5: Hyperbolic Geometry
Hyperbolic geometry can be modelled in many different ways. We will focus here on the Poincaré disk model, developed by Henri Poincaré (1854-1912) in around 1880. Poincaré did remarkable work in mathematics, though he was never actually a professor of mathematics. He was particularly interested in the relationship between mathematics, physics, and psychology. He began studying non-Euclidean geometry in detail after it appeared in his study of two apparently unrelated disciplines: differential equations and number theory. 1 Poincaré took Klein's view that geometries are generated by sets and groups of transformations on them. We consider a second model of hyperbolic geometry, the upper half-plane model, in Section 5.5.
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- 5.1: The Poincaré Disk Model
- The Poincaré disk model for hyperbolic geometry is the pair (D,H) where D consists of all points z in C such that |z|<1, and H consists of all Möbius transformations T for which T(D)=D. The set D is called the hyperbolic plane, and H is called the transformation group in hyperbolic geometry.
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- 5.2: Figures of Hyperbolic Geometry
- The Euclidean transformation group, E, consisting of all (Euclidean) rotations and translations, is generated by reflections about Euclidean lines. Similarly, the transformations in H are generated by hyperbolic reflections, which are inversions about clines that intersect the unit circle at right angles. This suggests that these clines ought to be the lines of hyperbolic geometry.