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