I recently came to know about the orthogonal parametrization of a surface, for which $F={\bf X_u}\cdot{\bf X_v}=0$ and $E={\bf X_u}\cdot{\bf X_u}=G={\bf X_v}\cdot{\bf X_v}$. Here, $(E,F,G)$ denote the coefficients of the first fundamental form of a surface $S:={\bf X}(u,v)$.
According to the discussion of this thread, it is always possible to parameterize a regular surface $S$ (2 dimensions) via isothermal coordinates that makes the parametrization orthogonal. I went through the wikipedia link suggested in this discussion, but it does not illustrate the principle with a concrete example like: orthogonal parametrization of an ellipsoid/hyperboloid etc. in terms of the isothermal coordinates.
If anyone here can refer me to some book/online resources to see such an example, that will be very helpful. The standard parametrization of ellipsoid ($a\sin\theta\cos\phi,b\sin\theta\sin\phi,c\cos\theta$) is not an orthogonal one, that I have checked. So, there must be some parametrization which I did not come across so far. Thanks in advance to anyone who can point me to some direction.
P.S. I am sorry if my question sounds stupid; I am an experimental physicists trying to learn few aspects of differential geometry.
Best Answer
Existence of isothermal parameters on a general surface is not explicit, and examples are rather hard to come by.
As in the answer linked by M.V., stereographic coordinates on the sphere are isothermal. The Poincaré metric on the open unit disk is conformally flat; that is, Euclidean coordinates are isothermal.
On an ellipsoid, or a hyperboloid of one or two sheets, there are explicit isothermal coordinate systems coming from confocal quadrics. On an ellipsoid, the coordinate curves look like this:
(Image from http://www.math.rug.nl/models/Serie_XVI/nr1.jpg)
It's a straightforward calculus exercise to construct isothermal parameters on a surface of rotation, or more generally in a "Clairaut patch", a coordinate system $(u, v)$ in which the metric is written $$ E(u)\, du^{2} + G(u)\, dv^{2}. $$ The construction involves inverting a definite integral, however, just as with finding explicit arc length parametrizations, so even for explicitly-parametrized surfaces there's no guarantee of explicit isothermal parameters.