mRSED What is Subdivision?

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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