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The NURBS basis functions, denoted by in the following, are defined from a B-spline basis of the same degree . In order to do so, it is necessary to introduce the weights wi for the control points xi. More precisely, the piecewise rational functions read as follows:

[1.15]

Then, NURBS curves are defined similarly to those in [1.9]:

[1.16]

Consequently, a NURBS curve takes the same inputs as a B-spline curve,

i.e. a degree p, a knot-vector Ξ₁ and control points Pi, plus an additional set of n₁ weights wi. The weights give more shape control to the designer. By changing their values, a curve can be modified, as shown in Figure 1.7, for example. In fact, the positions of the control points and the values of the associated weights can be adjusted in order to build conical sections exactly (circles, ellipses, etc.; see, for example, Cottrell et al. (2007, 2009)). Given equation [1.15] (and verifying that the B-spline functions satisfy the partition of unity), it may be noted that if all weights are equal, the NURBS entity turns out to be a B-spline entity.

Figure 1.7. An example of a NURBS curve: influence of the weights on the basis functions and on the final curve with comparison to the B-spline case (the weights of the first, third and sixth control points have been moved from 1 to 0.5 , 2 and respectively)