Where do the coefficients in the Catmull-Clark subdivision algorithm come from? I'm learning about subdivision surface algorithms. The "Catmull-Clark" algorithm seems to be one of the most widely-used classical algorithms.

The introduction of new face points and edges is straightforward, but I'm confused by the following formula for modifying the locations of original points

$$ \frac{F + 2R + (n-3)P}{n} $$

where:


$P$ is the original point
$R$ is the average of all $n$ edges midpoints touching $P$
$F$ is the average of all $n$ face points touching $P$


I understand the result is a weighted average of three points $F$, $R$ and $P$, but where do the coefficients $1$, $2$, and $(n-3)$ come from? Why $2R$? Why not $2.01R$? Or $1R$ and $2F$?

All the introductions I've seen just present the formula without presenting a justification.
The derivation is presented in the original paper that introduced CC subdivisions as a generalisation of B-Spline patches: https://people.eecs.berkeley.edu/~sequin/CS284/PAPERS/CatmullClark_SDSurf.pdf