# approximating a surface with a spline

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Hi everyone, I have a problem that I have been thinking about for some time. I, of course, do not expect a solution from here but wondering if people have any ideas about papers or work that deals with it. So, basically what I want to do is represent a delienated surface with a spline. So, I have a collection of points that represent a closed polygon. What I would like to do is closely represent that polygon with a spline with as few control points as possible. So it is basically decimation of the points and still be able to represent the surface boundary accurately. I am working on an editor that lets you edit these surfaces and visualize and edit them in an uncluttered way. So, I was wondering if people know of some papers, books or existing work that has been done in this area. Many thanks. I look forward to hearing from you folks. Cheers, xarg

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Hey Rob,

Yes, the class of spline is not important. The problem is basically that given a lot of points, how to generate a spline that passes through all of them, yet keep the number of control points as less as possible... I am not sure if I have stated the problem clearly enough...

Cheers,
xarg

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You may be interested in the oscillation theorem. There are many variants, but the basic 'continuous real-valued function' version roughly says that the best order n L polynomial approximation to a curve will be such that its error oscillates n-1 times between +E and -E, where E is the maximum error on the interval. It sounds like a mouthful, but it's fairly intuitive when you break the expression up and think about it.

This theorem directly leads to the exchange algorithm, which provides an iterative method of refining a spline-approximation by repositioning the abscissas. The theorem translates quite directly to approximation of surfaces, and I would be surprised if there is no way to adapt it to efficiently regress a surface with the node count as a parameter.

I also recommend you take a look at Numerical Recipes, as I recall it has a fairly strong section on this sort of business.

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Thanks a lot for this. I will look into this soon!

Cheers,
xarg

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