What do you call a linear map of the form $\alpha X$, where $\alpha\in\Bbb R$ and $X\in\mathrm O(V)$ is an orthogonal map ($V$ being some linear space with inner product)? Are there established names, historical names, some naming attempts that haven't caught on?


* "[Conformal](https://en.wikipedia.org/wiki/Conformal_map)" aka. "angle-preserving" feels rather close, but I believe these terms are more commonly used in the sense of "locally angle-preserving" (i.e. it is not implicitly understood to be linear). Also, $\alpha=0$ is explicitly allowed in my context, which is not quite angle-preserving.
* I first thought "[homotheties](https://en.wikipedia.org/wiki/Homothety)" are what I am looking for, but these only capture the scaling part, not the rotation part.
* Roto-scaling or scale-rotation is apparently also already taken and is more general than what I need (see the comment by Carlo).


At the risk of letting this become too "opinion-based", let me also say that I am open for suggestions.



What do you call a scaled orthogonal map?
Wikipedia suggests "conformal orthogonal group" for the group of all such maps; see the articles


<https://en.wikipedia.org/wiki/Conformal_group>
<https://en.wikipedia.org/wiki/Orthogonal_group#Conformal_group>


The same term is used in Magma handbook:


<http://magma.maths.usyd.edu.au/magma/handbook/text/317>


and in quite a few other reputable places, e.g.


[https://people.maths.bris.ac.uk/~matyd/GroupNames/linear.html](https://people.maths.bris.ac.uk/%7Ematyd/GroupNames/linear.html)


so it appear that "conformal orthogonal transformation" is, even if slightly tautological when taken literally, the way to go.


