Horosphere
Sign in to save220px|right|thumb|A horosphere within the Poincaré disk model tangent to the edges of a [[hexagonal tiling cell of a hexagonal tiling honeycomb]] thumb|Apollonian sphere packing can be seen as showing horospheres that are tangent to an outer sphere of a [[Poincaré disk model]] 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.
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4 sectionsContents
- History
- Models
- Curvature
- References
220px|right|thumb|A horosphere within the Poincaré disk model tangent to the edges of a [[hexagonal tiling cell of a hexagonal tiling honeycomb]] thumb|Apollonian sphere packing can be seen as showing horospheres that are tangent to an outer sphere of a [[Poincaré disk model]] 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.
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).