In order to automate the process of animation with a computer, the computer needs to be able to automatically update the location of moving objects at each frame. In practical terms, the computer needs a formula for computing the new location of a point. In other words, the computer requires a special kind of function which takes in the coordinates of a point, and spits out the coordinates of a new, transformed point. In math, such functions are called transformations.

In day to day life, we encounter transformations all the time; anytime you pick up your coffee cup and randomly set it down somewhere else, there is a transformation lurking behind the scenes that describes the move from the old position to the new position.

However, the transformations we encounter in the physical world are very special. Moving your coffee cup randomly may make it hard to find, but it will still be pretty much the same ol' cup it always used to be. By contrast, in mathematics or computer graphics, picking a transformation at random is almost certain to distort objects as they move.

Consequently, for computer animation, the transformations we are most interested in are the distortion-free ones, which correspond to motion in the physical world. Such transformations are called isometries, coming from the Greek, meaning roughly equal measures. However, knowing we want isometries is one thing, while actually finding formulas is another. In fact, even capturing the idea of "distortion-free" in quantifiable, mathematical terms leads to some interesting subtleties.

Next Time: some isometry issues in "Spacetime Subtleties"
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