Interactive Möbius transformation

This page shows a fractal pattern of circles and two small anchors (⚓). You can move the anchors with the mouse. You can also click and drag part of the pattern.

As you move the pattern, the anchored locations remain fixed. The pattern deforms to keep all the shapes circular and consistently positioned among their neighbors. This kind of deformation is a type of Möbius transformation.

You can approximate almost any Möbius transformation this way, but the simplest kinds—shifts, rotations, and scaling—require one or both anchors to be at infinity. (Note that flips are not Möbius transformations.)

To place an anchor at infinity, double-click it. It will disappear from the screen. To bring an anchor back from infinity, or to create a new anchor, double-click where there is not currently an anchor.

Having more than two visible anchors prevents transformation of the pattern. With fewer than two, the remaining anchor or anchors are considered to be at infinity.