Is there a way to specify position on a sphere's surface such that distance is constant?

Question

I know azimuth-elevation, or lat/long, works for defining locations on a sphere (such as a planetary body), but it has a problem where, the higher your latitude, the less actual distance is covered by a given change in longitude. Is there some coordinate system that allows you to define locations on the surface of sphere (such as a planetary body) in which, if any of the components of the coordinates of a point are changed, the actual distance the point moves is always a constant proportion of the change to the coordinate component? i.e. a change of X in the [blank]-coordinate will always result in a displacement of c * X, where c is some constant of proportionality.

Answer 1

Yes there are local coordinates with this property, but they are essentially absolutely useless.

Pick a totally convex ball $B$ on the sphere, and a point in this ball to be the origin $O$ of the coordinate system. Pick two nonparallel directions $\partial_1$ and $\partial_2$, and define $x^1,x^2$ axis in the obvious way following the geodesic. For every sufficiently small $x^1,x^2$, define the point $P$ with coordinates $(x^1,x^2)$ as the intersection (other than $O$) of the (spherical) circle of radius $x^1$ centred at point $(0,x^2)$ and the (spherical) circle of radius $x^2$ centred at the point $(x^1,0)$. Then this coordinate system, at least restrict to $(x^1,x^2)\in(0,\epsilon)\times(0,\epsilon)$, has the property you desired.

But as I said before, this is mostly useless. It doesn't tell you the angle $\partial_1$ and $\partial_2$ makes at any point inside this coordinate patch (indeed it must change with the point by Gauss-Bonnet), so there is no easy way to get the distance between $(x^1,x^2)$ and $(y^1,y^2)$ unless one coordinate is the same, or you go back and work in more reasonable coordinate system.