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Horosphere
In hyperbolic geometry, a horosphere (or parasphere) 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.
==History==
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.〔Roberto Bonola (1906), ''Non-Euclidean Geometry'', translated by H.S. Carslaw, Dover, 1955; p. 63〕 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.〔Roberto Bonola (1906), ''Non-Euclidean Geometry'', translated by H.S. Carslaw, Dover, 1955; p. 88〕 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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