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In Loop's subdivision scheme, what do the variables $\alpha$ and $n$ refer to? Knowing what the variables refer to will help to derive the fractions around the vertices.

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But, what do these fractions around the vertices, such as the one above, meant for?

The first image (above) should be this:

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and shows the extraordinary case where the vertex point has valence $n$ ($n \neq 6$), i.e. there are $n$ points sharing an edge with it.

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The fractions are the masks for generates the new control points of the mesh, in the below left image you have the mask for generates the new edge point and in the right the mask for generates the new vertex points.

Each new control point is computed as a convex combination of "some" old control points in his neighborhood (that's why if you add all the fractions you will get 1). In the case of your images (below) related to Loop subdivision scheme, the new points are represented as black and the old points are the vertex that you see with fractions beside (which are the weights associated with each old vertex). Take into account that in the vertex point case, i.e. in the above image and the below right image, the old point and the new point are represented one above the other, because that you see a black point and a fraction together. Those are the geometric rules of subdivision, whereas the topological rules tell you how to connect those new points.

The below images are what is known as the regular case: each vertex point has six neighbors (sharing an edge) in the graph, while the above image represent the extraordinary case: a point with more than six neighbors. In an arbitrary triangular mesh only the rule for new vertex points needs to be changed, the rule for new edge points don't need to be changed and is the same as the regular case. That's why you have 1 image above (extraordinary vertex) and two images below (regular case: edge and vertex rules).

Please don't hesitate to ask if you keep with doubts and need more help, but I should tell that the Loop's MSc thesis is a nice document to read and learn! ;-)

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