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.


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*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.
 A: 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.
A: Not entirely sure, but there are some elements of Mobius Transformation. 
A: I think that what you're looking for is simply called "affinity".  
From Wikipedia:

Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.

A: It could be  Dilation , Dialation, Dialatation etc.. It may be  general extension or reduction accompanied by  Rotations  applicable to vectors of a group of lines or objects .
(Googling in the subject of Mechanics of materials and compressible Fluid mechanics one may encounter more generalized terminology in  reference to Stress-Strain theorems.)
In the special 2 dimensions case it represents 


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*Quotient of two complex numbers:
$$  Z_1  = r_1 \cdot e^{t_1} ,\; Z_2  = r_2 \cdot e^{t_2},$$
$$ | Z_1 / Z_2| = (r1/r2) \cdot e^{ (t1-t2)}. $$
where quotient $(r_1/r_2)$ is the reduced scaling or dilation/dimunition factor and argument $ (t_1-t_2) $ is amount of rotation/Drehung as a difference.
In the same sense it represents what happens to 


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*Product of complex numbers as well.
$$ | Z_1 \cdot Z_2| = (r1\cdot r2) \cdot e^{ (t1+t2)}. $$
where now product  $(r_1 r_2)$ is the increased scaling or dilation/Streckung factor and argument $ (t_1 + t_2) $ is amount of rotation as a sum required to reach the new position of resultant vector in the Complex plane, Gauss or Argand diagram. 
May be term like Coupled Dilatory Rotation express it.
