In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant...

boundary circle is not part of the horocycle. It is an ideal point and is the hyperbolic center of the horocycle. It is also the point to which all the...

ideal point, the centre of the horocycle). Through every pair of points there are two horocycles. The centres of the horocycles are the ideal points of the...

to some given horocycle. These numbers are the hyperbolic distance x h {\displaystyle x_{h}} from P {\displaystyle P} to the horocycle, and the (signed)...

Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study of the dynamics of unipotent flows played a decisive...

regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle. The order-2 apeirogonal tiling represents an infinite dihedron in the...

terms horosphere and horocycle are due to Lobachevsky, who established various results showing that the geometry of horocycles and the horosphere in...

mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been...

of the sphere it projects generalized circles (geodesics, hypercycles, horocycles, and circles) in the hyperbolic plane to generalized circles (lines or...

The half pseudosphere of curvature −1 is covered by the interior of a horocycle. In the Poincaré half-plane model one convenient choice is the portion...

boundary circle are not distorted. All other circles are distorted, as are horocycles and hypercycles Chords that meet on the boundary circle are limiting parallel...

arcs of horocycles. The choice of log ⁡ 2 {\displaystyle \log 2} as the distance between the two horocycles causes one of the two arcs of horocycles (the...

and information theory. He has published contributions in the theory of horocycle flows and entropy. Marcus has written over seventy research papers, some...

circles can be interpreted as horocycles. In hyperbolic geometry any two horocycles are congruent. When these horocycles are circumscribed by apeirogons...

sequence of hyperbolic Laguerre transformations that map a circle to a horocycle to a hypercycle and converge towards a line. This uses the split-complex...

triangle has a circumscribed circle (see below). Its vertices can lie on a horocycle or hypercycle. Hyperbolic triangles have some properties that are analogous...

numberword.org. Two independent computations done by Clifford Spielman. Horocycles exinscrits : une propriété hyperbolique remarquable, cabri.net, retrieved...

given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity. Hypercycles in hyperbolic geometry...

typical of rigidity theory. In the 1930s G. A. Hedlund proved that the horocycle flow on a compact hyperbolic surface is minimal and ergodic. Unique ergodicity...

tangent configurations in hyperbolic geometry including hypercycles and horocycles, if k j {\displaystyle k_{j}} is the geodesic curvature of the cycle relative...

Epicycloid Cardioid Nephroid Deferent and epicycle Ex-tangential quadrilateral Horocycle Hypotrochoid Hypocycloid Astroid Deltoid curve Lune Pappus chain Peaucellier–Lipkin...

hyperbolas in Q correspond to lines parallel to the boundary of HP, they are horocycles in the metric geometry of Q. If one only considers the Euclidean topology...

ergodic. A classic example for this is the Anosov flow, which is the horocycle flow on a hyperbolic manifold. This can be seen to be a kind of Hopf fibration...

complex projective space into stable and unstable manifolds, with the horocycles appearing perpendicular to the geodesics. See Anosov flow for a worked...

circle GEOS circle Great circle Great-circle distance Circle of a sphere Horocycle Incircle and excircles of a triangle Inscribed circle Johnson circles...

2577 [math.DS]. Bainbridge, Matt; Smillie, John; Weiss, Barak (2016). "Horocycle dynamics: New invariants and eigenform loci in the stratum H(1,1)". arXiv:1603...

arbitrary large arithmetic progressions. He proved unique ergodicity of horocycle flows on compact hyperbolic Riemann surfaces in the early 1970s. The Furstenberg...

the central line and radius. A horocompass can be used to construct a horocycle through a specific point if the diameter and direction are also provided...

horocycles or hypercycles rather than circles. Regular apeirogons that are scaled to converge at infinity have the symbol {∞} and exist on horocycles...

impossible in Euclidean geometry, such as infinite trees, equidistants and horocycles, and straight lines which never cross. There is also one land that relies...