A function f:XY between two metric spaces X and Y is called an isometry if it preserves distances, i.e. if dX(a,b)=dY(f(a),f(b)) for all a,b in X.

Note that an isometry must be one to one, but needn't be onto. However, in many cases this is sufficient to ensure it must be onto, and indeed the term is often used in this more restricted manner even when this is not the case.

An isometry is a homomorphism of metric spaces.

Isometry is a powerful mathematical concept used when dealing with curved space. I list several possible ways of defining isometry, because each definition provides a unique perspective.

1) An isometry is a mapping from one surface* to another which preserves definitions of distance. This is the basic mathematical definition of isometry, from which the other definitions are easily derived. To be more precise, I could have used the prase "dot products" instead of "distance", but it generally amounts to the same thing.

2) An isometry is a transformation of a surface under which the intrinsic geometry is invariant; a being living within the surface would not know the difference. This is true because the only way a being inside the surface can infer information about its shape is by taking measurements of distances within the surface. If this seems like a silly abstract idea, consider that we humans live within a four-dimensional curved space, and have no concept of any larger-dimensional space that it's sitting in. Therefore, all we can do to determine the shape of our four dimensions is take some kind of distance measurements.

3) An isometry is a transformation which bends, rotates, or reflects a surface without actually distorting it. By "distorting", I mean expanding or contracting the space at given points within the surface. In other words, changing the distance between points. Imagine a flattened piece of silly putty. You can bend it into many different shapes, but if you want to transform it, say, from a flat circle into a hollow sphere, you will have to do some stretching and/or squishing (for example, the outer edge will need to be contracted into a point). In contrast, if you just wanted to transform the flat circle into a kind of "taco" shape, no stretching or squishing is necessary. Just bending.

Here are some properties which are preserved under isometries. Since they are invariant, they can only depend on the intrinsic geometry of the surface:

  • Distances between points (measured along the surface)
  • Angles between curves**
  • Gaussian Curvature (In two dimensions, this describes how much a surface curves and whether it looks like a "bump" or a "saddle" at given points. A very interesting and useful geometric quantity. In higher dimensions, this gets replaced by the curvature tensor, a much more complicated beast.)
  • Geodesics ("straight" lines)

Finally, here are some examples of isometries realizable between two-dimensional surfaces sitting in three dimensions:

  • The flat plane is isometric to any cylinder or cone***.
  • Any transformation which involves rotating a surface rigidly, reflecting the surface across any plane, and/or moving the entire surface rigidly to a different point in space is an example of an isometry. (this type of transformation is known as a Euclidean Isometry.)
  • The helicoid is isometric to the catenoid*** (probably the most interesting example of isometry; it shows the geometric equivalence of two surfaces which look nothing alike.)

*For the sake of clarity, I use the word "surface" quite liberally; I really mean any arbitrary n-dimensional smooth manifold, but a smooth manifold is really nothing more than a higher-dimensional generalization of a curved surface, so I think this terminology will do.
**The preservation of angles is not unique to isometry; any conformal mapping will also guarantee this.
***Actually, when I say "isometric" I mean "locally isometric". A two-dimensional being would be able to realize he was not on a flat plane (or helicoid) as soon as he went in a straight line and ended up back where he started! Thanks to krimson for catching me on this one.

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