Description
At the end of each section is an ample collection of exercises of varying difficulty that provides problems that both extend and clarify results of that section, as well as problems that apply those results. At the end of chapters 3–7, a summary list of the new definitions and theorems of each chapter is included.
“This is the only undergraduate text I know which has reasonable coverage of both hyperbolic and elliptic geometry, along with Euclidean geometry. I am also delighted that it is now available at a lower cost.” — Tevian Dray, Oregon State University
Historical Background / Axiomatic Systems and Their Properties / Finite Geometries / Axioms for Incidence Geometry
2. Many Ways to Go: Axiom Sets for Geometry
Introduction / Euclid’s Geometry and Euclid’s Elements / Modern Euclidean Geometry / Hilbert’s Axioms for Euclidean Geometry / Birkhoff’s Axioms for Euclidean Geometry / The SMSG Postulates for Euclidean Geometry / Non-Euclidean Geometry
3. Traveling Together: Neutral Geometry
Introduction / Preliminary Notions / Congruence Conditions / The Place of Parallels / The Saccheri-Lengendre Theorem / The Search for a Rectangle / Summary
4. One Way to Go: Euclidean Geometry of the Plane
Introduction / The Parallel Postulate and Some Implications / Congruence and Area / Similarity / Some Euclidean Results Concerning Circles / Some Euclidean Results Concerning Triangles / More Euclidean Results Concerning Triangles / The Nine-Point Circle / Euclidean Constructions / Laboratory Activities Using Dynamic Geometry Software / Summary
5. Side Trips: Analytical and Transformational Geometry
Introduction / Analytical Geometry / Transformational Geometry / Analytical Transformations / Inversion / Summary
6. Other Ways to Go: Non-Euclidean Geometries
Introduction / A Return to Neutral Geometry: The Angle of Parallelism / The Hyperbolic Parallel Postulate / Hyperbolic Results Concerning Polygons / Area in Hyperbolic Geometry / Showing Consistency: A Model for Hyperbolic Geometry / Classifying Theorems / Elliptic Geometry: A Geometry with No Parallels? / Geometry in the Real World / Laboratory Activities Using Dynamic Geometry Software / Summary
7. All Roads Leads to . . . : Projective Geometry
Introduction / The Real Projective Plane / Duality / Perspectivity / The Theorem of Desargues / Projective Transformations / Summary
Appendix A: Euclid’s Definitions, Postulates, and First Ten Propositions, Book I
Appendix B: Hilbert’s Axioms for Euclidean Plane Geometry
Appendix C: Birkhoff’s Postulates for Euclidean Plane Geometry / Undefined Terms and Relations
Appendix D: The SMSG Postulates for Euclidean Geometry
Appendix E: Using Dynamic Geometry Software to Explore the Poincare Model of Hyperbolic Geometry / Shortcuts for The Geometer’s SketchPad / Cabri Macros for the Poincare Model of Hyperbolic Geometry
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