The Hyperbolic Isometry

P is the Poincaré Disk. The arcs c1 and c2 belong to circles C1 and C2 with centers at O1 and O2 respectively. In this example we will first invert over c1, taking the point x to y. The second inversion over c2 takes y to z. Thus, the effect of the composition is to take x to z.

The line l is the common orthogonal between c1 and c2. (Given any two ultra parallel lines in hyperbolic space, there exists a unique line that intersects both parallel lines orthogonally.) We have selected a fairly simply example to illustrate this isometry. Keep in mind that given a different c1 and c2 l would not necessarily be a straight Euclidean line.

Although the endpoints of p, a and b are not fixed by the individual inversions, they are fixed by the hyperbolic isometry. That is, inversion over c1 maps a point at a to the interior of C2 and inversion over c2 then maps the point back to a. Similarly, b is mapped away from and then back to itself by inversion over c1 and c2, respectively. The fact that this isometry fixes two points on the boundary is often used to characterize/identify it.

The line p plays an important roll in determining the arcs along which the hyperbolic isometry will "translate" points. Consider the following figure which illustrates the "motion" of the isometry.

This image depicts a few of the arcs along which the isometry would "translate" points. Consider how this picture might change if the common orthogonal (of c1 and c2) was not l, but instead some other "line" of the Poincaré Disk. Notice that the points all "translate" to the right. The direction of translation depends on the order of the composition. If we had inverted first over c2 and then over c1 the isometry would move points from the right to the left.

This animation illustrates the isometry as a composition of two circle inversions over divergently parallel lines. The first two "flips" we observe are the circle inversions, and the final shift to the right is the hyperbolic isometry.

Equations

Up: Isometries

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