Horosphere

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.

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.

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

[1] 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.

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

A horosphere within the Poincaré disk model tangent to the edges of a hexagonal tiling cell of a hexagonal tiling honeycomb
Apollonian sphere packing can be seen as showing horospheres that are tangent to an outer sphere of a Poincaré disk model