Moebius Transformations

Moebius transformations include the basic hyperbolic transformations of rotations and translations. The set of Moebius transformations also includes other geometric transformations which are compositions of these two types of transformations.

It is an interesting fact of hyperbolic geometry that two translations, when combined together, do not necessarily make another translation, unlike in Euclidean Geometry where the composition of translations is always another translation. This fact was illustrated in the help section on holonomy and parallel transport in Hyperbolic Geometry.

To be technically accurate, what we are calling Moebius transformations are really elements of a larger class of transformations in the complex plane which are called Moebius transformations. However, in hyperbolic geometry we restrict these more general transformations to those which preserve the properties of Hyperbolic Geometry. For example, in the Poincare model, we only consider those transformations which fix the Poincare disk (the boundary circle in the Poincare canvas),  since points on the Poincare disk (points at ``infinity'') must stay on the disk. It is this restricted set of transformations on the complex plane that we will label ``Moebius'' transformations.

In general a Moebius transformation is an invertible transformation of points z=x+iy in the complex plane having the form

Here t is a real number and  z,z0 are points in the Poincare disk represented as complex numbers. A complex number z represents a point (x,y) in the plane by  z=x+iy, where i is the square root of -1. The conjugate of a complex number x+iy is the complex number x-iy.  The term e^(it) can be expanded as cos(t) + i sin(t).

Let's look at the geometric behavior of Moebius transformations using Geometry Explorer. In the Poincare Model we construct a quadrilateral to the right of the origin (0,0) as shown below. We also fill this quadrilateral.

Moebius 1

To define a Moebius transformation, we need to go to the ``Custom'' menu in the Transform Panel and select ``Moebius...''. A dialog box titled ``Build a Moebius Transform'' will pop up as shown below. We can put in values for t and the x and y values for z0 in the appropriate text fields in this dialog box. (Note that we can quickly go from one text field to another by hitting the Tab key on the keyboard.) Put in the following values: t=0.0, Real(z0)=0.5, Imag(z0)= 0.0. Then, name the transformation ``test1'' and hit the Okay button in the dialog box to finish the definition of the Moebius transformation.

moebius panel

At this point, the new transformation can be used to transform objects in the canvas. To transform the quadrilateral, first select the interior of the quadrilateral by clicking the select tool somewhere in the filled region. Then, click on the ``Custom'' menu. You will see the transformation ``test1''  listed near the bottom of the pull-down menu. Drag down to ``test1'' and select this menu item. The filled quadrilateral will be transformed as shown at the right.

moebius 2

If we repeat this process again and again on each new area formed, we get a sequence of transformed areas as shown.

moebius 3

Note that it appears that the transformation is actually translating the filled region along a line in the Poincare disk. If we measure the area of each transformed region we would see that it is identical to the original.  Moebius transformations in Hyperbolic Geometry preserve areas.  If we move the original quadrilateral up a bit, we see that we know longer have a simple translation. A translation would have to follow some hyperbolic line, but lines such as the one shown below bend the other way from the motion of the quadrilateral. Thus, the combination of our translation of the quadrilateral upwards using the mouse with the original Moebius translation is not a translation.

moebius 4

Undo the constructions that you have done to this point until you are back to the original filled quadrilateral. Now, choose ``Moebius Transform'' in the ``Custom"" menu again. and put in the value of t=0.5 and 0.0 for the other two values. Name this new Moebius transformation ``test2'' and hit return. Select the filled quadrilateral and select ``test2'' under the ``Custom'' menu several times to transform the region under this new Moebius transformation. What is happening now? Clearly, this is a rotation of the quadrilateral around the origin.

moebius 5
 

Generally, Moebius transformations that have t non-zero and z0 equal to zero will be rotations about the origin. Moebius transformations that have t zero and z0 non-zero will translate a point along a line through the origin,  if that point lies on the line to begin with.