Stahl S. A Gateway to Modern Geometry: The Poincaré Half-Plane

Sudbury: Jones and Bartlett Publishers, 2008. — 272 p.Stahl's Second Edition continues to provide students with the elementary and constructive development of modern geometry that brings them closer to current geometric research At the same time, repeated use is made of high school geometry, algebra, trigonometry, and calculus, thus reinforcing the students' understanding of these disciplines as well as enhancing their perception of mathematics as a unified endeavor This distinct approach makes these advanced geometry principles accessible to undergraduates and graduates alikeAn Introduction to Euclidean Geometry
Excerpts from Euclid's Elements
Hilbert's Axiomatization (Optional)
Variations on Euclid's Fifth
Rigid Motions
Translation, Rotation, Reflection
Glide Reflection
The Main Theorems
Rigid Motions and Absolute Geometry
An Interesting Nonrigid Transformation
Inversions Applied (Optional)
Hyperbolic Length
Hyperbolic Straight Lines
Hyperbolic Angles
Hyperbolic Rigid Motions
Riemannian Geometry (Optional)
Euclid's Postulates Revisited
Hyperbolic Geometry
Hyperbolic Rigid Motions
The Angles of the Hyperbolic Triangle
Regular Tessellations (Optional)
The General Definition of Area
The Area of the Hyperbolic Triangle
The Trigonometry of Hyperbolic Line Segments
Hyperbolic Right Triangles
The General Hyperbolic Triangle
Euclidean Rigid Motions
Hyperbolic Rigid Motions
Hyperbolic Flow Diagrams I: Special Cases
Hyperbolic Flow Diagrams II—The General Case
Hyperbolic Rigid Motions: Construction
The Sum of the Angles of the Triangle
Geodesics on the Sphere
Spherical Trigonometry
Spherical Area
A Digression Into Geodesy (Optional)
Elliptic Geometry
Differential Geometry
A Review of Length and Area on Surfaces
Gauss's Formula for the Curvature at a Point
Riemannian Geometry Revisited
Conformal Transformations
The Cross-Ratio
The Unit Disk Model and Its Flow Diagrams
Explicit Rigid Motions of the Unit Disk Model
The Riemann Metric of the Unit Disk Model
Regular Tessellations of the Unit Disk Model
The Beltrami-Klein Model
History
Spheres and Horospheres
Hyperbolic Space and Its Rigid Motions
Hyperbolic Geodesics
The Stereographic Projection
The Geometry of Spheres and Horospheres
Appendix A: Proofs of Some of Euclid's Propositions
Appendix B: Formulas for Hyperbolic Trigonometric Functions
Sources, References, and Suggested Readings