Definition: Rational B-spline curve (NURBS: non-uniform rational B-spline)
Given
Then: \[ F(u)=\frac{\sum_{i=0}^{m-1}\,N_i^n(u)\,w_i\,\mathbf{d}_i}{\sum_{i=0}^{m-1}\,N_i^n(u)\,w_i} \] is called rational B-spline curve.
de Boor Algorithm
Given control points \(\mathbf{d}_0,\ldots,\mathbf{d}_{m-1}\), the weights \(w_0,\ldots,w_{m-1}\geq 0\) with \(w_0=w_{m-1}=1\), and the knot vector \(T=(t_i)\) of a rational B-spline curve \( F(u)=\frac{\sum_{i=0}^{m-1}\,N_i^n(u)\,w_i\,\mathbf{d}_i}{\sum_{i=0}^{m-1}\,N_i^n(u)\,w_i} \) and a parameter interval \(t_j\leq{} u < t_{j+1}\).
Evaluate \(F\) at \(u\) \[ \newcommand{\d}{\mathbf{d}} \begin{align} \d_i^0 &:= w_i \cdot \d_i\,,\qquad i = j-n, \ldots , j\\ w_i^0 &:= w_i\,,\qquad i = j-n, \ldots , j\\ \d_i^k &:=\frac{t_{i+n+1-k}-u}{t_{i+n+1-k}-t_i}\,\d_{i-1}^{k-1}+\frac{u-t_i}{t_{i+n+1-k}-t_i}\,\d_i^{k-1}\,,\\ w_i^k &:=\frac{t_{i+n+1-k}-u}{t_{i+n+1-k}-t_i}\,w_{i-1}^{k-1}+\frac{u-t_i}{t_{i+n+1-k}-t_i}\,w_i^{k-1}\,,\\ &\mbox{with} \quad k = 1, \ldots, n \quad\mbox{and} \quad i = j-n+k, \ldots, j \end{align} \]
The: \(F(u) = \frac{\mathbf{d}_j^n}{w_j^n}\).
Select and drag control points in order to interact with the graphic representation. Use the sliders to change the values of the weights, and observe how the curve behaves.