Hyperbolic triangle

Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane.

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

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.

A triangle with a zero angle is impossible in Euclidean geometry for straight sides lying on distinct lines.

The relations among the angles and sides are analogous to those of spherical trigonometry; the length scale for both spherical geometry and hyperbolic geometry can for example be defined as the length of a side of an equilateral triangle with fixed angles.

This choice for this length scale makes formulas simpler.

[2] In terms of the Poincaré half-plane model absolute length corresponds to the infinitesimal metric

In terms of the (constant and negative) Gaussian curvature K of a hyperbolic plane, a unit of absolute length corresponds to a length of In a hyperbolic triangle the sum of the angles A, B, C (respectively opposite to the side with the corresponding letter) is strictly less than a straight angle.

The area of a hyperbolic triangle is equal to its defect multiplied by the square of R: This theorem, first proven by Johann Heinrich Lambert,[3] is related to Girard's theorem in spherical geometry.

In all the formulas stated below the sides a, b, and c must be measured in absolute length, a unit so that the Gaussian curvature K of the plane is −1.

In other words, the quantity R in the paragraph above is supposed to be equal to 1.

The relations are: Whether C is a right angle or not, the following relationships hold: The hyperbolic law of cosines is as follows: Its dual theorem is There is also a law of sines: and a four-parts formula: which is derived in the same way as the analogous formula in spherical trigonometry.