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Re: Trimmed NURBS and subdivision

Posted by Scott Schaefer on January 19, 2005 at 15:16:05:

In Reply to: Trimmed NURBS and subdivision posted by Jianbing Huang on January 19, 2005 at 00:38:13:

: Considering the fact that NURBS has been the de-facto standard representation for 3D models for a long time, doesn't it make sense to think about the differences and relations between NURBS and subdivision? For example, what is the advantages/disadvantages of subdivision over NURBS and how the conversion can be done between them.

: Subdivision representation seems to be much more compact than trimmed NURBS representation: no trimming is necessary, control vertices are shared across the trim boundaries, and uniform knot vector may be ignored assuming uniform subdivision scheme.

Subdivision surfaces and NURBS share a common ancestor... namely the tensor product, uniform cubic B-spline. NURBS really only come in four-sided patches due to their reliance on tensor product structures (I'm excluding things like S-patches for now). NURBS patches are easy to reason about because they're simply rational polynomial patches of a fixed degree.

Subdivision surfaces, on the other hand, leave the world of B-splines behind much faster. These surfaces are uniform B-splines (usually cubic) in tensor product regions of the mesh, but they become non-polynomial at extra-ordinary vertices. At these vertices the surface continuity usually drops to C^1. However, no explicit manipulation of the controls points is necessary to achieve this smoothness. In contrast, NURBS have many constraints on the control points at these extra-ordinary vertices to generate a smooth surface.

: However, there seem to be some issues. For example, is the subdivision always more compact. It is possible that subdivision may not be so compact for some shapes like a plane face with 1000 holes. While this face can represented as a trimmed NURBS with 1000 trimming curves, the control mesh for such simple shape clearly needs to have thousands of vertices. Also, current method of representing crease edges by adjusting local subdivision rules seems to be ad-hoc, while non-uniform rules with weights would compromise the size advantage. Exact representation of quadric surfaces may be realized by adding a parameter in addition to uniform subdivision - is there any link between this parameter and nurss? Nice representation of crease edges and quadric surfaces is critical for the adoption of subdivision scheme in some application areas (such as CAD).

Here you're really lamenting the fact that subdivision schemes do not have trimming operators. Notice that Peter Schroder and Adi Levin have worked on this problem and published "Trimming for Subdivision Surfaces" to help address this very issue. Nonetheless, trimming is certainly more difficult with subdivision surfaces. However, the notion of creases and sharp curves has been incorporated into subdivision surfaces as well and has made them a viable modeling tool for the animation industry.

Subdivision schemes can also exactly represent different degree polynomial surfaces. However, they do so only in the tensor product setting. The most common degree is 3 for subdivision surfaces, but you can build schemes that reproduce polynomials of any fixed degree (though the support for the rules may become large).

: Can somebody point out relevant work that may help me to further my understanding here? Thanks.

What in particular are you interested in for subdivision surfaces? I already included a pointer to trimming with subdivision surfaces. If there's something else you're interested in, just ask and I'll see what I can do.

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