An isometry of the hyperbolic plane is a mapping of the hyperbolic plane to itself that preserves the underlying hyperbolic geometry (e.g. distances and angles).
An isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. We first focus on orientation-preserving isometries.
There are three types of orientation-preserving isometries of the hyperbolic plane: hyperbolic, elliptic, and parabolic. (The terminology can be confusing at times because "hyperbolic isometry" can be used to mean both any isometry of the hyperbolic plane, as well as only those isometries that have hyperbolic type. In general, a reader will have to pay attention to the context to know which meaning is being used.)
The main way to tell the type of an isometry is by its fixed points. We will discuss each type below.
Hyperbolic isometry
A hyperbolic isometry has the property that it has exactly two fixed points. These two points are always at the boundary at infinity. A hyperbolic isometry is sometimes called a "translation." A hyperbolic isometry is determined by (1) its two fixed points, or equivalently the geodesic it fixes, (2) a direction, or orientation of that geodesic, and (3) its translation length.
Elliptic isometry
An elliptic isometry has the property that it has one fixed point in the interior of the hyperbolic plane. An elliptic isometry is sometimes called a "rotation."
Parabolic isometry
A parabolic isometry has the property that it has one fixed point on the boundary of the hyperbolic plane.
