Surprisingly, the geometry of curved surfaces throws light on the geometry of the plane. More than 2000 years after Euclid formulated axioms for plane geometry, differential geometry showed that the parallel axiom does not follow from the other axioms of Euclid. It had long been hoped that the parallel axiom followed from the others, but no proof had ever been found. In particular, no contradiction had been derived from the contrary hypothesis, P 2, that there is more than one parallel to a given line through a given point. In the 1820s, Bolyai and Lobachevsky proposed that the consequences of P 2 be accepted as a new kind of geometry—non-Euclidean geometry. To prove that no contradiction follows from P 2, however, one needs to find a model for P 2 and the other axioms of Euclid. One seeks a mathematical structure, containing objects called “points” and “lines,” that satisfies Euclid’s axioms with P 2 in place of the parallel axiom. Such a structure was first found by Beltrami (1868a), in the form of a surface of constant negative curvature with geodesics as its “lines.” By various mappings of this surface, Beltrami found other models, including a projective model in which “lines” are line segments in the unit disk, and conformal models in which “angles” are ordinary angles. Finally, Poincaré (1882) showed that Beltrami’s conformal models arise naturally in complex analysis. Papers had already been published with pictures of patterns of non-Euclidean “lines,” most notably Schwarz (1872). Thus, non-Euclidean geometry was actually a part of existing mathematics, but a part whose geometric nature had not previously been understood.
KeywordsHyperbolic Plane Hyperbolic Geometry Conformal Model Constant Negative Curvature Asymptotic Line
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