The next case is that the two lines of reflection meet at infinity. The result of reflecting first across l1, then across l2, where l1 and l2 meet at infinity is a parabolic transformation. These are in a sense a "rotation about a point at infinity".
Click on the image to download an interactive demonstration. (To use the demonstrations, you must have Geometer's Sketchpad. A demo version is available here)
Parabolic transformations follow horocycles meeting a point at infinity;
Click on the image to download an interactive demonstration.
You can download a Quicktime animation of the following image, which shows points moving along a family of horocycles by parabolic transformations:
large version (800K)
small version (200K)
(Be sure to set the movies to Loop in the Movie Menu in your Quicktime player)
Chaim Goodman-Strauss firstname.lastname@example.org