# classification of isometries of Euclidean space

any isometry of Euclidean vector space $$\mathbb{R}^3$$ has the form

$$\mathbb{R}^3\ni x\mapsto A\cdot x + b\in\mathbb{R}^3,$$

where $$A\in O(3)$$ is an orthonormal matrix, and $$b\in\mathbb{R}^3$$. However, geometrically there are only a few types of isometries, namely: translations (iff $$A$$ is the unit matrix), rotations, reflections, and compositions of those three types of geometric maps.

I am wondering, how to classify those geometric mappings in terms of $$A$$ and $$b$$? Are there any properties of $$A$$ and $$b$$, such that one can classify the isometries of Euclidean space? I looked in many books, but I could not find anything satisfactory.

• A translation corresponds to $A=I$, the identity matrix, and an arbitrary translation vector $b$, etc. – Dietrich Burde Sep 7 at 14:52
• For rotation around the origin, we have $b=0$ and and an arbitrary $A\in SO_3(\Bbb R)$, see 3D rotation group. In general, we have the group $O(3)\ltimes \Bbb R^3$, so the semidirect product, see the wikipedia article for the corresponding geometric types. – Dietrich Burde Sep 7 at 14:57
• Why is that the case? Cant be a map with $b\neq 0$ become a rotation? – Oliver Watt Sep 7 at 14:59
• @OliverWatt No. Not around the origin. If $T$ is a rotation around $b$, however, then $Tx = A(x-b)+b = Ax + (b-Ab)$, where $A$ is a rotation around the origin. – amsmath Sep 7 at 16:03
A reflection is an inverse isometry, hence $$\det A=-1$$, whereas $$\det A=1$$ for a rotation.
In addition, a rotation has an invariant line, while a reflection has an invariant plane. It follows that, for a rotation, $$A$$ has only one unit eigenvalue, while for a reflection $$A$$ has two unit eigenvalues. The corresponding eigenvectors span the rotation axis or the reflection plane.