In Cartesian coordinates on Euclidean space, the coordinates are essentially distances. If we move from the point (x,y) to (x+s,y), or to (x,y+s), we travel a distance s.

For coordinates on curved spaces, this is hardly ever true! Using latitude and longitude on a sphere a given change in latitude always carries you the same distance along a meridian, but a change in longitude takes you a *different* distance around different circles of latitude.

What about the hyperbolic plane? There are many ways of drawing the hyperbolic plane and choosing coordinates for it, but none of the usual ones measure distances.

The first image here shows the disk model of the hyperbolic plane, and a grid of coordinates, (u,v), with

(x,y) =
(2(u-v), cosh(u+v)^2 + (u-v)^2 - 1) /
[(1+cosh(u+v))^2 + (u-v)^2]

The grid doesn’t fill the whole plane, but in the region covered, changing either u or v by s will carry you precisely a distance s along a grid line.

I found these coords by mapping the “asymptotic curves” on the pseudosphere [second image] onto the disk model.

https://en.wikipedia.org/wiki/Asymptotic_curve

These are curves with the property that every plane spanned by their tangent at a point, and the normal to the surface, slices the surface along a line with zero curvature at that point.

But it’s still a bit mysterious to me why these curves can form a distance-based grid.