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In hyperbolic geometry, a horosphere is a specific hypersurface 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.
The concept has its roots in a notion expressed by F. L. Wachter in 1816 in a letter to his teacher Gauss. 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 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, 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. The term "horoball" is due to William Thurston, who used it in his work on hyperbolic 3-manifolds. The terms horosphere and horoball are often used in 3-dimensional hyperbolic geometry.
In the conformal ball model, a horoball is represented by a ball whose boundary sphere is tangent to the horizon sphere. In the upper half-space model, a horoball can appear either as a ball whose boundary sphere is tangent to the horizon plane, or as a half-space whose boundary is parallel to the horizon plane. In the hyperboloid model, a horoball is the region above a plane whose normal lies in the asymptotic cone.
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 (a hyperhypercycle).
- Roberto Bonola (1906), Non-Euclidean Geometry, translated by H.S. Carslaw, Dover, 1955; p. 63
- Roberto Bonola (1906), Non-Euclidean Geometry, translated by H.S. Carslaw, Dover, 1955; p. 88