An Isometry in the isomorphism of metric spaces, meaning it preserves all properties of a metric space. Is there a generalization of an Isometry which talks about metric spaces that are the same up to a factor by a scalar? For example, if we look at a circle with radius one and a circle with radius two as sub matric spaces of $R^2$, they are 'almost the same'. They have basically the same structure just that one is factored by two. This is a stronger similarity than just saying that their topological space is isomorphic.
Is there a definition that captures this similarity?
apparently it is called a similitude:
from: https://www.wikiwand.com/en/Similarity_(geometry)#/Similarity_in_general_metric_spaces