I'm trying to grasp how NURBS curves work. I think I understand the principle, but I'm unsure about the formula. Here, I base myself on this course.

From a DXF file, I got the following parameters:

knots (7): [0,0,0,0.5,1,1,1]
degree: 2
closed: false
control points (4): [(0,0), (-50,112.5), (150,112.5), (100,0)]

The knots vector looks good : non-decreasing, length of #controls + degree + 1, max multiplicity of degree+1.

Here are the formulas from the course:

$C(u) = \frac{\sum_{i=0}^n{w_i P_i N_{i,k}(u)}}{\sum_{i=0}^n{w_i P_i}}$

  • $w_i$ : weights
  • $P_i$ : control points (vector)
  • $N_{i,k}$ : normalized B-spline basis functions of degree k

and the recursive B-spline basis function:

$N_{i,k}(u) = \frac{u-t_i}{t_{i+k} - t_i} N_{i,k-1}(u) + \frac{t_{i+k+1}-u}{t_{i+k+1}-t_{i+1}} N_{i+1,k-1}(u)$

$N_{i,0}(u) = 1$ if $t_i \leq u < t_{i+1}$ or $0$ otherwise.

What I don't get is the fraction in the $N_{i,k}(u)$ function.

Let's take $i=0$ (calculating the influence of the first control point over the curve, if I understand correctly): because of the multiplicity, I've got $t_{i+k} = t_2 = 0$ and $t_i = t_0 = 0$, which makes me divide by zero ... What am I missing?

EDIT: As usual, writing down the problem helps to solve it. By re-reading the material linked above, I realized I missed one line: 0/0 fractions should be considered as 0, not NaN. Is this right ? At least on the few toy examples I tried, my drawing matches the original DXF version.

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.