I am interested in what in computer graphics is called
*morphing* between two topologically equivalent shapes $S_0$
and $S_1$ in 3D.
This is a continuous "path" of shapes $S_t$, each embedded and
all with the same genus, for $t \in [0,1]$.
Let us assume the shapes are closed surfaces;
I am particularly interested in polyhedra, but likely could adapt from
a method for smooth surfaces.
For example, I would like to morph between these
two genus-7 polyhedra:

$V{=}30, E{=}84, F{=}42$

It would make a nice movie to show that the first is truly genus 7.

There is an extensive literature on this topic, as it is needed in many graphics contexts. A sample of some work in this area is provided in the references below. As far as I know, all have some ad hoc heuristics to obtain "nice" morphs. What I am seeking to learn here is whether there might be some attractive embedding theorems that could lead to a clean, perhaps more principled morph.

Here is what I have in mind, a simple algorithm for
*convex* shapes. Embed $S_0$ in the 3-flat of
$\mathbb{R}^4$ with 4-th coordinate $x_4=0$,
and embed $S_1$ in the 3-flat with $x_4=1$.
Let $H$ be the convex hull of $S_0 \cup S_1$ in $\mathbb{R}^4$.
Now intersect $H$ with $x_4 = t$ to obtain $S_t$.
Of course this only works for convex shapes.

Here, finally, is my question:

Is there some mapping that would send $S_0$ and $S_1$ into some space, and a definition of a canonical parametrized path between those shapes as endpoints, so that the intermediate shapes $S_t$ along the path (a) are all embedded in $\mathbb{R}^3$, (b) all have the same genus?

It is likely too much to hope for, but if there were a mapping/space combination that would map genus-$g$ shapes naturally to convex genus-0 shapes, then the above convex algorithm would apply. Or if there were a construct akin to the convex hull that accommodated nonconvexity and holes... I know my question is vague, but I hope its intent is clear. Any ideas, however speculative or tentative, would be appreciated. Thanks!

**References.**

[GSLML98] Gregory, State, Lin, Manocha, Livingston, "Feature-based surface decomposition for correspondence and morphing between polyhedra," 1998. link

[ACL00] Alexa, Cohen-Or, Levin, "As-rigid-as-possible shape interpolation," 2000. link

[KSK00] Kanai, Suzuki, Kimura, "Metamorphosis of Arbitrary Triangular Meshes," 2000. link