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.

Keep an eye out for these distinguishing characteristics when viewing the following sections.

Isometries in:

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