Regions represented as foliated forms and natural smooth maps onto them

The author is interested in regions surrounded by hypersurfaces and natural smooth maps onto them respecting the canonical projections of the unit spheres and so-called special generic maps and moment maps, more generally. We consider situations where these regions are foliated via 1-dimensional families of functions and their zero sets (smoothly).
We including the author are also interested in explicit and nice functions obtained by composing the canonical projections and their topological or combinatorial properties. This is of singularity theory of differentiable maps and applications to differential topology and various real geometry. Explicitly, here, as a new challenge, we discuss the 1st derivative of such a function and critical sets of this.