This is a clear, concise, and fascinating overview of how some very advanced mathematical ideas are making their way into 3-D animation.

http://www1.ams.org/samplings/feature-column/fcarc-harmonic

Here’s the basic setup.   In order to efficiently model a character, you approximate it with a frame that is built around a few important points.  To move the character, you focus on moving just those points that define the frame.  Thus, moving the character from point A to point B boils down to understanding where those handful of crucial points go.

The tricky part is figuring out a way to smoothly bring all those in-between points along for the ride, and that’s where the math comes in.  The secret is to think of those in-between points as averages of the points that define the frame.  The article explains how barycentric coordinates, harmonic functions. and a surprising amount of calculus are being used to pull off this movie magic!