The rules to assign new vertex positions here only corresponds to interior points of an open mesh. For those vertices and edges on the boundary, the calculation of weights is slightly different. The position of old vertices is calculated as (1−2∗β)∗v+β∗v1+β∗v)(1 - 2*\beta)*v + \beta*v_1 + \beta*v)(1−2∗β)∗v+β∗v1+β∗v) with v1v_1v1 and v2v_2v2 as the nearest vertices on the boundary. More reading: http://www.pbr-book.org/3ed-2018/Shapes/Subdivision_Surfaces.html.
I wonder if I understand the bottom diagrams correctly. We create new vertices in the middle of the edge, and three for each triangle? And we then update the white vertices on the left. Since the old vertices on the boundary might be a center vertex of another hexagon, what are the orders to update all such vertices within a mesh?
Do we always define the triangle mesh using equilateral triangles? In all the examples so far, the triangles have been equilateral and the weighting seems to be specific to equilateral triangles as well.
I am still quite confused here for the bottom two diagrams. The left diagram tells that when we first create two trangles, we will mark this new vertice on the edge, and when another new traingle coming, there will be another new vertice on another edge. Then the bottom right diagram is telling when a vertice has one more degree (update once degree is updated, or?),then the vertice's position will also update?? Can someone tell whether I approach this correctly or not?
Yes, the bottom left diagram is describing how the new vertices in subdivision are getting created. The bottom right diagram describes how the old vertices shift when we apply subdivision! Hope that helps