# generalization of isometry 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?

• en.wikipedia.org/wiki/Quasi-isometry Aug 16, 2020 at 20:10
• In normed vector spaces, you have the group of dilations (homothety+translation). Aug 16, 2020 at 20:10
• @MoisheKohan I think that is a weaker definition than what I am talking about, but that is the name I would use :) Aug 16, 2020 at 20:12
• @Bernard that does seem like the same idea of what I am talking about, I wonder why it is defined for a normed space and not a metric one Aug 16, 2020 at 20:16
• I see. As I am not a specialist of topology, I don't know whether it has extensive developments, but I guess it is mainly useful in contexts where there's at least an implicit vector structure. Aug 16, 2020 at 20:46

apparently it is called a similitude: 