Is there a concise, specific name for a transform that consists of rotate, scale and translate?

Question

I'm working on software that involves transforming between different mapping coordinate systems. In one part of the maths/logic, I have to derive, then apply a transform between two cartesian coordinate systems using a rotate, uniform-scale, translate, and an optional flip operation.

The resulting transformation will preserve angles as well as the aspect-ratio (due to the uniform-scale).

Is there a specific mathematical name for this combined transform? I want to make sure I get the terminology in the code as correct (and succinct) as possible.

Answer 1

Such a thing is commonly called a similarity, as it sends a geometric object $X$ to an object that's similar to $X$.

To quote the great Wikipedia,

A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection $f$ from the space onto itself that multiplies all distances by the same positive real number $r$ [...]

You can even distinguish between direct similarities, that preserve orientation, and opposite similarities, that reverse orientation.

This convention is followed in Needham's Visual Complex Analysis, for example:

A similarity $\mathcal{S}$ is a mapping of the plane to itself that preserves ratios of distances.