i
i
i
i
i
i
i
i
616 13. Curves and Curved Surfaces
Figure 13.38. A tetrahedron is subdivided five times with Loop’s, the
√
3, and the
modified butterfly (MB) scheme. Loop’s and the
√
3-scheme are both approximating,
while MB is interpolating.
The normal is then n = t
u
× t
v
. Note that this often is less expen-
sive [1415] than the methods described in Section 12.3, which need to com-
pute the normals of the neighboring triangles. More importantly, this gives
the exact normal at the point.
A major advantage of approximating subdivision schemes is that the
resulting surface tends to get very fair. Fairness is, loosely speaking, re-
lated to how smoothly a curve or surface bends [903]. A higher degree of
fairness implies a smoother curve or surface. Another advantage is that ap-
proximating schemes converge faster than interpolating schemes. However,
this means that the shapes often shrink. This is most notable for small,
convex meshes, such as the tetrahedron shown in Figure 13.38. One way
to decrease this effect is to use more vertices in the control mesh. i.e., care
must be taken while modeling the control mesh. Maillot and Stam present
a framework for combining subdivision schemes so that the shrinking can
be controlled [809]. A characteristic that can be used to great advantage
at times is that a Loop surface is contained inside the convex hull of the
original control points [1413].
The Loop subdivision scheme generates a generalized three-directional
quartic box spline.
6
So, for a mesh consisting only of regular vertices, we
could actually describe the surface as a type of spline surface. However,
this description is not possible for irregular settings. Being able to generate
smooth surfaces from any mesh of vertices is one of the great strengths
of subdivision schemes. See also Sections 13.5.5 and 13.5.6 for different
extensions to subdivision surfaces that use Loop’s scheme.
13.5.2 Modified Butterfly Subdivision
Here we will present the subdivision scheme by Zorin et al. [1411, 1413],
which is a modification of the butterfly scheme by Dyn et al. [291], and
6
These spline surfaces are out of the scope of this book. Consult Warren’s book [1328],
the SIGGRAPH course [1415], or Loop’s thesis [787].