Subdivision surface

Remarks

Example: the curve analogue, Chaikin's scheme, maps a polygon with vertices $P_i$ to the polygon with vertices $\tfrac34 P_i + \tfrac14 P_{i+1}$ and $\tfrac14 P_i + \tfrac34 P_{i+1}$. These are affine combinations with weights summing to $1$ that depend only on adjacency, so clauses 2 and 3 hold. The limit curve is the uniform quadratic B-spline with control points $P_i$. The surface schemes of Doo–Sabin, Catmull–Clark and Loop are subdivision schemes in this sense; on regular parts of the mesh their limits are biquadratic, bicubic and box-spline surfaces. Non-example: the rule $P \mapsto 2P$ applied to all vertices has weights summing to $2$, so clause 2 fails. The meshes grow without bound, and the result depends on the choice of origin. The condition $\sum_v w_v = 1$ makes the scheme commute with affine maps of space, so the limit surface does not depend on the coordinate system. Near extraordinary vertices, those with a nonstandard number of neighbors, the smoothness of the limit is studied through the eigenvalues of the local subdivision matrix.