Author | Stillwell, John. author |
---|---|
Title | Geometry of Surfaces [electronic resource] / by John Stillwell |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1992 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0929-4 |
Descript | XI, 236 p. online resource |
1. The Euclidean Plane -- 1.1 Approaches to Euclidean Geometry -- 1.2 Isometries -- 1.3 Rotations and Reflections -- 1.4 The Three Reflections Theorem -- 1.5 Orientation-Reversing Isometries -- 1.6 Distinctive Features of Euclidean Geometry -- 1.7 Discussion -- 2. Euclidean Surfaces -- 2.1 Euclid on Manifolds -- 2.2 The Cylinder -- 2.3 The Twisted Cylinder -- 2.4 The Torus and the Klein Bottle -- 2.5 Quotient Surfaces -- 2.6 A Nondiscontinuous Group -- 2.7 Euclidean Surfaces -- 2.8 Covering a Surface by the Plane -- 2.9 The Covering Isometry Group -- 2.10 Discussion -- 3. The Sphere -- 3.1 The Sphere S2 in ?3 -- 3.2 Rotations -- 3.3 Stereographic Projection -- 3.4 Inversion and the Complex Coordinate on the Sphere -- 3.5 Reflections and Rotations as Complex Functions -- 3.6 The Antipodal Map and the Elliptic Plane -- 3.7 Remarks on Groups, Spheres and Projective Spaces -- 3.8 The Area of a Triangle -- 3.9 The Regular Polyhedra -- 3.10 Discussion -- 4. The Hyperbolic Plane -- 4.1 Negative Curvature and the Half-Plane -- 4.2 The Half-Plane Model and the Conformai Disc Model -- 4.3 The Three Reflections Theorem -- 4.4 Isometries as Complex Functions -- 4.5 Geometric Description of Isometries -- 4.6 Classification of Isometries -- 4.7 The Area of a Triangle -- 4.8 The Projective Disc Model -- 4.9 Hyperbolic Space -- 4.10 Discussion -- 5. Hyperbolic Surfaces -- 5.1 Hyperbolic Surfaces and the Killing-Hopf Theorem -- 5.2 The Pseudosphere -- 5.3 The Punctured Sphere -- 5.4 Dense Lines on the Punctured Sphere -- 5.5 General Construction of Hyperbolic Surfaces from Polygons -- 5.6 Geometric Realization of Compact Surfaces -- 5.7 Completeness of Compact Geometric Surfaces -- 5.8 Compact Hyperbolic Surfaces -- 5.9 Discussion -- 6. Paths and Geodesies -- 6.1 Topological Classification of Surfaces -- 6.2 Geometric Classification of Surfaces -- 6.3 Paths and Homotopy -- 6.4 Lifting Paths and Lifting Homotopies -- 6.5 The Fundamental Group -- 6.6 Generators and Relations for the Fundamental Group -- 6.7 Fundamental Group and Genus -- 6.8 Closed Geodesic Paths -- 6.9 Classification of Closed Geodesic Paths -- 6.10 Discussion -- 7. Planar and Spherical Tessellations -- 7.1 Symmetric Tessellations -- 7.2 Conditions for a Polygon to Be a Fundamental Region -- 7.3 The Triangle Tessellations -- 7.4 Poincarรฉโs Theorem for Compact Polygons -- 7.5 Discussion -- 8. Tessellations of Compact Surfaces -- 8.1 Orbifolds and Desingularizations -- 8.2 Prom Desingularization to Symmetric Tessellation -- 8.3 Desingularizations as (Branched) Coverings -- 8.4 Some Methods of Desingularization -- 8.5 Reduction to a Permutation Problem -- 8.6 Solution of the Permutation Problem -- 8.7 Discussion -- References