Subdivision surface

In computer graphics, subdivision surfaces are used to create smooth surfaces out of arbitrary meshes. Subdivision surfaces are defined as the limit of an infinite refinement process. They were introduced simultaneously by Edwin Catmull and Jim Clark, and by Daniel Doo and Malcom Sabin in 1978. Little progress was made until 1995, when Ulrich Reif solved subdivision surfaces behaviour near extraordinary vertices.

The fundamental concept is refinement. By repeatedly refining an initial polygonal mesh, a sequence of meshes is generated that converges to a resulting subdivision surface. Each new subdivision step generates a new mesh that has more polygonal elements and is smoother.

Subdivision methods

There are several refinement schemes:

Catmull-Clark is a generalization of bi-cubic uniform B-splines
Doo-Sabin is a generalization of bi-quadratic uniform B-splines
Loop, by Charles Loop, is a generalization of quartic triangular box splines (works with triangular meshes)
Butterfly named after the scheme's shape
Midedge
Kobbelt is a variational subdivision method that tries to overcome uniform subdivision drawbacks

Advantages over NURBS modelling

Subdivision surface modeling is now preferred over NURBS modeling in major modelers because subdivision surfaces have many benefits:

work with more complex topology
numerically stable
easier to implement
local continuity control
local refinement
no tesselation issue
switch between coarser and finer refinement

B-spline relationship

B-spline curves are refinable: their control point sequence can be refined and the iteration process converges to the actual curve. This is a useless property for curves, but its generalization to surfaces yields subdivision surfaces.

Refinement process

Interpolation inserts new points while original ones remain undisturbed.

Refinement inserts new points and moves old ones in each step of subdivision.

Extraordinary points

The Catmull-Clark refinement scheme is a generalization of bi-cubic uniform B-splines. Any portion of the surface that is equivalent to a 4x4 grid of control points represents a bi-cubic uniform B-spline patch. Surface refinement is easy in those areas where control points valence is equal to four. Defining a subdivision surface at vertices with valence other than four was historically difficult; such points are called extraordinary points. Similarly, extraordinary points in the Doo-Sabin scheme have a valence other than three.

Most schemes don't produce extraordinary vertices during subdivision.

External links

Resources about subdvisions
Geri's Game : Oscar winning animation by Pixar completed in 1997 that introduced subdivision surfaces (along with cloth simulation)
Subdivision for Modeling and Animation tutorial, SIGGRAPH 2000 course notes
A unified approach to subdivision algorithms near extraordinary vertices, Ulrich Reif (Computer Aided Geometric Design 12(2):153-174 March 1995)

3D computer graphics