This article is about triangles in hyperbolic geometry. For triangles in a hyperbolic sector, see Hyperbolic sector § Hyperbolic triangle.
A hyperbolic triangle embedded in a saddle-shaped surface
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices.
Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles possible in any higher dimension of hyperbolic spaces.
An order-7 triangular tiling has equilateral triangles with 2π/7 radian internal angles.
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In hyperbolic geometry, a hyperbolictriangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three...
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate...
the former. When in standard position, a hyperbolic sector determines a hyperbolictriangle, the right triangle with one vertex at the origin, base on the...
triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolictriangle. Each triangle group is the symmetry group of a tiling...
positive). For an ideal triangle, a generalization of hyperbolictriangles, this sum is equal to zero. For a spherical triangle, the sum of the angles...
sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. In complex analysis, the hyperbolic functions...
straight segments also determine a triangle, for instance a spherical triangle or hyperbolictriangle. A geodesic triangle is a region of a general two-dimensional...
In hyperbolic geometry an ideal triangle is a hyperbolictriangle whose three vertices all are ideal points. Ideal triangles are also sometimes called...
cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolictriangles: cosh c R =...
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular...
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic...
premised on hyperbolic analogies to the corresponding circular trigonometric functions by regarding a hyperbolic angle as defining a hyperbolictriangle. The...
In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar...
the hyperbolic plane by congruent hyperbolictriangles known as the V6.6.∞ Infinite-order triangular tiling is created. Note that each such triangle has...
triangles Hyperbolic triangle, a triangle that has straight sides in hyperbolic geometry, but is drawn as circular in some models of hyperbolic geometry Lune...
as Euclidean polygons. In particular, the sum of the angles of a hyperbolictriangle is less than 180 degrees. Coxeter decompositions are named after...
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example,...
Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an...
As the triangle gets larger or smaller, the angles change in a way that forbids the existence of similar hyperbolictriangles, as only triangles that have...
defect of a hyperbolictriangle; and the excess also arises in two ways: the excess of a toroidal polyhedron. the excess of a spherical triangle; In the Euclidean...
precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group...
hyperbolic geometry, angle of parallelism Π ( a ) {\displaystyle \Pi (a)} is the angle at the non-right angle vertex of a right hyperbolictriangle having...
In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of...
may refer to: Angle of parallelism, in hyperbolic geometry, the angle at one vertex of a right hyperbolictriangle that has two hyperparallel sides Axial...
hyperbolic spaces as they are 0-hyperbolic (i.e. all triangles are tripods). The 1-skeleton of the triangulation by Euclidean equilateral triangles is...
mathematics, hyperbolic trigonometry can mean: The study of hyperbolictriangles in hyperbolic geometry (traditional trigonometry is the study of triangles in plane...
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