miscellaneous stuff

Here Links to an external site. is the youtube video "Mobius transformations revealed". Can you think of a Mobius transformation that is not illustrated in the video by a movement of the sphere?

 

Here Links to an external site. are some animated jpegs showing Mobius transformations. Actually, they show isometries of hyperbolic 3-space, but the 2-d pictures on the "floor" are exactly Mobius transformations.

On the right side are rotations, dilations, and translations. On the left are the same transformations, but with infinity moved to a finite point of the plane. The left side is the result of applying a Mobius transformation to the right side.

 

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is a nice article about the Mercator projection, which is a conformal mapping of the earth. The conformal property makes it useful for navigation. Straight-line paths on the map correspond perfectly to paths on the earth which follow a fixed compass bearing. In the movies, whenever the Captain plots a course using a marker and a straightedge, that map is a Mercator projection.