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 isoparametric 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 isoparametric
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 subpatch, 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 wellknown divideandconquer strategy which is very popular
for display processing as well as geometric processing.
