Curves
Based on both bases curves are defined.
Definitions
B-Spline Curves
Based on the B-splines $B_{i,p}$ the $p$-th degree curve [1, p. 80]
\[\bm{c}(u) = \sum_{i=1}^N B_{i,p}(u) \bm{p}_i\]
is defined with the constant controlpoints $\bm{p}_i \in \mathbb{R}^3$.
NURBS Curves
Analogously, the $p$-th degree NURBS curve [1, p. 117].
\[\bm{c}(u) = \sum_{i=1}^N R_{i,p}(u) \bm{p}_i = \cfrac{\sum_{i=1}^N B_{i,p}(u) w_i \bm{p_i}}{\sum_{i=1}^N B_{i,p}(u) w_i}\]
is defined.
$\bm{c}(u): u \mapsto (x,y,z)$, that is, the curve is a mapping from the parametric space $u \in [0,1]$ to the physical space $\mathbb{R}^3$.
Derivatives
B-Spline Curves
The $k$-th derivative of a B-spline curve is given as [1, p. 91]
\[\bm{c}^{(k)}(u) = \sum_{i=1}^N B_{i,p}^{(k)}(u) \bm{p}_i \,.\]
NURBS Curves
The $k$-th derivative of a NURBS curve is given as
\[\bm{c}^{(k)}(u) = \sum_{i=1}^N R_{i,p}^{(k)}(u) \bm{p}_i \,.\]
It can be computed as [1, p. 125]
\[\bm{c}^{(k)}(u) = \cfrac{\bm{a}^{(k)}(u) - \sum_{i=1}^k \begin{pmatrix} k \\ i \end{pmatrix} w^{(i)}(u) \bm{c}^{(k-i)}(u) }{w^{(0)}(u)}\]
with the auxiliary
\[\bm{a}^{(k)}(u) = \sum_{i=1}^N B_{i,p}^{(k)}(u) w_i \bm{p}_i\]
and
\[w^{(k)}(u) = \sum_{i=1}^N B_{i,p}^{(k)}(u) w_i \,.\]