Mid-edge subdivision

Mid-edge subdivision replaces each mesh edge by its midpoint, joins successive midpoints around every old face, and joins successive midpoints around every old vertex. For an interior manifold mesh, each new vertex has valence four. Each geometric update is a convex combination, so every refined control point remains in the convex hull of the original control mesh. The regular limit is associated with the quadratic four-direction box spline.

The descendant of a face with cyclic vertices $p_0,\dots ,p_{n-1}$ has vertices $p_i^{\prime }=(p_i+p_{i+1})/2$. Their arithmetic mean is unchanged. On the cyclic Fourier basis, a mode of frequency $j$ is multiplied by $(1+e^{2\pi ij/n})/2$. Every nonconstant mode has modulus less than one, so the descendant face shrinks to the original vertex centroid. This puts that centroid on the subdivision limit, without requiring the original face to be planar.