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Type of curve in hyperbolic geometry
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight
Hypercycle_(geometry)
Topics referred to by the same term
Hypercycle may refer to: Hypercycle (chemistry), a kind of reaction network prominent in a theory of the self-organization of matter Hypercycle (geometry)
Hypercycle
Type of non-Euclidean geometry
Euclidean geometry, each hyperbolic triangle has an incircle. In hyperbolic space, if all three of its vertices lie on a horocycle or hypercycle, then the
Hyperbolic_geometry
Curve whose normals converge asymptotically
geometry have some properties similar to those of circles in Euclidean geometry: No three points of a horocycle are on a line, circle or hypercycle.
Horocycle
Study of geometries as axiomatic systems
in hyperbolic geometry (they form a hypercycle.) Advocates of the position that Euclidean geometry is the one and only "true" geometry received a setback
Foundations_of_geometry
infinite cell consists of a pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli
Order-5-3_square_honeycomb
Equation for radii of tangent circles
also holds for mutually tangent configurations in hyperbolic geometry including hypercycles and horocycles, if k j {\displaystyle k_{j}} is the geodesic
Descartes'_theorem
infinite cell is an order-4 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli
Order-4-3 pentagonal honeycomb
Order-4-3_pentagonal_honeycomb
Distance from one geometric object to another along a line perpendicular to both
fitting and for defining offset surfaces. Distance between sets Hypercycle (geometry) Moment of inertia Signed distance Ballantine, J. P.; Jerbert, A
Perpendicular_distance
infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli
Order-3-5 heptagonal honeycomb
Order-3-5_heptagonal_honeycomb
infinite cell consists of a pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli
Order-4-4 pentagonal honeycomb
Order-4-4_pentagonal_honeycomb
infinite cell consists of a hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli
Order-6-3_square_honeycomb
Model of hyperbolic geometry
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside
Poincaré_disk_model
Concept in geometry
all points. In hyperbolic geometry the set of points that are equidistant from and on one side of a given line form a hypercycle (which is a curve, not a
Equidistant
infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli
Order-7-3 triangular honeycomb
Order-7-3_triangular_honeycomb
infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli
Order-3-6 heptagonal honeycomb
Order-3-6_heptagonal_honeycomb
1959 woodcut by M. C. Escher
the squares and triangles are formed by arcs of hypercycles, which are not straight in hyperbolic geometry, but which connect smoothly to each other without
Circle_Limit_III
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate
Constructions in hyperbolic geometry
Constructions_in_hyperbolic_geometry
Model of hyperbolic geometry
not distorted. All other circles are distorted, as are horocycles and hypercycles. Chords that meet on the boundary circle are limiting parallel lines
Beltrami–Klein_model
infinite cell consists of an octagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli
Order-8-3 triangular honeycomb
Order-8-3_triangular_honeycomb
infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere. The Schläfli
Order-3-4 heptagonal honeycomb
Order-3-4_heptagonal_honeycomb
Hypersurface in hyperbolic space
the surface would be an (N − 1)-dimensional hypercycle. Roberto Bonola (1906), Non-Euclidean Geometry, translated by H.S. Carslaw, Dover, 1955; p. 63
Horosphere
Möbius geometry, where lines and circles can be mapped to each other, but neither can be mapped to points. Both Möbius geometry and Laguerre geometry are
Laguerre_transformations
Upper-half plane model of hyperbolic non-Euclidean geometry
projection of the sphere it projects generalized circles (geodesics, hypercycles, horocycles, and circles) in the hyperbolic plane to generalized circles
Poincaré_half-plane_model
Fundamental result in geometry
pairs of curves called hypercycles, and the foliation is non-singular. In Taxicab Geometry, a type of non-Euclidean geometry where distance is measured
Sum_of_angles_of_a_triangle
Triangle in hyperbolic geometry
horocycle or hypercycle. Hyperbolic triangles have some properties that are analogous to those of triangles in spherical or elliptic geometry: Two triangles
Hyperbolic_triangle
Characterizes spherical triangles with fixed base and area
also be proven for hyperbolic triangles, for which the apex lies on a hypercycle. Given a fixed base A B , {\displaystyle AB,} an arc of a great circle
Lexell's_theorem
Isometric automorphisms of a hyperbolic space
perpendicular to the given line; points off the given line move along hypercycles; three degrees of freedom. Orientation reversing reflection through a
Hyperbolic_motion
boundary. Lines parallel to the boundaries of the band within the band are hypercycles whose common axis is the line through the middle of the band. Mercator
Band_model
Uniform tiling of the hyperbolic plane
angle at each vertex, while in Escher's woodcut they appear to be smooth hypercycles. Circle Limit III Square tiling Uniform tilings in hyperbolic plane List
Alternated_octagonal_tiling
and with all vertices on the ideal surface. It contains and that tile 2-hypercycle surfaces, which are similar to these paracompact order-4 apeirogonal tilings
Order-4 square tiling honeycomb
Order-4_square_tiling_honeycomb
and with all vertices on the ideal surface. It contains and that tile 2-hypercycle surfaces, which are similar to the paracompact tilings and (the truncated
Order-6 hexagonal tiling honeycomb
Order-6_hexagonal_tiling_honeycomb
Polyhedron with regular congruent polygons as faces
honeycomb {7,3,3}; they are inscribed in an equidistant surface (a 2-hypercycle), which has two ideal points. Another group of regular polyhedra comprise
Regular_polyhedron
pairs of ultraparallel mirrors: . This honeycomb contains that tile 2-hypercycle surfaces, which are similar to the paracompact order-3 apeirogonal tiling
Square_tiling_honeycomb
honeycomb. The order-4 hexagonal tiling honeycomb contains , which tile 2-hypercycle surfaces and are similar to the truncated infinite-order triangular tiling
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
the Euclidean plane, with the vertices circumscribed by horocycles or hypercycles rather than circles. Regular apeirogons that are scaled to converge at
List_of_regular_polytopes
Category of coordinate systems
(see Lambert quadrilateral). Also in hyperbolic geometry there are no equidistant lines (see hypercycles). This all has influences on the coordinate systems
Coordinate systems for the hyperbolic plane
Coordinate_systems_for_the_hyperbolic_plane
All Latin and Greek roots beginning with G
epicycle, epicycloid, hemicycle, hemicyclium, heterocyclic, homocyclic, hypercycle, hypocycloid, isocyclic, mesocyclone, monocyclic, polycyclic, pseudocyclosis
List of Greek and Latin roots in English/A–G
List_of_Greek_and_Latin_roots_in_English/A–G
domain, [((3,∞,3)),((3,∞,3))]: . This honeycomb contains and that tile 2-hypercycle surfaces, which are similar to the paracompact infinite-order triangular
Order-4_octahedral_honeycomb
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