hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

, a horosphere (or parasphere) is a specific

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...

in hyperbolic ''n''-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. For ''n'' = 2 a horosphere is called a

horocycle In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosph ...

. A horosphere can also be described as the limit of the hyperspheres that share a tangent hyperplane at a given point, as their radii go towards infinity. In Euclidean geometry, such a "hypersphere of infinite radius" would be a hyperplane, but in hyperbolic geometry it is a horosphere (a curved surface).

History

The concept has its roots in a notion expressed by F. L. Wachter in 1816 in a letter to his teacher

Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

. Noting that in Euclidean geometry the limit of a sphere as its radius tends to infinity is a plane, Wachter affirmed that even if the

fifth postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...

were false, there would nevertheless be a geometry on the surface identical with that of the ordinary plane. The terms ''horosphere'' and ''horocycle'' are due to

Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kn ...

, who established various results showing that the geometry of horocycles and the horosphere in hyperbolic space were equivalent to those of lines and the plane in Euclidean space.Roberto Bonola (1906), ''Non-Euclidean Geometry'', translated by H.S. Carslaw, Dover, 1955; p. 88 The term "horoball" is due to

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston ...

, who used it in his work on

hyperbolic 3-manifold In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It ...

s. The terms horosphere and horoball are often used in 3-dimensional hyperbolic geometry.

Models

In the conformal ball model, a horosphere is represented by a sphere tangent to the horizon sphere. In the upper half-space model, a horosphere can appear either as a sphere tangent to the horizon plane, or as a plane parallel to the horizon plane. In the

hyperboloid model In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloid ...

, a horosphere is represented by a plane whose normal lies in the asymptotic cone.

Curvature

A horosphere has a critical amount of (isotropic) curvature: if the curvature were any greater, the surface would be able to close, yielding a sphere, and if the curvature were any less, the surface would be an (''N'' − 1)-dimensional hypercycle.

References

* ''Appendix, the theory of space'' Janos Bolyai, 1987, p.143 {{Manifolds 3-manifolds Curves Hyperbolic geometry