# Decomposing Isometries

1. Jan 23, 2007

### sat

It's fairly well known that isometries in Euclidean space are composed of only translations, reflections and rotations. However, I'm finding it difficult to locate a proof of that. As usual, it's "intuitively obvious" but formally I'm not sure where to start.

Does anyone know of a good reference on geometry that might have one?

(My question is actually set in the context of reading about the Galilean group, with every element of that group being a composition of a rotation, translation and motion with uniform velocity. (Arnold's book on classical mechanics.))

Edit: There seems to be a uniqueness aspect to this too. It seems to me to be connected to the direct/semidirect product nature of the Euclidean group.

Last edited: Jan 23, 2007
2. Jan 23, 2007

### Hurkyl

Staff Emeritus
An explicit factorization into translation * rotation (* reflection) is actually pretty easy to find:

Choose a point. Where does it go? That's your translation.
Choose another point. Where does it go? That fixes the rotation.
Choose a third point. Where does it go? That (usually) fixes whether or not you reflect.

Now, you just have to prove that this composition gets all other points right.

3. Jan 23, 2007

### sat

Seems quite reasonable. Thanks. I'll look into this tomorrow.