Möbius Transformation of a Doyle Spiral

Malin Christersson ◊◊◊

The circles making up the Doyle spiral to the left in the top canvas are transformed to the cluster of circles to the right through the transformation

\[ z \mapsto \ \frac{az+b}{cz+d}. \]
Drag right/left to transform from/to Doyle spiral.

The values of b and d are fix but a and c can be changed by dragging in the canvas below.

Drag up or down to change transformation.

A Möbius transformation preserves angles and therefore tangency is also preserved. A circle that is not on the boundary of the displayed Doyle spiral is tangent to six circle neighbours. When such a circle is transformed, it will still be tangent to six transformed circle neighbours.

Circles on the boundary of the Doyle spiral have fewer neighbours. When transformed, such circles will be on the inside of one of the two spirals in the cluster.

A circle is transformed to either a circle or a line. Cases where a transformed circle is a line or "almost" a line (having a really large radius) are not displayed due to the numerical errors caused by the large radius. Circles with radii big enough to obscure the image are either drawn as "unfilled" (in the top canvas) or not at all (in the bottom canvas).

Animated gifs

Rotating Möbius transformation on tumblr.

under an Attribution-NonCommercial-ShareAlike CC license
Creative Commons License