Isometry Classes of Hyperbolic 3-Space
Jeppe Klit Nielsen

There are four classes of transformations: elliptic, hyperbolic, loxodromic, and parabolic. These are also the classes of Möbius transformations of the complex plane. Any Möbius transformation applied to the boundary plane in the model will extend to an isometry of hyperbolic 3-space. The first three classes fix two ideal points on the boundary plane.