3
$\begingroup$

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" 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" 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.

$\endgroup$
5
  • $\begingroup$ Please let me know if this should be CommunityWiki or is even too opinion based for that! $\endgroup$
    – M. Winter
    May 23, 2022 at 17:39
  • 1
    $\begingroup$ Similarity transformation comes close. $\endgroup$ May 23, 2022 at 17:54
  • 3
    $\begingroup$ I think "conformal" is the most commonly used term, and it's made clear that only linear transformations are being considered. $\endgroup$
    – Deane Yang
    May 23, 2022 at 18:05
  • 2
    $\begingroup$ linear conformal ............... $\endgroup$
    – Ben McKay
    May 23, 2022 at 19:37
  • 1
    $\begingroup$ In Italian I've heard the word "roto-omotetia" to denote the composition of a rotation and a homotethy (mostly in Euclidean affine geometry though). "Roto-homothety" returns a few results in English, too (and apparently not from Italian authors). $\endgroup$ May 23, 2022 at 22:32

2 Answers 2

4
$\begingroup$

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

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

$\endgroup$
3
$\begingroup$

"Orthogonal similitude" would be consistent with a very-common use of "symplectic similitude" (in automorphic forms and repn theory) for $g\in GL_n$ such that $g^\top J g=\nu(g)\cdot J$ for skew-symmetric matrix $J$.

And $GSp(J)$ is the "symplectic similitude group", and $GO(S)$ (with $S$ the non-degenerate quadratic form) is the orthogonal similitude group...

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.