Coordinate systems for the hyperbolic plane

In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used.

This article tries to give an overview of several coordinate systems in use for the two-dimensional hyperbolic plane.

In the descriptions below the constant Gaussian curvature of the plane is −1. Sinh, cosh and tanh are hyperbolic functions.

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, or polar angle.

From the hyperbolic law of cosines, we get that the distance between two points given in polar coordinates is

Let ${\displaystyle r=r_{1}=r_{2},\theta =\theta _{2}-\theta _{1}}$, differentiating at ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} {\theta }}}=0}$:

we get the corresponding metric tensor: ${\displaystyle (\mathrm {d} s)^{2}=(\mathrm {d} r)^{2}+\sinh ^{2}r\,(\mathrm {d} \theta )^{2}\,.}$