Tensor product non-uniform rational B-splines (NURBS) have become the
standard representation for complex smoothly varying surfaces.
However, a major drawback of NURBS is the requirement that control
nets consist of a regular rectangular grid of control points.
Subsequently, a NURBS surface can only represent limited surface
topologies (planes, cylinders, and tori), they lack a local refinement
procedure, and surface discontinuities cannot be locally introduced or
Subdivision surfaces -- defined as the limit of an infinite refinement
process -- overcome many of these deficiencies. For instance, the
images below show an initial control mesh, the mesh after one refinement
step, after two refinements, and in the limit of infinite refinement,
Subdivision surfaces are easy to implement, they can model surfaces of
arbitrary topological type, and as shown above, the continuity of the
surface can be controlled locally. Although subdivision surfaces have
been known for nearly fifteen years, their use has been hindered by
the lack of a closed form -- they are defined only as the limit of an
We believe that subdivision surfaces are a likely candidate to replace
NURBS in future graphics and CAD/CAM applications. Our believe is
partly based on recent work
that shows how to compute exact properties (such as positions, normal
vectors, and thin plate energies) despite a closed form. We have also
discovered a connection between subdivision surfaces and
multiresolution analysis (aka, wavelets). See Lounsbery,
DeRose, and Warren and
Eck et al.
for more details.
- PIs: Tony DeRose,
John McDonald, Werner Stuetzle.
- Graduate students: Adam Finkelstein, Jean Schweitzer, Eric Stollnitz.
Multiresolution Analysis of Arbitrary Meshes,
M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and
W. Stuetzle, Technical Report #95-01-02, January 1995.
Smooth Surface Reconstruction, Hoppe et al., SIGGRAPH '94.
Reconstruction from Unorganized Points, Hugues Hoppe,
Ph.D. Thesis, 1994.
Analysis for Surfaces of
Arbitrary Topological Type, Michael Lounsbery, Ph.D. Thesis, 1994.
Condensed version available as TR 93-10-05b.