Hyperbolic Geometry - Triangles

Triangles

Distances in the hyperbolic plane can be measured in terms of a unit of length, analogous to the radius of the sphere in spherical geometry. Using this unit of length a theorem in hyperbolic geometry can be stated which is analogous to the Pythagorean theorem. If a, b are the legs and c is the hypotenuse of a right triangle all measured in this unit then:

The cosh function is a hyperbolic function which is an analog of the standard cosine function. All six of the standard trigonometric functions have hyperbolic analogs. In trigonometric relations involving the sides and angles of a hyperbolic triangle the hyperbolic functions are applied to the sides and the standard trigonometric functions are applied to the angles. For example the law of sines for hyperbolic triangles is:

For more of these trigonometric relationships see hyperbolic triangles.

Unlike Euclidean triangles whose angles always add up to 180° or π radians the sum of the angles of a hyperbolic triangle is always strictly less than 180°. The difference is sometimes referred to as the defect. The area of a hyperbolic triangle is given by its defect multiplied by R² where . As a consequence all hyperbolic triangles have an area which is less than R²π. The area of an ideal hyperbolic triangle is equal to this maximum.

As in spherical geometry the only similar triangles are congruent triangles.