Hyperbolic triangle

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 :$\{(x,y):x,y\; in\; mathbb\; R\}$,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.

There is an analogue of the cosine rule for a hyperbolic triangle with angles "A", "B","C" and sides of hyperbolic length "a", "b", "c":

:$cosh(c)=\; cosh(a)cosh(b)\; -\; sinh(a)sinh(b)cos(A).$

The hyperbolic area of the triangle is equal to π - "A" - "B" - "C".

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.

A property of hyperbolic triangles in hyperbolic geometry is that the sum of the angles of the triangle is less than 180°. Given any angle less than 180°, a hyperbolic triangle can be found with angle sum equal to the given angle. An ideal hyperbolic triangle has an angle sum of 0°.

ee also

* Triangle group

References

*citation|first=Wilson|last=Stothers|title=Hyperbolic geometry|url=http://www.maths.gla.ac.uk/~wws/cabripages/hyperbolic/hyperbolic0.html|publisher=University of Glasgow|year=2000, interactive instructional website.
* 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.)"