You can have as many perpendicular lines as you want on a Poincaré disk. http://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model http://en.wikipedia.org/wiki/Poi

You can have as many perpendicular lines as you want on a Poincaré disk. http://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model http://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model (credit goes to my coworker dmvaldman)