Subdivision schemes work by considering the vertices and their connectivity information to calculate averaging weights.
However, other than specifying which vertices are connected, and perhaps which weights to use (in the case of loop subdivision for example), edges don;t really play a role.
Same as faces. I am wondering if there is a subdivision scheme that attaches "weights" (and weights could be either scalars or matrices or similar vector fields) and has the faces and edges have a controllable effect on the convergence.
For example in the case of meshes, consider a cube. If the weights of the faces are 0, then the subdivision is just a regular subdivision. But if the weights of each face go towards infinity, then the limit of the convergence should be the original cube.
I am trying to find something like this, but I cannot find any associated literature with the topic.