Does a 3-Dimensional coordinate transformation exist such that its scale factors are equal?

Question

Let $\vec r=(x,y,z) $ be the position vector expressed in Cartesian coordinates. Let us define the coordinate transformation as

$\vec r(u,v,w)=(x(u,v,w),y(u,v,w),z(u,v,w)) $

The scale factors are defined by

$h_u=\vert \partial \vec r/\partial u \vert, h_v=\vert \partial \vec r/\partial v \vert, h_w=\vert \partial \vec r/\partial w \vert$

I wonder if a transformation can be defined such that

$h_u=h_v=h_w$

Now a pair of examples in the two dimentional case.

The transformation between elliptic and cartesian coordinates:

$\vec r(u,v)=(cosh(u)cos(v)/2,sinh(u)sin(v)/2) $

$h_u=h_v=\sqrt{cosh^2(u)-cos^2(v)}/2$

The transformation between parabolic and cartesian coordinates.

$\vec r(u,v)=((u^2-v^2)/2,u v) $

$h_u=h_v=\sqrt{u^2+v^2}$

Answer 1

Suppose we add another condition: not only $$\left\lVert\frac{\partial\mathbf r}{\partial u}\right\rVert=\left\lVert\frac{\partial\mathbf r}{\partial v}\right\rVert=\left\lVert\frac{\partial\mathbf r}{\partial w}\right\rVert,$$ but also $$\frac{\partial\mathbf r}{\partial u}\cdot\frac{\partial\mathbf r}{\partial v}=\frac{\partial\mathbf r}{\partial v}\cdot\frac{\partial\mathbf r}{\partial w}=\frac{\partial\mathbf r}{\partial w}\cdot\frac{\partial\mathbf r}{\partial u}=0.$$ That is, the coordinate system is orthogonal, which is usually desirable. (In particular, both of your two-dimensional examples have this property.) This means that the Jacobian matrix of the transformation is a multiple of the identity, and the transformation $(u,v,w)\mapsto(x,y,z)$ is a conformal map.

Liouville's theorem states that in three or more dimensions, all such maps are compositions of translations, similarities, orthogonal transformations and inversions. So the space of such coordinate systems is much more restricted than in two dimensions. Nevertheless, we do have a non-Cartesian example, namely inversion: $$(x,y,z)=\frac{(u,v,w)}{u^2+v^2+w^2}.$$ This has $\lVert\partial\mathbf r/\partial u\rVert=\lVert\partial\mathbf r/\partial v\rVert=\lVert\partial\mathbf r/\partial w\rVert=1/(u^2+v^2+w^2)$.