitutes a solution, and where to look for a strategy. Instructors should welcome questions ; that's what we're here for. The author generally prefers to see several attempts at most exercises, each one more detailed or progressing farther than its predecessor. In the end students will have portfolios of correct solutions, an accomplishment they can build on later in applying the techniques of this book.
A complete solution manual is available to instructors. You can obtain one from John Wiley & Sons, Inc., Mathematics Editorial Office, if you properly identify your academic affiliation. Please use the manual only for class preparation. The solutions are copyrighted, and their further dissemination is not authorized.
When pursuing geometry — or any other subject — it's often instructive for students to study a related area using some of the text's methods, then write a paper on it in a style modeled on the text. That gives practice in independent research, organization, and writing. One strategy for topic selection is to choose a subject that doesn't depend much on what's done in the course after the first weeks. That way, students aren't forced to postpone research and outlining until it's nearly too late. The first two chapters of this book raise a number of questions that could lead to excellent papers, questions to which it gives no specific attention later. Some of those are listed in sections 1.5 and 2.10_. You're invited to add to the list.
There's more than enough material in this text for a semester's course. There's always more than enough material! A major goal is to prepare students to go further on their own. Chapters 1 and 2 are intended for independent reading. The author tries to present the chapter 3 précis of elementary geometry in about three weeks, and assigns selected chapter 4 exercises at the onset. Some of these may occupy students for the rest of the semester. Alternatively, chapter 4 exercises could be assigned at the beginning, but chapter 3 covered only at the very end. From chapter 5 an instructor is expected to pick favorites. Chapter 6 is essential. Only a little of chapter 7 is required for chapter 8, and that could be covered in a couple of lectures. From chapter 8, pick favorites again. Warning: The author has found it only barely within his capacity to present the classification of frieze groups (section 8.2) in lecture. Attempting to classify all the wallpaper groups (8.3) in lecture is unfeasible. A summary or limited selection is possible, however.Foundations
Elementary Euclidean geometry
Exercises on elementary geometry
Some triangle and circle geometry
Plane isometrles and similarities
Three dimensional isometries and similarities
Symmetry