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Moebius Transformation Revealed

I would just like to share this interesting video explaining Moebius Transformation. In geometry, a Moebius transformation is a function defined by F(z) = (az+b)/(cz+d), where z, a, b, c, d are complex numbers satisfying the condition that ad-bc ≠ 0. (See Moebius transformation in Wikipedia). In simpler terms, this transformation sends each point on a plane to another point somewhere else on the plane by a combination of simpler transformations including translation, rotation, inversion, and dilation. The resulting transformation can appear very complicated. But when viewed with an added dimension (in this case, a third dimension), the transformation becomes surprisingly simple! 

This reminds me of disco mirror balls. By reflecting light directed at it in many directions, it can produce complex display patterns. A disco transformation? By the way, this video was created by Douglas N. Arnold and Jonathan Rogness of the University of Minnesota and was awarded an honorable mention in the Science and Engineering Visualization Challenge 2007 in the noninteractive multimedia category.  For a complete list of the visualization challenge winners, visit Science magazine’s feature story here!


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