Term for similarity transformation which is not a translation

Question

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.

Answer 1

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.

Answer 2

Not entirely sure, but there are some elements of Mobius Transformation.

Answer 3

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.

Answer 4

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

• 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

• 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.