Doo–Sabin subdivision surface

In 3D computer graphics, a Doo–Sabin subdivision surface is a type of subdivision surface based on a generalization of bi-quadratic uniform B-splines, whereas Catmull-Clark was based on generalized bi-cubic uniform B-splines. The subdivision refinement algorithm was developed in 1978 by Daniel Doo and Malcolm Sabin.^[1][2]

A Doo-Sabin mesh after 2 levels of refinement. The new faces come from vertices, edges and faces of the original mesh (colored dark, white, and midtone respectively).

The Doo-Sabin process generates one new face at each original vertex, ⁠${\displaystyle n}$⁠ new faces along each original edge, and ⁠${\displaystyle n^{2}}$⁠ new faces at each original face. A primary characteristic of the Doo–Sabin subdivision method is the creation of four faces and four edges (valence 4) around every new vertex in the refined mesh. A drawback is that the faces created at the original vertices may be triangles or n-gons that are not necessarily coplanar.

Evaluation

Doo–Sabin surfaces are defined recursively. Like all subdivision procedures, each refinement iteration, following the procedure given, replaces the current mesh with a "smoother", more refined mesh.^[2] After many iterations, the surface will gradually converge onto a smooth limit surface.

Just as for Catmull–Clark surfaces, Doo–Sabin limit surfaces can also be evaluated directly without any recursive refinement, by means of the technique of Jos Stam.^[3] The solution is, however, not as computationally efficient as for Catmull–Clark surfaces because the Doo–Sabin subdivision matrices are not (in general) diagonalizable.

Two Doo–Sabin refinement iterations on a ⊥-shaped quadrilateral mesh

See also

• Expansion (equivalent geometric operation) - facets are moved apart after being separated, and new facets are formed
• Conway polyhedron notation - a set of related topological polyhedron and polygonal mesh operators
• Catmull-Clark subdivision surface
• Loop subdivision surface