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Curve whose normals converge asymptotically
In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant
Horocycle
Type of non-Euclidean geometry
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
Hyperbolic_geometry
Model of hyperbolic geometry
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
Poincaré_disk_model
Category of coordinate systems
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)
Coordinate systems for the hyperbolic plane
Coordinate_systems_for_the_hyperbolic_plane
Hypersurface in hyperbolic space
terms horosphere and horocycle are due to Lobachevsky, who established various results showing that the geometry of horocycles and the horosphere in
Horosphere
mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been
Ergodic_flow
Tiling of the hyperbolic plane
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
Binary_tiling
Model of hyperbolic geometry
boundary circle are not distorted. All other circles are distorted, as are horocycles and hypercycles. Chords that meet on the boundary circle are limiting
Beltrami–Klein_model
Periodic tiling of the hyperbolic disk
regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle. The order-2 apeirogonal tiling represents an infinite dihedron in the
Order-3_apeirogonal_tiling
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
Ratner's_theorems
American mathematician
and information theory. He has published contributions in the theory of horocycle flows and entropy. Marcus has written over seventy research papers, some
Brian_Marcus
Number, approximately 1.618
numberword.org. Two independent computations done by Clifford Spielman. Horocycles exinscrits : une propriété hyperbolique remarquable, cabri.net, retrieved
Golden_ratio
Type of curve in hyperbolic geometry
given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity. Hypercycles have some properties
Hypercycle_(geometry)
Upper-half plane model of hyperbolic non-Euclidean geometry
of the sphere it projects generalized circles (geodesics, hypercycles, horocycles, and circles) in the hyperbolic plane to generalized circles (lines or
Poincaré_half-plane_model
Independent video game
impossible in Euclidean geometry, like infinite trees, equidistants and horocycles, and straight lines which never cross. There is also one land that relies
HyperRogue
Equation for radii of tangent circles
tangent configurations in hyperbolic geometry including hypercycles and horocycles, if k j {\displaystyle k_{j}} is the geodesic curvature of the cycle relative
Descartes'_theorem
American mathematician (born 1953)
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
John_Smillie_(mathematician)
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
Laguerre_transformations
Auditory illusion
"Temple of Cthulhu". Since the latter is an infinite sequence of concentric horocycles, the music conveys the feeling of the player continually descending, but
Shepard_tone
this honeycomb The honeycomb is analogous to the H2 order-4 apeirogonal tiling, {∞,4}, shown here with one green apeirogon outlined by its horocycle
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
Diffeomorphism that has a hyperbolic structure on the tangent bundle
{\displaystyle h_{t}^{*}} and h t {\displaystyle h_{t}} are called horocycles. Horocycles correspond to the motion of the normal vectors of a horosphere on
Anosov_diffeomorphism
Italian mathematician (1835–1900)
unit disk and that the singularity of the pseudosphere corresponds to a horocycle on the non-Euclidean plane. On the other hand, in the introduction to
Eugenio_Beltrami
Geometric surface
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
Pseudosphere
Branch of mathematics that studies dynamical systems
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
Ergodic_theory
Property of measure-preserving dynamical systems
is closely related to Anosov flows. Other geometric examples include horocycle flows, flows on translation surfaces, and billiard flows. In these settings
Ergodicity
Triangle in hyperbolic geometry
triangle has a circumscribed circle (see below). Its vertices can lie on a horocycle or hypercycle. Hyperbolic triangles have some properties that are analogous
Hyperbolic_triangle
horocycles or hypercycles rather than circles. Regular apeirogons that are scaled to converge at infinity have the symbol {∞} and exist on horocycles
List_of_regular_polytopes
Geometrical term
angles}}\Rightarrow \angle aAB+\angle bBA>2{\text{ right angles}}} . Contradiction. horocycle, In Hyperbolic geometry a curve whose normals are limiting parallels angle
Limiting_parallel
American-Israeli mathematician
large arithmetic progressions. Furstenberg proved unique ergodicity of horocycle flows on compact hyperbolic Riemann surfaces in the early 1970s. The Furstenberg
Hillel_Furstenberg
List of concrete topologies and topological spaces
manifold − A cusped hyperbolic 3-manifold of finite volume. Horosphere Horocycle Picard horn Seifert–Weber space Lakes of Wada − Three disjoint connected
List_of_topologies
American mathematician (1938–2017)
with "some controversy" following within the department. Ratner studied horocycle flows, proving that they are "loosely Bernoulli", and their Cartesian
Marina_Ratner
Rational circle tangent to the real line
circles can be interpreted as horocycles. In hyperbolic geometry any two horocycles are congruent. When these horocycles are circumscribed by apeirogons
Ford_circle
dynamics in smooth ergodic theory. His research also includes results on horocycle flows and multifractal analysis. Michael Brin Prize in Dynamical Systems
Omri_Sarig
Epicycloid Cardioid Nephroid Deferent and epicycle Ex-tangential quadrilateral Horocycle Hypotrochoid Hypocycloid Astroid Deltoid curve Lune Pappus chain Peaucellier–Lipkin
List_of_mathematical_shapes
Regular paracompact honeycomb
seen as similar to the order-3 apeirogonal tiling, {∞,3} of H2, with horocycles circumscribing vertices of apeirogonal faces. It has a total of five reflectional
Hexagonal_tiling_honeycomb
Möbius transformation generalized to rings other than the complex numbers
complex projective space into stable and unstable manifolds, with the horocycles appearing perpendicular to the geodesics. See Anosov flow for a worked
Linear fractional transformation
Linear_fractional_transformation
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
Constructions in hyperbolic geometry
Constructions_in_hyperbolic_geometry
Tiling of the plane by pentagons
with its tiles bounded by hyperbolic line segments rather than arcs of horocycles, forms pentagonal tilings that must be non-periodic, in the sense that
Pentagonal_tiling
slices of spherePages displaying short descriptions of redirect targets Horocycle – Curve whose normals converge asymptotically Incircle and excircles of
List_of_circle_topics
Geometric mean and hyperbolic angle as coordinates in quadrant I
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
Hyperbolic_coordinates
On tangency patterns of circles
tangent to all the circles on the outer boundary of the disk (which become horocycles of infinite radius in the second packing). They prove that in the second
Circle_packing_theorem
Point at infinity in hyperbolic geometry
point or ideal point is infinite. The centres of horocycles and horoballs are ideal points; two horocycles are concentric when they have the same centre
Ideal_point
Isometric automorphisms of a hyperbolic space
reflections through lines leading to the ideal point; points move along horocycles centered on the ideal point; two degrees of freedom. translation along
Hyperbolic_motion
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