Hyperbolic plane

A two-dimensional geometry of constant negative curvature in which, through a point outside a given line, multiple lines do not intersect the given line.

Euclidean geometry: Geometry characterized by zero curvature and Euclid’s parallel postulate. Its unique-parallel rule is the alternative that hyperbolic geometry rejects.
Poincaré disk model: A model of the hyperbolic plane represented inside a disk, with geodesics shown as circular arcs or diameters. It displays the entire plane within a bounded circle while preserving angles.
Euclid: An ancient Greek mathematician whose Elements organized geometry axiomatically. His formulation of the parallel postulate framed the problem later resolved by hyperbolic geometry.
Pseudosphere: A surface of revolution with constant negative curvature, formed by rotating a tractrix. It locally models hyperbolic geometry, though it cannot represent the complete plane smoothly.
Intrinsic geometry: The study of geometric properties measurable within a space, independent of its embedding. Hyperbolic curvature belongs to the plane’s metric, not how a diagram is drawn.
Gaussian curvature: A measure of intrinsic curvature at each point on a surface. The hyperbolic plane has the same negative Gaussian curvature everywhere.
Poincaré half-plane model: A model of the hyperbolic plane represented by the upper half of the Euclidean plane. Its vertical lines and boundary-orthogonal semicircles are hyperbolic geodesics.
Giovanni Girolamo Saccheri: An Italian mathematician who investigated alternatives to Euclid’s parallel postulate in the eighteenth century. His attempted proof of the postulate derived results now recognized as hyperbolic.
Hyperbolic surface: A surface equipped with a metric of constant negative curvature. Such surfaces provide finite or compact settings built from hyperbolic geometry.
Theorema Egregium: Gauss’s theorem that Gaussian curvature is preserved by local isometries. It explains why negative curvature is an intrinsic geometric feature.

Linked from 9 pages