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.

I would appreciate your help!

  • $\begingroup$ A translation corresponds to $A=I$, the identity matrix, and an arbitrary translation vector $b$, etc. $\endgroup$ – Dietrich Burde Sep 7 at 14:52
  • $\begingroup$ Thank you! The case of translations is not the problem here. I am interested in the "etc." Do you know a reference, where the other isometries are classified similarly to the translations (A=I) $\endgroup$ – Oliver Watt Sep 7 at 14:54
  • $\begingroup$ 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. $\endgroup$ – Dietrich Burde Sep 7 at 14:57
  • $\begingroup$ Why is that the case? Cant be a map with $b\neq 0$ become a rotation? $\endgroup$ – Oliver Watt Sep 7 at 14:59
  • $\begingroup$ @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. $\endgroup$ – 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.


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