Poincare Hyperbolic Disk Tiling

This sketch illustrates an algorithm for drawing checkered patterns on the the Poincare disk. The Poincare disk is a model of hyperbolic geometry. The pattern is defined by reflections in the circles shown. For each point we check weather it is inside any of the circles. If not, it must be in that triangular space in the middle, the only bit thats not inside any of the circles. So we draw that pixel white. Otherwise we repeatedly “reflect” the point until it finds its way to the middle triangle. We then colour it depending on weather it took an even or odd number of reflections.