Weighted average of original vertices in a Loop's Subdivision Mask

Question

In Loop's Subdivision, I don't understand what is often written as "Choose locations for new vertices as weighted average of original vertices in local neighbourhood".

The new vertices are derived from adding new points in the middle of each edge. How is this related to a "weighted average" of the original vertices? How do the

In the first place, what is "weighted average of original vertices"? Where is the "weight" seen in the mesh?

Answer 1

Adding a point in the middle of an edge is actually just a first phase of the process. Subdivision algorithms usually have two:

1. Topology phase - adding new vertices and edges
2. Geometry phase - adjusting the newly added vertices so that they "form a smoother surface" with their neighbors

The following slides give some more details on the weight scheme (the beta parameter).