Why use Subdivision?
Today's geometric modeling systems almost always use NURBS
representations (non-uniform rational B-splines) for freeform surface
design. Such surfaces consist of arrangements of piecewise polynomial
patches, each parameterized over the unit square. Unfortunately, NURBS
have great difficulty with arbitrary topology surface modeling: when
assemblying a number of such patches to form a more complex shape,
cross boundary continuity conditions have to be carefully
managed. This is easy to do when four rectangular patches meet at a
corner, but fundamental problems arise when other than four patches
meet. Such corners are referred to as extraordinary vertices. The
Euler characteristic of a graph tells us that such vertices cannot be
avoided. A similar constraint holds when modeling with triangular
patches and other than six of them meet at a corner.
Subdivision addresses these problems in an elegant way (if you want to
learn more, take a look at the Subdivision
Course Notes).
What is Subdivision?
Subdivision defines smooth surfaces as the limit of a sequence of
successively refined polyhedra. A given triangle (or quadrilateral) is
split through the insertion of new midpoints along the edges (and a
center point in the case of quadrilaterals). New point positions are
computed as weighted averages of nearby old point positions. Very few
assumptions have to be made about the global nature of the objects
modeled. Data structures need support only operations such as neighbor
finding in a graph. Special constraints, e.g., boundary conditions,
can be incorporated through local modification of the subdivision
weights. The weights themselves are typically derived from spline
based knot insertion algorithms.
Even though the surface is defined through a limit process, it has a
well defined parameterization and important functionals of the limit
surface, such as evaluation and partial differentiation, can be
computed exactly through eigen analysis.
The Bottomline
Subdivision schemes provide an unprecedented opportunity to unify
geometric modeling, physical simulation, and design under one basic
paradigm. Additionally, subdivision algorithms naturally support such
essentials as progressive transmission and single source scalable
content creation among many others. In the PC arena subdivision is
already taking over. We predict that this disruption of current
practice will extend all the way to high-end industrial CAD
applications.
To learn more about individual subdivision research components see Progressive Geometry Compression,Curvature Smoothness of Subdivision Surfaces, Dual Subdivision, and Trimming of Subdivision Surfaces
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