I've a special question between mathematics and 3d. I struggle for two days understand relation between B-splines and Catmull-Clark subdivs. Everywhere wrote the Catmul-Clark subdivs is based on bicubic B-spline surface but I can't understand how. I can't find direct dependence. Only here I met a mention about three strange coefficient - 1/8,6/8,1/8, but I not seen that in B-spline theory. I did a experiment with cubic B-spline and three poligons with CC and they was equal with the same cage. I'm not expert in spline, but I seen many papers about that theme and I can't find info. :((
UPD For example this my comparision for 3 poligons with CC subdiv and cage (yellow) + not clamped (clamped has other curvature) B-spline degree 3 (black):
It looks as the same! But why?
The coefficients [1,6,1]/8 for even control points, as well as [4,4]/8 for odd ones, appear in the rows of the local subdivision matrix $H$, which is derived from the matrix form of the parametric bicubic B-Spline patch S(u,v). See the derivation in the original 1978 paper:
https://people.eecs.berkeley.edu/~sequin/CS284/PAPERS/CatmullClark_SDSurf.pdf
Eq. 1 shows the matrix form of the parametric definition of S(u,v), which is reformulated to a recursive (subdivision) formulation that maps a 4x4 control point matrix $G$ to a matrix of a refined control point mesh $G_1$ according to Eq. 3, using the local subdivison matrix $H$ (in the paper it's called "splitting matrix").
This matrix also appears (in normalized form including the factor $1/8$) in the Stanford lecture slides you linked (Slide 74).