Symbolic Computation Group at FSU

CARS - Defining and deforming Möbius transformations

We are mostly interested in hyperbolic Möbius transformations mapping the unit disk (or the upper half-plane) onto itself. Such mappings are of the form

A hyperbolic Möbius transformation g mapping the unidt disk D onto itself has always two fixed-points on the unit circle. Furthermore, g maps the non-euclidean line, having the fixed points g as end-points, onto itself. This non-euclidean line is the axis of g.

A hyperbolic Möbius transformation g with a given axis moves all points of the unit disk towards one of the fixed-points of g. This fixed-point is called attractive, the other one is called repulsive.

A hyperbolic Möbius transformation becomes uniquely defined if we give its axis and the speed with which it moves points towards the attracting fixed-point.

This is what the program CARS does. It lets us to define the axis and the speed with mouse actions. Here is an image of the display of CARS.

This display shows the axis of two Möbius transformations, w1, and w2. The axes are the light blue circular arcs. The reddish circular arcs intersecting the axes are the respective isometric circle and its image under the transformation. The point where the isometric circle intersects the axis can be moved by a mouse action. In this way we can define and modify the speed with which the transformation moves points towards the attracting fixed-point.

The program assigns names for the transformations and lets one to edit the coefficients directly through a parameter window displayed below.

In this case the parameter window corresponding to the Möbius transformation, whose axis is the lower one, has been opened. Now one could simply edit any of the parameters shown in the parameter window and push the apply button to get a display of the deformed Möbius transformation.

One can also display Möbius transformations that are words of other transformations. That is done using the parameter window and simply entering the word that one wants to be displayed. In the figure below I have entered the commutator of w1 and w2.

Here the circular arc in between the axes of w1 and w2 is the axis of the commutator of these two transformations.

If one now deforms the parameters of w1 or those of w2 by mouse actions one can see what happens to the commutator, or to any other word in w1 and w2 that the user may have defined.

In the picture below I was looking for a situation where the axis of the commutator intersects that of w1. That situation was achieved by lowering the speed with which w2 moves points towards its fixed-point. For clarity I have deleted the isometric arcs.

One of the two white points on the axis of w2 denotes the intersection point of the axis of w2 and the isometric circle. The other one denotes its image under w2. One sees immediately that w2 does not move points very fast.

Working in the upper half-plane

One can also choose to work in the upper half-plane only, if that is preferable.