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Wa are talking about Non-uniform rational B-spline. We have some simple 3 dimentional array like

{1,1,1}
{1,2,3}
{1,3,3}
{2,4,5}
{2,5,6}
{4,4,4}

Which are points from a plane created by some B-spline

How to find controll points of spline that created that plane? (I know its a hard task because of weights that need to be calculated but I really hope it is solvable)

alt text For thouse who did not got idea of question - sory my writting is wwbad - we have points that are part of plane rendered here and we need to find controll points that form a spline which solution is that rendered plane.

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BTW: I am new here so feel free to edit my question and its tags. –  Ole J Dec 14 '10 at 12:22
    
Personally, I think whoever downvoted owes an explanation. Aside from the poster's non-native English, which is eminently fixable, this doesn't strike me as a bad question. On the other hand, I don't know enough of the words in the question to know this for sure -- hence the appropriateness of a comment accompanying a downvote. –  Cam McLeman Dec 14 '10 at 14:51
    
@Ole-J, have you looked at this problem in a smaller dimensionality as a starting point? What if you had a set of points in 2-d space $L=${$(x_1,y_1),(x_2,y_2),...(x_n,y_n)$} and were trying to find a B-spline that approximated those points $L$? Do you have a way to find the (or any) B-spline that could fit those points? This seems more like a question that is appropriate for stack-overflow in the current format of the question. It's not really an interpolation problem as a "reverse-mapping" or "best-fit" problem. –  sleepless in beantown Dec 14 '10 at 15:19
    
@OJ: I don't understand this: "Which are points from a plane." Perhaps you didn't intend your example to be interpreted literally, but those points don't lie on a plane, not even nearly. Maybe "plane" = "surface"? –  Joseph O'Rourke Dec 14 '10 at 15:24
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Without fixing the number of control points before-hand, you have the problem of this being an "under-determined" set of equations, or an "over-determined" set of equations. With enough control points, you can have an exact fit, at the expense of bizarre swings and interpolated regions. It's the same as trying to find a best-fit polynomial to a set of points: there could be no solution in the form you're looking for; there could be multiple equally good solutions; there could be an overly-good fit because you use so many terms (control points) to fit it perfectly. How did you get to this Q? –  sleepless in beantown Dec 14 '10 at 15:27
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1 Answer 1

up vote 1 down vote accepted

Use a black box optimizer where the parameters are the locations and weights of the control points and the function to minimize is something like the error volume between the surfaces.

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could you provide some links to articles on this black box optimizer topic? –  Ole J Dec 14 '10 at 12:46
    
    
'the function to minimize is something like the error volume between the surfaces.' ... hm - I need and have data from 1 surface. so how can I calculate such error function? –  Ole J Dec 14 '10 at 13:10
    
The other surface is the NURBS surface defined by the control point locations and weights. –  h10 Dec 14 '10 at 13:20
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If your next comment is that you don't yet have the NURBS surface, then I'm going to give up and let someone else help you. –  h10 Dec 14 '10 at 13:26
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