Isometries are transformations that don't distort the shapes of objects in the process of moving them. The mathematical definition of an isometry is a transformation which preserves distances.

That is, with an isometry, the distance between any two points p and q in the original image is the same as the distance between the corresponding points T(p) and T(q) in the transformed image. By constract, with a generic transformation, there will be some points p and q for which the distance between the corresponding transformed points either shrinks (as illustrated above) or grows.

Our everyday experience gives us a pretty good intuitive idea of what "distortion free" ought to mean. However, one of the major achievements of 19th century geometry was the realization that a lot goes into really pinning down what you mean by "distortion free". Subtle changes in your idea of distortion quickly lead to completely new, non-Euclidean geometries.

As an example, one good way of looking at Einstein's theory of special relativity is to regard it as a reformulation of what a "distortion free" transformation of space is. The basic idea is that your sense of "distortion" depends on how fast you are moving relative to the object you are trying to measure, and thus both space and time are involved. One ends up considering transformations of "spacetime."

Of course, one doesn't need to understand special relativity to animate a teapot. However, some subtleties are unavoidable. Even convincing yourself that a transformation which "preserves lengths" is the same as a "distortion-free" mapping takes a bit of thought. For example, why can't you have a transformation which preserves lengths but distorts angles?

Next Time: "Would the real isometry please stand up?"
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