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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.

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  • $\begingroup$ Please let me know if this should be CommunityWiki or is even too opinion based for that! $\endgroup$
    – M. Winter
    Commented May 23, 2022 at 17:39
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    $\begingroup$ Similarity transformation comes close. $\endgroup$
    – Michael Renardy
    Commented May 23, 2022 at 17:54
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    $\begingroup$ I think "conformal" is the most commonly used term, and it's made clear that only linear transformations are being considered. $\endgroup$
    – Deane Yang
    Commented May 23, 2022 at 18:05
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    $\begingroup$ linear conformal ............... $\endgroup$
    – Ben McKay
    Commented May 23, 2022 at 19:37
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    $\begingroup$ 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). $\endgroup$
    – Federico Poloni
    Commented May 23, 2022 at 22:32

2 Answers 2

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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.

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"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...

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