"Rigorous" definition of Cartesian coordinates I, like most, first learned about Cartesian coordinates very early on in my educational career, and so the most instructional way to think about them was that you place down some perpendicular lines and measure the perpendicular distance from each line to get your coordinates; in other words, an operational definition. Now that I'm much farther along in my education, though, I wonder whether or not there's another, more "rigorous" way to define Cartesian coordinates, perhaps using the idea of a coordinate system as a mapping? Or is my question a bit pointless, since we already have the operational definition?

I'm also surprised not to find any definition of Cartesian coordinate, so let's suggest this (personal and most simple) definition:

A cartesian coordinate on a Euclidean space E (finite dimensional $\mathbb{R}$ vector space, with a scalar product) is a "global chart" = Euclidean space isomorphism = isometry from $E$ to $\mathbb{R}^n$ 

