Binary Subdivision

Quadratic Subdivision

Recursive Subdivision

Adaptive Subdivision


Binary Subdivision --

Similar to the curves, we can refine the knots to subdivide a surface. Let us look at the knots in U direction. We can insert knot any where along an iso-parametric curve, but preserve exactly the same point set as the surface before knot refinement. A new mesh is to be calculated for the new knot vector. It can be done by treating the mesh as rows of control polygons of curves. When a knot is inserted here, these polygons are replaced by the new ones, such that they form a new mesh (with eventually an extra column of vertices) for the new surface that has exactly the same point as the original surface. To subdivide this surface at this knot position, we need to keep inserting knots here, until the multiplicity at this knot is equal to the order, which is three here. The last knot insertion actually created a double mesh column. This surface can be subdivided by splitting the mesh along this double column. Knot refinement and surface subdivision in V direction can also be done along an iso-parametric curve. We keep inserting knots at this position until its multiplicity equals to the order, which is four in this case. This surface is further subdivided in V direction. The same surface could have been subdivided in Vfirst, and then in U.

Quadratic Subdivison --

Subdivision could have been quadruple instead of binary.

Recursive Subdivision

In fact, surface subdivision can be done recursively, that is, a surface is subdivided into four subpatches, and each sub-patch, as a well defined surface itself, can be recursively subdivided into smaller patches, and so on, and so forth.

Adaptive Subdivision --

Certainly the subdivision doesn't have to be quadruple all the time. For instance, if the aspect ratio is important, we need to subdivide in one direction for a few times, then mixed with subdivisions in the other direction. Sometimes, we put different subdivision criteria together to determine the direction of subdivisions and the level of recursion. This is called the adaptive recursive subdivision. It is usually coupled with a well-known divide-and-conquer strategy which is very popular for display processing as well as geometric processing.