To do the knot refinement, one important thing is to not change the geometry. Also, it is done in the parametric domain. However, for Catmull–Clark subdivision surface, it is done on the physical domain, and the geometry is changed. I am wondering why the Catmull–Clark subdivision surface called refinement scheme.
Note, if Catmull–Clark subdivision is done on the control points. Then my question is whether the geometry changes combining the basis function.