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Spline

Upper and lower bound for NURBS length

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Hi all,

although NURBS seem to be used widely, I can find only little information on length approximation of NURBS curves on the web. It can very well be that I simply didn't feed google with the right search terms, in which case I appologize for the noise and kindly ask for a hint as to what these terms might be.

To my actual problem... given a NURBS curve C of degree d, defined on the real (well, in terms of implementation, rational) interval [0;1]. For 0<=a,b<=1 and ?>0 find l, u: l <= length of C in the interval [a;b] <= u and u-l <= 2?.

Differential geometry says that I have to compute the integral of the speed vector at each point. Well, nice try, but NURBS cannot be differentiated analytically. So a numerical solution is required. However using the "default" algorithms like Gaussian quadrature and difference quotiont (likely to require an exact rational arithmetic to work at all) seems to be a bit wasteful. There must be better algorithms out there, or am I missing something?

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Uh, you need to help me out here... how are these guys going to help me? Save asking them, I mean...?

Perhaps I should re-state my intentions. What I currently do in order to determine the length of a nurbs curve is to split it into the non-empty intervals defined by the knot vectors and perform a piece-wise numerical integration (my special thanks go to Gauss & Legendre...). Now I split the intervals in half and perform another iteration. The difference between the larger and the sum of the two corresponding smaller intervals are my error estimates. Until the sum of error estimates is smaller than some ?, I recursively split the interval with the largest error estimate.

For all curves I have tested so far this works well and all, however, I lack a (mathematically sound!) argument as to the "real" error bound. Or another approach that has such a guarantee - although my current method seems to give good results, an error bound would be rather large due to my derivate estimation alone.

And while De Casteljau has defined a numerically sound algorithm to evaluate Bezier curves, and the NURBS basis can, for some degenerate cases, degenerate to Bernstein polynomials, I fail to see the connection to my problem and would appreciate any more concrete hints.

Kind regards
Spline

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