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## Main Question or Discussion Point

Disclaimer: I am a physics student and I have very little knowledge of topology or differential geometry. I don't necessarily expect a complete answer to this question, but I haven't really found any reference that approaches what I'm trying to ask, so I'd be quite happy to simply be pointed in the right direction, even just advice on what terms to google.

Suppose I have a typical Cartesian xy coordinate system. Now suppose I replace the x-axis and the y-axis with curves (but just gentle curves, they still extend to infinity and do not intersect or anything crazy). Given the shapes of these new axes, is there some natural or "nice" way to define new coordinates for the entire 2D space? I realize you could probably do many things arbitrarily depending on what properties you want your coordinates to have (first thought that came to mind was to use the minimum distance to each curve), but I was wondering if people who are familiar with topology would know of any choices that are particularly intuitive or have particularly nice properties, and some literature behind them.

I'm actually interested in higher dimensional versions of this, where you deform all of the coordinate planes, but I can't even satisfy myself in the 2D case, so I thought I'd focus on that.

Sorry this is such a vague question, but I've been searching for anything related to this and I can't seem to find a foothold anywhere, except that I should learn some basic topology/differential geometry, but I'm looking for a little more direction than that so I know what to work towards.

Suppose I have a typical Cartesian xy coordinate system. Now suppose I replace the x-axis and the y-axis with curves (but just gentle curves, they still extend to infinity and do not intersect or anything crazy). Given the shapes of these new axes, is there some natural or "nice" way to define new coordinates for the entire 2D space? I realize you could probably do many things arbitrarily depending on what properties you want your coordinates to have (first thought that came to mind was to use the minimum distance to each curve), but I was wondering if people who are familiar with topology would know of any choices that are particularly intuitive or have particularly nice properties, and some literature behind them.

I'm actually interested in higher dimensional versions of this, where you deform all of the coordinate planes, but I can't even satisfy myself in the 2D case, so I thought I'd focus on that.

Sorry this is such a vague question, but I've been searching for anything related to this and I can't seem to find a foothold anywhere, except that I should learn some basic topology/differential geometry, but I'm looking for a little more direction than that so I know what to work towards.