Sources of Hyperbolic GeometryAmerican Mathematical Soc., 1996 - 153 pages This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincare that brought hyperbolic geometry into the mainstream of mathematics. It sets out to provide recognition of Beltrami comparable to that given the pioneering works of Bolyai and Labachevsky, not only because Beltrami rescued hyperbolic geometry from oblivion by proving to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics. The models subsequently discovered by Klein and Poincare brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensional geometry and topology. By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics. The unified setting and historical context reveal the insights of Beltrami, Klein, and Poincare in their full brilliance. |
Table des matières
Introduction to Beltramis | 1 |
Introduction to Beltramis | 35 |
Introduction to Kleins | 63 |
Introduction to Poincarés | 113 |
Translation of Poincarés | 127 |
| 147 | |
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Expressions et termes fréquents
angle sum arbitrary basic figure Beltrami Cayley chords circle with centre coincide conic section constant curvature constant negative curvature construction coordinates corresponding cosh cross-ratio curve defined determined disc distance dn² ds² dx² dy² elliptic geometry equal equation euclidean geometry expression finite formula Fuchsian fuchsian groups functions fundamental conic fundamental elements fundamental points fundamental surface Gauss geodesic circle geodesic triangle given hence horocycles hyperbolic geometry hyperbolic plane infinitely intersection inversions Klein latter limit circle line element line pencil linear transformations Lobachevsky logarithm memoir noneuclidean geometry obtain ordinary orthogonal parabolic geometry parallel planimetry Poincaré points at infinity polygon projective geometry projective measure pseudosphere radius real points represented RIEMANN sinh spaces of constant special measure sphere spherical surface of constant surface of revolution theorem theory Translator's note variables vertices zero