Is there a reason why we aren't assigning equal weights (e.g. 1/4 for the diamond-shaped case) for the neighboring vertices to interpolate the new position for the center vertex?

@SeanW0823 Good question! I would bounce this question back: why would you expect the weights to be equal?

@SeanW0823, not sure if I am correct, but I think it ultimately boils down to this new vertex is not the same distance from all the other vertices. Conceptually speaking, the new vertex should be affected more by the closer vertex, and less by the further away ones. I am not sure if my intuition is correct.

If the neighboring vertices were equally distant from the center vertex, then I would reason that because no neighboring vertex should be more significant than the other so they should be assigned equal weights. Therefore, I agree with geos98's comment that maybe we have to assign unequal weights for the case of a diamond-shaped polygon.

This article seems to show a formula for the weights. There are multiple options but I guess this one just works well so it is used:

https://graphics.stanford.edu/~mdfisher/subdivision.html

I found that the loop subdivision is based on quartic box spline