Euclidean and Non-Euclidean Geometries

Euclidean and Non-Euclidean Geometries

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Description

For use in upper-level undergraduate courses in Geometry.This text develops a one or two term, self-contained treatment of classical Euclidean geometry through both axiomatic and analytic methods. The analytical methods help students visualize abstract theorems. Much of the content has been written to the needs of students with a secondary teaching option, yet the text will also appeal to undergraduate mathematics majors interested in geometry. Concise and well-organized, Euclidean and Non-Euclidean Geometry prompts students to build geometry into their calculus and linear algebra knowledge.show more

Product details

  • Hardback | 409 pages
  • 152.4 x 226.1 x 20.3mm | 657.72g
  • Pearson Education (US)
  • Pearson
  • United States
  • English
  • 013033717X
  • 9780130337177

Table of contents

1. Neutral Geometry. Introduction. Axioms of Incidence and Betweenness. Convex Sets. Measuring Segments and Angles. Congruence of Triangles. The Circle. Principles of Continuity. The Basic Geometric Constructions.2. Euclidean Plane Geometry. Euclid's Fifth Postulate. Euclidean Triangles. Similar Triangles. Euclidean Circles. Trigonometric Functions. Euclidean Constructions.3. Geometric Transformations. Rigid Motions. Similarities. Inversions. Coordinate Systems. Appendix.4. Euclidean 3-Space. Axiom System for 3-Dimensional Geometry. Perpendicular Lines and Planes. Rigid Motions in 3-Space. Vectors in 3-Space.5. Euclidean n-Space. The n-Space. Basis and Change of Coordinates. Linear Transformations. Orthogonal Transformations. Orthogonal transformations of R2 and R3. Orientation.6. Perimeter, Area and Volume. Perimeter and Circumference. Area of Polygonal Regions. Area of Circles. Volumes.7. Spherical Geometry. Arc Length. Metric Spaces. Spherical Distance and Its Isometries. Geodesics and Triangles on Spheres. Conformal Maps. Appendix.8. Hyperbolic Geometry. Introduction. Neutral Geometry Revisited. The Hyperbolic Parallel Postulate. Classification of Parallels. The Angle of Parallelism. Defect of Hyperbolic Triangles.9. Models for Plane Geometries. Introduction. The Poincare Models. A Note on Elliptic Geometry.10. The Hyperbolic Metric. Introduction. The Complex Plane. Analytic Functions. Geometric Transformations. Classification of Isometries. The Hyperbolic Metric. Hyperbolic Triangles. Hyperbolic Circles.List of Axioms. Index.show more