For twenty years we have lived with the rectangular tyranny of NURBS. We now have a couple of surface technologies which look as if they could realistically liberate us from it. Both offer NURBS-like definition in terms of a polyhedron with control vertices not necessarily lying on the surface, but permitting irregular vertices and faces while retaining the guarantee of the defined surface having minimum continuity properties.

One of these is the technology of subdivision surfaces, which have been the subject of increasing interest since they were used for major film animations.

The other is the technology of tensor-product-border patches, based essentially on the theory of toric varieties.

There are also hybrids in which subdivision is used to isolate the extraordinarities before patches are fitted, and variants in which the n-sided tensor-border patch is built from n quads meeting at a central vertex.

What all of these offer is the ability to make the control flow follow the features in the surface without loss of convergence or continuity. What they have in common is the operational process in which a relatively coarse polyhedron can be manipulated to give control over a smooth surface, and the broad B-spline-like nature of that control.

In all cases, we know a fair amount about how the surface is related to the polyhedron, but this is only the second half of the story. The first half is the question of how to get a good polyhedron from the desired surface, and this has received much less attention. It is time to rectify this.

**Devise an algorithm which, given a smooth bivariate manifold with or without boundary (there are two sub-challenges here), represented in any of the standard surface representations, will produce a good coarse polyhedron.**This description is both sharp and woolly at the same time. This is deliberate, since the objective of the challenges is to stimulate the creation of lots of good new ideas. I am already aware of at least three plausible approaches, but there may be many more.

The description of the input data merely says that it is some representation which an algorithm can get its teeth into, of a surface. Because we already have ways of converting each of the standard representations into the others, if a good method is proposed with any one format of starting data it can be applied to data in any other.

The question of what is a good polyhedron is similarly loose. In my mind a good algorithm will produce a polyhedron which gives good accuracy with few control points. A surface with a lot of detail will require a lot of control points, one with very little should require only a few.

I do not specify which method is to be used to convert the polyhedron back into a surface for comparison with the original. To a first order it doesn't matter.

**Devise a procedure which can be applied to the mental image of a surface in the head of a designer, to give a good coarse polyhedron.**This is probably the more important of the two. Those of us in the NURBS correction business know what a lot of terrible NURBS have been designed because we have not had good procedures for getting good polyhedra from mental images. There has been no widely publicised street wisdom.

2002: Think about the challenges and invent algorithms and procedures.

2003: Implement them and experiment. Submit a paper to the Tromso meeting.

2004: Present your ideas at the Tromso meeting.

Then in 2006 I will persuade somebody to make a survey and present, at the next AFA Curves and Surfaces meeting, some comparison and maybe even some unification of the ideas presented.

This page is maintained by Malcolm Sabin. email

It was last updated on 24th July 2002