Let $(M_1,d_1)$ and $(M_2,d_2)$ be two metric spaces. I am wondering if the class of functions $f:M_1\to M_2$ that satisfy the following property is known in math (and if so -- what these class of functions is called):
$d_1( a_1,b_1 )=d_1( a_1',b_1' ) \ \ \Rightarrow \ \ d_2( f(a_1),f(b_1) )=d_2( f(a_1'),f(b_1') )$
In words: If $(a_1,b_1)$ and $(a_1',b_1')$ have the same distances in $M_1$, then their images $(f(a_1),f(b_1))$ and $(f(a_1'),f(b_1'))$ must have the same distances in $M_2$.
In literature, I could find the concept of Isometry, which seems to be close to the above but still different: Isometries compare distances over the spaces $M_1$ and $M_2$, while the class of functions above compares distances only in the individual spaces.
Note that this is the same as asking for the map $f$ between two metric spaces to satisfy the following property: There exists some map $\eta: \mathbb{R_+} \to \mathbb{R_+}$ such that for all $x,y \in M_1$: $d_2(f(x),f(y)) = \eta(d_1(x,y))$. If the map $\eta$ is additionally continuously monotonously increasing, then functions as $f$ are usually called scale metric transfomations. There are some examples in Deza-Deza, Encyclopedia of Distances (Chapter 4).