I need to find a partition of $\mathbb{R}^n$ into smooth $(n-1)$ dimensional manifolds that is optimal in a sense that is not relevant to this post.
To do this practically on a computer, I can simply use the ansatz $f=\sum_i c_i\psi_i(x)$ for some basis functions $\psi_i\colon \mathbb{R}^n\to\mathbb{R}$, which determines a partition via the level sets $S_x=\{f=x\}$.
However, this ansatz is very underdetermined, so it is very difficult to perform the search for the optimum using this ansatz. For example, for $n=1$ there is only one possible partition of $\mathbb{R}$ into $0$-dimensional connected manifolds, namely $\mathbb{R}=\{\{x\}: x\in \mathbb{R}\}$, yet there are infinitely many (all monotonic) functions whose level sets yield this partition. (I don't really need the restriction to connected manifolds in the partition to be connected, this is just for illustratory purposes, but I do know that the optimal partition in my problem will consist of connected manifolds) As a way out, I could enforce the gradient of $f$ to have norm $1$. In one dimension, this would mean that the only allowed function would be $x\mapsto x$, which is exactly what I want. The problem with this is that I do not know how to parametrize functions with unit length gradient in higher dimensions, hence my question:
Is there a large family of functions $F\subset\{f\colon \mathbb{R}^n\to\mathbb{R}, |\nabla f|=1\}$ that can easily be parametrized on a computer? It is okay for $F$ to be a strict subset. Therefore, a start would be $F=\{a\cdot x, |a|=1\}$, but that's a bit too restrictive for my purpose.