# Coordinates in a torus eversion

(This may be a somewhat vague question.)

If the circle $(y-R)^2+z^2=1$ ($R>1$) is revolved about the $z$-axis, the surface generated is a torus that can be parametrized by longitude $\alpha$ and latitude $\beta$ as follows: \begin{align} x & = (R+\cos\beta)\cos\alpha, \\ y & = (R+\cos\beta)\sin\alpha, \\ z & = \sin\beta. \end{align} The names of the two parameters make sense since curves of constant latitude are circles parallel to each other and curves of constant longitude are circles in planes intersecting in a line through the center of the torus and orthogonal to planes containing circles of constant latitude.

Now consider the surface \begin{align} u & = (R-\cos\alpha)\cos\beta, \\ v & = (R-\cos\alpha)\sin\beta, \\ w & = \sin\alpha. \end{align} On this surface $\beta$ is the longitude and $\alpha$ is the latitude rather than vice-versa. A one-to-one correspondence between this surface and the other one is identified by this transposition of variables. We have turned the torus inside-out (or "everted" it, if you like that word). Also, the minus sign where a plus sign had appeared means that a point on the inner equator is now on the outer equator (this aspect of the situation didn't need to be there to justify the conclusion that we have turned the torus inside-out, but I want it because of a geometric problem I'm thinking about).

My question is whether some particular one of the six correspondences between the variables $x,y,z$ and the variables $u,v,w$ should be preferred by reason of convention or of aesthetics or of mathematics?