Apollonian Circle Packings via Möbius Transformations of Hyperbolic Polygons

Hyperbolic polygons are an interesting aspect of non-Euclidean geometry. They can be created as parametric plots using two pairs of sine and cosine functions, corresponding to the horizontal and vertical axes. Each function in the pair has an amplitude equal to the inverse of the argument. Because Möbius transformations preserve straight lines and circles, the transformations associated with a single hyperbolic polygon yield an arrangement of circular cutouts and straight line segments. When no straight lines are present, the transformations trace portions of Apollonian circle packings.