Horoball
Encyclopedia
In hyperbolic geometry
, a horoball is an object in hyperbolic n-space
: the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. Its boundary is called a horosphere. For n = 2 a horosphere is called a horocycle
.
This terminology is due to William Thurston
, who used it in his work on hyperbolic 3-manifold
s. Thus horosphere/horoball often has a connotation of referring to 3-dimensional hyperbolic geometry.
In the conformal ball model, a horoball is represented by a ball whose surface is tangent to the horizon sphere. In the upper half-space model
, a horoball can appear as such a sphere, or as a half-space whose lower 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).
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, a horoball is an object in hyperbolic n-space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
: the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. Its boundary is called a horosphere. For n = 2 a horosphere is called a horocycle
Horocycle
In hyperbolic geometry, a horocycle is a curve whose normals all converge asymptotically. It is the two-dimensional example of a horosphere ....
.
This terminology is due to William Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
, who used it in his work on hyperbolic 3-manifold
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
s. Thus horosphere/horoball often has a connotation of referring to 3-dimensional hyperbolic geometry.
In the conformal ball model, a horoball is represented by a ball whose surface is tangent to the horizon sphere. In the upper half-space model
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry....
, a horoball can appear as such a sphere, or as a half-space whose lower boundary is parallel to the horizon plane. In the hyperboloid model
Hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model , is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in -dimensional Minkowski space and m-planes are...
, 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
Hypercycle (geometry)
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line.Given a straight line L and a point P not on L,...
(a hyperhypercycle).