Author | Kelly, Paul. author |
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Title | The Non-Euclidean, Hyperbolic Plane [electronic resource] : Its Structure and Consistency / by Paul Kelly, Gordon Matthews |
Imprint | New York, NY : Springer New York, 1981 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-8125-9 |
Descript | 333p. online resource |
I Some Historical Background -- 1. The Parallel-Postulate Problem -- 2. The Lost Centuries -- 3. Saccheri and the โNear Missโ -- 4. The Correct PerspectiveโGauss, Bolyai, Lobatchevsky -- 5. Not-Euclidean Geometry, Absolute Geometry -- II Absolute Plane Geometry -- 1. Linear Sets and Linear Order -- 2. Half-Planes, Angles, and Angle Measure -- 3. Triangle Relations, Congruence, Foot in a Set -- 4. Non-intersecting Lines -- 5. Dedekind Cut, Continuity, A Basic Circle Property -- 6. Motions and Symmetries -- III Hyperbolic Plane Geometry -- 1. Hyperbolic Parallels -- 2. Biangles, Hyperparallels -- 3. Saccheri and Lambert Quadrilaterals, Polygon Angle Sums -- 4. Angle of Parallelism Function, Triangle Defect, Distance Variations -- 5. The 3-Point Property, Cycles -- 6. Hyperbolic Compass and Straight Edge Constructions -- 7. Existence Problems; the Method of Associated Right Triangles -- IV A Euclidean Model of the Hyperbolic Plane -- 1. An Overview of the Model -- 2. Circular Inversions in E2 -- 3. Angle and Cross Ratio Invariance Under Inversion -- 4. Linear Order and Motions in the Model -- 5. Half-Planes, Angles and Angle Measure in the Model -- 6. Triangle Congruence in the Model, the Consistency of Hyperbolic Geometry -- Appendix Distance Geometrics -- Topic I. Metric Space and Metric Geometry -- Topic II. A Spherical Metric -- Topic III. Elliptic Geometry -- Topic IV. Barbilian Geometries, the Cross Ratio Metric