Moebius Transformations Revealed

Moebius (or Mobius) Transformations Revealed is a short video by Douglas Arnold and Jonathan Rogness of the University of Minnesota which depicts the beauty of Moebius transformations and shows how moving to a higher dimension reveals their essential unity.

In geometry, a Möbius transformation is a function:

"Moebius transformations are among the most fundamental mappings in geometry, with applications from brain mapping to relativity theory.
A Mobius transformation acts on the plane, sending each point to a corresponding point. There are four basic types: 1. The simple translations 2. Dilations 3. Rotations 4. and Inversions, which turn the plane inside out.
Lines on the plane either remain lines or transform to circles, and right angles stay true. In general a Moebius transformation can be a complicated combination of all four effects.
The true unity of Moebius transformations is revealed by moving into the next dimension. Taking a cue from Bernhard Riemann, we place a sphere above the plane. A light at the top shines through the spherical surface, illuminating the plane. As the sphere moves, the points on the plane follow. When the sphere translates, so does the plane. Raising the sphere gives dilation. Spin the sphere like a top, and the plane rotates. Rotation about a horizontal axis corresponds to inversion. Even the most complicated Moebius transformations are revealed to be simple motions of the sphere. " D. Arnold and J. Rogness..

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