It is interesting to note that the isometries form a group under composition.

Furthermore, the individual isometries can be distinguished by their fixed point behavior and whether or not they preserve orientation.

- Circle Inversion/Reflection: infinitely many fixed points (on the arc of inversion or line of reflection) and orientation reversing.
- Hyperbolic Isometry: fixes two points on boundary and preserves orientation.
- Parabolic Isometry: fixes one point on boundary and preserves orientation.
- Elliptic Isometry: fixes one point in interior and preserves orientation.

Isometries in:

- The Poincaré Disk -- this site is an
in depth exploration of the four hyperbolic isometries in the
Poincaré Disk including graphics, animations and interactive
applications.
- The Upper Half Plane -- graphic and interactive
applications illustrating the four isometries in the UHP.
- The Klein-Beltrami Model -- with animations of the four hyperbolic isometries.

**Note:** The most in depth discussion is devoted to the
isometries in the Poincaré Disk. The isometries in the remaining
two models are briefly illustrated via comparison with those in the
Poincaré Disk.

Created: Jul 15 1996 --- Last modified: Jul 15 1996