# What are the four rules, that the 2D subdivision matrix describes?

I got the following 2D subdivision mask (which is used to subdivide a given quad-mesh):

$$M = \frac{1}{8} \begin{bmatrix} 1 & 2 & 1\\ 3 & 6 & 3\\ 3 & 6 & 3\\ 1 & 2 & 1\\ \end{bmatrix}$$ Now I need to know, which four rules this matrix describes, to compute one new vertex for each rule. I know, that I need to have a look at the odd/even (row/column) indices in the matrix.

Row Column Rule
Odd Odd $$M = \frac{1}{8} \begin{bmatrix} 1 & X & 1\\ X & X & X\\ 3 & X & 3\\ X & X & X\\ \end{bmatrix}$$
Odd Even $$M = \frac{1}{8} \begin{bmatrix} X & 2 & X\\ X & X & X\\ X & 6 & X\\ X & X & X\\ \end{bmatrix}$$
Even Odd $$M = \frac{1}{8} \begin{bmatrix} X & X & X\\ 3 & X & 3\\ X & X & X\\ 1 & X & 1\\ \end{bmatrix}$$
Even Even $$M = \frac{1}{8} \begin{bmatrix} X & X & X\\ X & 6 & X\\ X & X & X\\ X & 2 & X\\ \end{bmatrix}$$

Can someone help me and explain, what the rules mean? I already encountered a subdivision mask (square matrix) of size 5x5 and where I knew, what the rules mean. For example with

$$M = \frac{1}{64} \begin{bmatrix} 1 & 4 & 6 & 4 & 1\\ 4 & 16 & 24 & 16 & 4\\ 6 & 24 & 36 & 24 & 6\\ 4 & 16 & 24 & 16 & 4\\ 1 & 4 & 6 & 4 & 1\\ \end{bmatrix}$$ the odd/odd index rule defines the calculation of a new position of the given vertices of the mesh. The odd/even rule defines the calculation of new vertices on the horizontal edges between vertices, while the even/odd rule defines the calculation of new vertices on the vertical edges. The even/even rule is then important for placing new vertices in the middle of the quad faces of the mesh.

But with a matrix, which is not squared, I am a bit confused, how to read the rules.

• Where did you get these rules? It should be explained in your text. What are you trying to achieve here ? Sep 14 at 23:39
• I tried to concretize the question. Maybe now it is a bit clearer. Sep 15 at 18:41
• computergraphics.stackexchange.com might be a better place to ask. Sep 17 at 14:27