horocycle
Sign in to saveAlso known as oricycle, limit circle
220px|right|thumb| A blue horocycle in the Poincaré disk model and some red normals. The normals converge asymptotically to the upper central [[ideal point.]]
~9 min read
Article
13 sectionsContents
- Properties
- Properties similar to those of Euclidean circles
- Other properties
- Horocycles in a hyperbolic plane with standardized Gaussian curvature
- Representations in models of hyperbolic geometry
- Poincaré disk model
- Poincaré half-plane model
- Hyperboloid model
- Metric
- Horocycle flow
- See also
- References
- Further reading
220px|right|thumb| A blue horocycle in the Poincaré disk model and some red normals. The normals converge asymptotically to the upper central [[ideal point.]]
In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant curvature where all the perpendicular geodesics (normals) through a point on a horocycle are limiting parallel, and all converge asymptotically to a single ideal point called the centre of the horocycle. In some models of hyperbolic geometry, it looks like the two "ends" of a horocycle get closer and closer to each other and closer to its centre, but this is not true; the two "ends" of a horocycle get further and further away from each other and stay at an infinite distance off its centre. A horosphere is the 3-dimensional version of a horocycle.