Let $\mathbf{x}:U\subset\mathbb{R}^2\longrightarrow\mathbb{R}^3$ be a regular parametric surface with the following properties:
- conjugate net: $\langle \mathbf{x}_{uv}, \mathbf{x}_u\times\mathbf{x}_v\rangle = 0$,
- planar coordinate curves: $\det\left(\mathbf{x}_u,\mathbf{x}_{uu},\mathbf{x}_{uuu} \right) = 0$ and $\det\left(\mathbf{x}_v,\mathbf{x}_{vv},\mathbf{x}_{vvv} \right) = 0$,
- orthogonal carrier planes: $\langle \mathbf{x}_u\times\mathbf{x}_{uu},\mathbf{x}_v\times\mathbf{x}_{vv}\rangle = 0$.
For simplicity, one would probably also require the nondegeneracy conditions:
$$\mathbf{x}_u\times\mathbf{x}_{uu} \neq 0\quad\quad\quad \mathbf{x}_v\times\mathbf{x}_{vv} \neq 0.$$
Example:
The scale-rotation surfaces:
$$
\mathbf{x}(u,v) =
\left[ \begin {array}{ccc}
\mu\left( v \right) \cos \left( \theta
\left( v \right) \right) &-\mu \left( v \right) \sin \left( \theta
\left( v \right) \right) &0\\ \mu \left( v
\right) \sin \left( \theta \left( v \right) \right) &\mu\left( v
\right) \cos \left( \theta \left( v \right) \right) &0
\\
0&0&1\end {array} \right]\,
\left[
\begin{array}{c}
f(u)\\
0\\
g(u)
\end{array}
\right],
$$
where $\mu: \mathbb{R}\longrightarrow \mathbb{R}^+ $ and $f(u) \neq 0 $,
fits into this class of surfaces.
My Question:
I would like to determine the parametric surfaces possessing the above properties. My knowledge of PDE is very limited and I would appreciate answers putting me on the right track.
1st Edit (July 2022):
Does a generic regular surface in $\mathbb{R}^3$, locally support such a parametrization?
Best Answer
Apparently, the answer to this question was more than just a simple quick one. One can find the answer in the following paper:
Isometric Deformations of Smooth and Discrete T-surfaces
In this paper the authors describe such surfaces in detail. They also mention their polygonal counterparts (as discrete surfaces) and study the isometric deformations of both scenarios.