Moebius transformation

Moebius transformation demo. This applet also supports inverse transformations: you can draw on both planes. The transformed/inverse transformed shape will appear in another plane.
Moebius (or Möbius) transformation is a complex mapping that can be written in the form

where a,b,c,d (also z and the result of the function) are complex numbers. The special case ad-bc = 0 is not considered a Möbius transformation as the function becomes a constant[1]. Moebius transformation should not be confused with Moebius transform or Moebius function terms that have other meanings. These transformations are completely determined by three points and they transformed images[2]

Depending on the value of coefficients, these transformations can translate (shift), scale and rotate the mapped shapes on the complex plane. Moebius transformations can also perform inversion that is defined as "turning the plane inside out" (see the animation at [3] to understand the "inside out". They can also be a complicated combination of all these four effects.

Moebius transformations preserve angles and always map generalized circles to generalized circles. "Generalized circle" is either circle or line, understanding a line as a special kind of circle that contains the point of infinity. For instance, Smith map (available under "Map" tab in the applet) maps imaginary axis into the circle around the origin. While circles are mapped into circles, the center of the initial circle may not map into the center of the mapped circle, and the disk does not necessarily maps into the disk. However even while two lines may be mapped into the two circles, they would still have the same crossing angles as they did before transformation.


  1. 1 Prof Terence Tao page that contains multiple exercises, possible with the previous and current versions of this applet.
  2. 2
  3. 3 Arbitrary Moebius Transformation of the Poincare Disk