Hyperbolic geometry is a non-Euclidean geometry of constant negative sectional curvature, also called Lobachevsky-Bolyai-Gauss geometry. The sectional curvature is often normalized to ; after a rescaling by , it is . For curvature , the complete simply connected model in dimension is hyperbolic space . Its two-dimensional case is the hyperbolic plane (Mann 2017, p. 5).

In the hyperbolic plane, the parallel postulate is replaced by the statement that for a line and a point not on it, infinitely many lines through do not intersect .

The angle sum of a hyperbolic triangle is less than . For curvature with , the Gauss-Bonnet formula gives its area as

where , , and are its interior angles in radians. The smaller the angle sum, the larger the area (Anderson 1999; Seidel 2023, p. 262). Hyperbolic triangles with the same angles are congruent and therefore have the same area, so there are no noncongruent similar triangles. Two standard disk models are the Poincaré hyperbolic disk and the Klein-Beltrami model. Other standard models include the Poincaré half-plane model and hyperboloid model. In the Poincaré half-plane model, the upper half-plane represents the hyperbolic plane, with hyperbolic lines represented by vertical Euclidean lines and semicircles centered on the real axis. For fixed curvature, the standard models of a given dimension are related by isometries and describe the same intrinsic geometry (Anderson 1999).

Daina Taimina's crocheted models illustrate the hyperbolic plane (Wertheim et al. 2004-2005).

In dimension 3, a useful matrix model identifies 3-dimensional hyperbolic space with the positive definite Hermitian matrices of determinant 1. Writing for the conjugate transpose, this gives

The special linear group acts by

The stabilizer of the identity matrix is the special unitary group, giving the displayed realization as a symmetric space (Helgason 1978).

Relations among the sides and angles of hyperbolic triangles are given by the hyperbolic law of cosines, hyperbolic law of sines, and hyperbolic law of tangents (Anderson 1999, pp. 146-151).

In the Klein-Beltrami model, an open disk in the Euclidean plane represents the hyperbolic plane, and its open chords represent hyperbolic lines. Felix Klein constructed an analytic hyperbolic geometry in 1870 in which a point is represented by a pair of real numbers with

(i.e., points of an open disk in the complex plane) and the distance between two points is given by

The geometry generated by this formula satisfies all of Euclid's postulates except the fifth. The metric of this geometry is given by the Cayley-Klein-Hilbert metric,

Hilbert extended the definition to general bounded sets in a Euclidean space.

See also

Elliptic Geometry, Euclidean Geometry, Hyperbolic Law of Cosines, Hyperbolic Law of Sines, Hyperbolic Law of Tangents, Hyperbolic Metric, Hyperbolic Space, Hyperbolic Triangle, Hyperboloid Model, Klein-Beltrami Model, Lobachevsky-Bolyai-Gauss Geometry, Non-Euclidean Geometry, Poincaré Half-Plane Model, Pseudosphere, Schwarz-Pick Lemma, Upper Half-Plane

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References

Anderson, J. W. Hyperbolic Geometry. New York: Springer-Verlag, 1999.Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 57-60, 1990.Eppstein, D. "Hyperbolic Geometry." https://ics.uci.edu/~eppstein/junkyard/hyper.html.Helgason, S. Differential Geometry, Lie Groups, and Symmetric Spaces. New York: Academic Press, 1978.Mann, K. "Groups, Geometry, and Rigidity." Notes from a minicourse at MIT, Mar. 2017. https://e.math.cornell.edu/people/mann/classes/GGRminicourse.pdf.Seidel, P. "Arclengths and Areas." Lecture 35 in "Geometry and Topology in the Plane." MIT OpenCourseWare, Spring 2023. https://ocw.mit.edu/courses/18-900-geometry-and-topology-in-the-plane-spring-2023/mit18_900s23_lec35.pdf.Stillwell, J. Sources of Hyperbolic Geometry. Providence, RI: Amer. Math. Soc., 1996.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 109-110, 1991.Wertheim, M.; Henderson, D.; and Taimina, D. "Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina." Cabinet 16, Winter 2004-2005. https://www.cabinetmagazine.org/issues/16/wertheim_henderson_taimina.php.

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Hyperbolic Geometry

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Weisstein, Eric W. "Hyperbolic Geometry." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperbolicGeometry.html

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