Hyperbolic triangle

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In mathematics, the term hyperbolic triangle has more than one meaning.

In the foundations of the hyperbolic functions sinh, cosh and tanh, a hyperbolic triangle is a right triangle in the first quadrant of the Cartesian plane

[\],

with one vertex at the origin, base on the diagonal ray [y=x], and third vertex on the hyperbola

[xy=1].

The length of the base of such a triangle is

[\sqrt 2 \cosh a],

and the altitude is

[\sqrt 2 \sinh a],

where [a] is the appropriate hyperbolic angle.

In hyperbolic geometry, a hyperbolic triangle is a figure in a hyperbolic plane, analogous to a triangle in Euclidean geometry. It consists of three distinct points, which are the vertices of the triangle, and three hyperbolic line segments, which are the sides of the triangle. Each pair of vertices is joined by exactly one of these segments.

The vertices are usually considered to be in the hyperbolic plane, but sometimes one considers some of the vertices to be at the circle at infinity. These are called ideal vertices and if all vertices are ideal, then the resulting figure is called an ideal hyperbolic triangle.

References

Svetlana Katok, Fuchsian Groups (1992), University of Chicago Press, Chicago ISBN 0-226-42583-5 (Provides a brief but simple, easily readable review in chapter 1.)

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Hyperbolic triangle

See also

References