# How to morph a enclosing mesh according to the enclosed mesh?

I am quite new to meshing and mesh manipulation. I am working on a problem consiting of meshes $$A$$, $$B$$, and $$C$$. The mesh $$C$$ completely encloses the meshes $$A$$ and $$B$$ as shown in the attached figure. I applied a rigid rotation to the mesh $$B$$, and want to morph the mesh $$B$$ accordingly. I have provided a simplified version of the meshes here as cylinder. (They can be complicated.)

Lets combine the meshes $$A$$ and $$B$$ into one mesh (say $$AB$$). For each point $$P_0$$ on $$C$$, I calculated their closest distances - $$d_{AB}$$ (and the corresponding faces $$F_{AB}$$) from the mesh $$AB$$. I also calculated and stored the translation vector of each point of $$AB$$ from its initial and final points positions.

Since for each $$P_0$$, I have the face $$F_{AB}$$, and for each of its vertices, I have a translation vector; I averaged these translation vectors for the face $$F_{AB}$$. Now, I have a translation vector $$T_{{P}_0}$$ for each point $$P_0$$ of $$C$$.

Now I created a translation vector $$T$$ for each $$P_0$$ as a function of $$d_{AB}$$ and $$T_{{P}_0}$$ such that $$\lvert{T}\rvert \propto d_{AB} T_{{P}_0}$$.

This is a very naive solution and didn't work and also cannot work for more complicated meshes of such kind. I am starting to learn some geometrical algorithms. I am working with VTK (Visualization Toolkit Library) to work with the meshes.

It will be very helpful if I get some resources for solving this problem. Also, can we get point to point correspondences between the meshes? If yes, then how can we approach the problem with that ?

Since, I have only started learning computer graphics, please let me know if the solution I have is utterly wrong. Thanks in advance. Following are the sample images for initial and desired orientations.