hyperbolic triangle

As a consequence all hyperbolic triangles have an area which is less than R2π.

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

The corresponding group is an example of a hyperbolic triangle group.

Gyrotrigonometry is the use of gyroconcepts to study hyperbolic triangles.

For more of these trigonometric relationships see hyperbolic triangles.

Lambert devised a formula for the relationship between the angles and the area of hyperbolic triangles.

A finite subset of hyperbolic triangle groups are arithmetic groups.

This region is a hyperbolic triangle.

For example the law of sines for hyperbolic triangles is:

The area of a hyperbolic triangle is given by its defect multiplied by R2 where .

See also hyperbolic triangle.

The hyperbolic triangle is a special right triangle used to define the hyperbolic functions.

In hyperbolic geometry the sum of angles in a hyperbolic triangle must be less than 180 degrees.

The hyperbolic triangle groups are notable NEC groups.

The same inequality holds for hyperbolic triangles more generally; in a non-ideal triangle, the distance to the second-closest side is strictly less than d.

This tiling can be extended to the Poincaré disk, where every hyperbolic triangle has one vertex on the boundary of the disk.

The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle.

This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles:

The analogous result holds for hyperbolic triangles, with "excess" replaced by "defect"; these are both special cases of the Gauss-Bonnet theorem.

The area of a hyperbolic triangle, conversely is proportional to its defect, as established by Johann Heinrich Lambert.

In the limit as the vertices go to infinity, there are even ideal hyperbolic triangles in which all three angles are 0 .

In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all lie on the circle at infinity.

In the 18th century, Johann Heinrich Lambert introduced the hyperbolic functions and computed the area of a hyperbolic triangle.

The vertical composition is divided, fragmented or faceted into series of hyperbolic triangles, rectangles, squares and planes or surfaces delineated by contrasting form.

There are infinitely many hyperbolic Coxeter groups describing reflection groups in hyperbolic space, notably including the hyperbolic triangle groups.