Up: Hyperbolic Geometry
Isometries
An isometry is a bijective transformation
that sends distances and angles to congruent counterparts. Recall
that there
are four kinds of isometries in
Euclidean 2-space - reflection, translation, rotation and glide
reflection. Similarly, there are four kinds of isometries in
hyperbolic space: circle inversion/reflection, the hyperbolic
isometry, the parabolic isometry and the elliptic isometry.
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.
Keep an eye out for these distinguishing
characteristics when viewing the following sections.
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.
Next: Isometries In the Poincaré Disk
Up: Hyperbolic Geometry
Created: Jul 15 1996 ---
Last modified: Jul 15 1996