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).
The isometries of the hyperbolic plane form a group under composition.
An isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. We first focus on orientation-preserving isometries.
Orientation-preserving isometries
The orientation-preserving isometries of the hyperbolic plane form a subgroup. This subgroup is equal to \(PSL(2, \mathbf{R})\) (e.g. by \(PSL(2, \mathbf{R})\) acting on the half-plane model of the hyperbolic plane).
There are three types of orientation-preserving isometries of the hyperbolic plane: hyperbolic, elliptic, and parabolic. (This terminology can be confusing at first because "hyperbolic isometry" can be used both to mean any isometry of the hyperbolic plane, as well as only those isometries that have hyperbolic type. In general, readers will have to pay attention to the context to know which meaning is being used.)
There is also a fourth type of isometry that is often not mentioned because it is trivial: the identity isometry. Alternatively, the identity isometry can be viewed as a degenerate form of the other three types.
Whether an isometry is hyperbolic, elliptic, parabolic, or the identity is invariant under conjugacy in the group.
The main way to tell the type of an isometry is by its fixed points. For example, the identity isometry is the isometry that fixes all points. We will discuss the remaining types below.
Hyperbolic isometry
A hyperbolic isometry is distinguished by the fact that it has exactly two fixed points. These two points are always at the boundary at infinity. A hyperbolic isometry fixes (as a set) the geodesic connecting those two points.
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.
Orientation-reversing isometries
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