When I was at school I learned about "implicit coordinates" for curves in a plane. Essentially these mapped the path in terms of arc length $s$ from a fixed point, and direction of travel $\psi$ measured against a fixed line (so the curve has to be sufficiently smooth). The point was that these data could be measured by a person travelling along the path. They were handy for computing curvature, and solving chase problems, I recall.
A question was posed about curves parameterised by arc length, which put me in mind of this. I did a search online, as I wanted to mention it in a comment, and came up with nothing resembling what I remembered.
I may have the name wrong, of course, or such co-ordinates may now be called something else. But a good reference would be helpful.
As noted in comments below, the term is intrinsic coordinates. Good references still welcome, though the correct terminology does open up various online resources. I learned about these at school, and afterwards they went nowhere, or so it seemed. Yet they always seemed interesting. I suppose the way that Riemann Surfaces are made up from local bits (the direction gives you a piece which is locally nearly straight) - and generalisations of the same in various manifolds and geometries are where it went. Yet the particular expression here, which is effectively how to map a curve using a car with a milometer and a compass always seemed very real.