# NURBS basis functions

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Hi everyone, I'm having trouble understanding the basis functions required to calculate a NURBS surface. On wikipedia, the calculation is defined here: http://en.wikipedia.org/wiki/NURBS#Construction_of_the_basis_functions_.5B3.5D as: Ni,n = fi,nNi,n − 1 + gi + 1,nNi + 1,n − 1 However, since the calculation for Ni,n involves seemingly already knowing Ni,n I am confused. It seems like circular logic to me. Could anyone please help me understand the basis function? I understand Ni,n(u) is the value for the basis function for control point i of degree n for time value u. But I do not understand the full calculation: Ni,n = fi,nNi,n − 1 + gi + 1,nNi + 1,n − 1 I'd really appreciate a hint or explanation of how to do this calculation. Specifically given i,n and u, how do I calculate Ni,n(u)? Best Regards, David

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I've no experience with NURBS in general but noone else has answered so hopefully just looking at this mathematically will help.

They key point is that the equation is Recursive:

$N_{i,n} = f_{i,n} N_{i,n-1} + g_{i+1,n} N_{i+1,n-1}$

$N_{i,n}$ is only dependent on $N_{i,n-1}$ and $N_{i+1,n-1}$

So since you are given that at n=0 $N_{j,0}$ is piecewise constant you can calculate $N_{j,1}$.

Page 4 of
http://libnurbs.sourceforge.net/nurbsintro.pdf
gives a better explanation and more explicit information on what the function N actually is where n=0.

p.s. I am a mathematician by training not a computer scientist so apologies if the explanation is overly mathematical.

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Thanks very much - apologies for the late reply.
But that's exactly what I needed - once I understood the function as recursive it was straightforward.

Thanks again :)

David

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