Term for similarity transformation which is not a translation What's the best (i.e. most concise) term to refer to an orientation-preserving similarity transformation which is not a translation? Here are some descriptions I could think of, but all of them feel rather bulky. I hope they are as equivalent to one another as I think they are, and I hope there is something simpler equivalent to all of them.


a direct similarity transformation which is not a translation (by a non-zero displacement)
a homothety (possibly with factor $1$) followed by rotation (with possibly zero angle)
an orientation-preserving similarity with at least one fixed point
a direct similitude with a well-defined similitude center, or the identity
what $z\mapsto a(z-c)+c$ describes in the complex plane, for some fixed $a,c\in\mathbb C$ (with $a\neq 0$)


As a native German, I tend to think about this using the German term “Drehstreckung” which literally translates to “rotation-dilation”. I'm somewhat surprised by the difficulty I have in finding an exact English translation for this concept.
I'm pretty sure your literal translation of Drehstreckung, "rotation-dilation", is the term that my high-school textbook used for this kind of transformation. Sure enough, I see the exact same terminology in lecture notes for a course in Linear Algebra with Probability at Harvard and in a "MATLAB help page" for linear-algebra students at Johns Hopkins.
Not entirely sure, but there are some elements of Mobius Transformation.