Hyperbolic geometry

A non-Euclidean geometry in which infinitely many lines through a point outside a given line do not meet that line. Its standard models have constant negative curvature.

Euclidean geometry: The geometry of flat space, governed by Euclid’s postulates, including a unique parallel through an external point. Its parallel postulate is the assumption hyperbolic geometry replaces.
Poincaré disk model: A model of the hyperbolic plane inside a disk, with geodesics represented by circular arcs orthogonal to its boundary or by diameters. It represents the entire hyperbolic plane within a finite circular diagram.
Euclid: An ancient Greek mathematician whose Elements organized geometry around definitions, axioms, and propositions. His parallel postulate became the assumption later geometers tried to prove.
Hyperbolic surface: A surface equipped with a metric of constant negative curvature. Such surfaces provide finite-area settings for studying hyperbolic geometry.
Spherical geometry: The geometry of a sphere, where great circles serve as geodesics and parallel lines do not exist. It has positive curvature, contrasting with hyperbolic geometry’s negative curvature.
Gaussian curvature: A measure of a surface’s intrinsic curvature at a point, defined through the behavior of nearby geodesics. Hyperbolic surfaces have negative Gaussian curvature at every point.
Poincaré half-plane model: A model of the hyperbolic plane in the upper half-plane, with vertical lines and boundary-orthogonal semicircles as geodesics. It offers another conformal model with explicit geodesics and a simple metric.
Girolamo Saccheri: An Italian Jesuit mathematician who investigated alternatives to Euclid’s parallel postulate in the eighteenth century. His work derived many results of hyperbolic geometry while aiming to refute them.
Fuchsian group: A discrete group of orientation-preserving isometries acting on the hyperbolic plane. These groups generate quotient surfaces and encode their symmetries.
Elliptic geometry: A non-Euclidean geometry in which no parallel lines exist and geodesics eventually intersect. It rejects the parallel postulate in the opposite direction from hyperbolic geometry.

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