Generalization of Euclidean Similarity In Euclidean geometry two polygons are said to be similar if, by rotation and scaling, one can be transformed into the other, and vice versa. If we consider a general metric space, does this notion exist? I can think of how this works by scaling: you consider $d(a_i , a_{I+1} )$, where $a_i$ are the vertices of the polygon, then two figures are similar if each pair can be scaled with the same constant to yield the distances of the second figure. My problem is more with rotation and angle. The notion if homeomorphic is too general because all (non-self intersecting) polygons are homeomorphic. 

In a metric space $(X,d)$ a similitude can be defined as a function $f:X\to X$ such that
$
d(f(x),f(y))=rd(x,y)
$
for $r>0$.
If $X$ is a vector space and the metric is derived from an inner product (so that the notion of angle can be defined) than similarities preserve the angles.
