Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 77 / 5 / i / Solution

The supplied subdivision mask describes the regular, valence-six triangular grid. Its coefficients split into four parity classes after binary refinement: one class for old vertices and three classes for the three edge directions. For an old vertex $V$ with cyclic neighbors $V_1,\dots ,V_6$, the even stencil isFor an edge with endpoints $A,B$ and opposite vertices $C,D$ in its two incident triangles, the new odd vertex isRotating this edge stencil supplies the other two odd parity classes. All stencil weights sum to one, giving affine invariance; their first moments place an affine regular grid at the old positions and edge midpoints. Apply the same weights separately to $x,y,z$. Boundary edges and extraordinary vertices need additional rules, which are not specified by this uniform mask. The subsequent arguments concern the regular interior grid and compatible extruded boundary treatment.