Hyperbolic triangle

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.

A hyperbolic triangle consists of three non-collinear points and the three segments between them.

Hyperbolic triangles have some properties that are analogous to those of triangles in Euclidean geometry:

Hyperbolic triangles have some properties that are analogous to those of triangles in spherical or elliptic geometry:

Hyperbolic triangles have some properties that are the opposite of the properties of triangles in spherical or elliptic geometry

Hyperbolic triangles also have some properties that are not found in other geometries:

The definition of a triangle can be generalized, permitting vertices on the ideal boundary of the plane while keeping the sides within the plane. If a pair of sides is limiting parallel (i.e. the distance between them approaches zero as they tend to the ideal point, but they do not intersect), then they end at an ideal vertex represented as an omega point.

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