Knot vector: degree \(n\), a sequence of knots \(t_0 < t_1 < t_2 < \ldots{} < t_{m+n}\) where \(m\) is the number of control points of the curve. The knot vector \(T = (t_i)_{i \in \mathbb{Z}}\) defines parameter intervals \([t_i , t_{i+1})\).
Definition: Normalized B-Splines \[ \begin{align} N_i^0(u):&= \begin{cases}1 & t_i\leq u < t_{i+1}\\ 0 & \text{otherwise} \end{cases},\quad i = 0, \ldots, n+m-1\\[2em] N_i^k (u):&=\frac{u-t_i}{t_{i+k}-t_i}\,N_i^{k-1} (u)+\frac{t_{i+k+1}-u}{t_{i+k+1}-t_{i+1}} \,N_{i+1}^{k-1} (u),\quad k = 1, \ldots, n \end{align} \]