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Subdivision surface 201
Overview
The subdivision surfaces are defined recursively. The process starts
with a given polygonal mesh. A refinement scheme is then applied to
this mesh. This process takes that mesh and subdivides it, creating new
vertices and new faces. The positions of the new vertices in the mesh
are computed based on the positions of nearby old vertices. In some
refinement schemes, the positions of old vertices might also be altered
(possibly based on the positions of new vertices).
This process produces a denser mesh than the original one, containing
more polygonal faces. This resulting mesh can be passed through the
same refinement scheme again and so on.
The limit subdivision surface is the surface produced from this process
being iteratively applied infinitely many times. In practical use
however, this algorithm is only applied a limited number of times. The
limit surface can also be calculated directly for most subdivision
surfaces using the technique of Jos Stam, [1] which eliminates the need
for recursive refinement.
Refinement schemes
First three steps of Catmull–Clark subdivision of
a cube with subdivision surface below
Subdivision surface refinement schemes can be broadly classified into two categories: interpolating and
approximating. Interpolating schemes are required to match the original position of vertices in the original mesh.
Approximating schemes are not; they can and will adjust these positions as needed. In general, approximating
schemes have greater smoothness, but editing applications that allow users to set exact surface constraints require an
optimization step. This is analogous to spline surfaces and curves, where Bézier splines are required to interpolate
certain control points, while B-splines are not.
There is another division in subdivision surface schemes as well, the type of polygon that they operate on. Some
function for quadrilaterals (quads), while others operate on triangles.
Approximating schemes
Approximating means that the limit surfaces approximate the initial meshes and that after subdivision, the newly
generated control points are not in the limit surfaces. Examples of approximating subdivision schemes are:
• Catmull and Clark (1978) generalized bi-cubic uniform B-spline to produce their subdivision scheme. For
arbitrary initial meshes, this scheme generates limit surfaces that are C 2 continuous everywhere except at
extraordinary vertices where they are C 1 continuous (Peters and Reif 1998).
• Doo–Sabin - The second subdivision scheme was developed by Doo and Sabin (1978) who successfully extended
Chaikin's corner-cutting method for curves to surfaces. They used the analytical expression of bi-quadratic
uniform B-spline surface to generate their subdivision procedure to produce C 1 limit surfaces with arbitrary
topology for arbitrary initial meshes.
• Loop, Triangles - Loop (1987) proposed his subdivision scheme based on a quartic box-spline of six direction
vectors to provide a rule to generate C 2 continuous limit surfaces everywhere except at extraordinary vertices
where they are C 1 continuous.
• Mid-Edge subdivision scheme - The mid-edge subdivision scheme was proposed independently by Peters-Reif
(1997) and Habib-Warren (1999). The former used the mid-point of each edge to build the new mesh. The latter