Single new-knot insertion for a B-spline curve

Use the zero-based reindexing $t_i={\xi }_{i+1}$ of the clamped knot vector in B-spline spaces and knot conventions, so $t_0\le \cdots \le t_{n+p}$ with $n\ge p+1$. Let $p\in {\mathbb{N}}_0$, $d\in \mathbb{N}$, and let the controls be $P_0,\dots ,P_{n-1}\in {\mathbb{R}}^d$. Let $k$ be a nonempty span index with $t_k<u<t_{k+1}$, where $u$ is not already a knot. Insert $u$ between $t_k$ and $t_{k+1}$ to obtain $t^{\mathrm{\prime }}$. The output controls $Q_0,\dots ,Q_n$ are defined by

$Q_i=\{\begin{array}{ll}{P}_i,& 0\le i\le k-p,\\ {\alpha }_i{P}_i+(1-{\alpha }_i)P_{i-1},& k-p+1\le i\le k,\\ P_{i-1},& k+1\le i\le n,\end{array}{\alpha }_i=\frac{u-t_i}{t_{i+p}-t_i}\mathrm{.}$

The middle range is empty when $p=0$. The curve uses the same degree and the B-spline basis for the new knot vector. Each denominator in the middle range is positive and each ${\alpha }_i$ lies in $[0,1]$, so the update is an exact affine blend. This procedure inserts one knot that is new to the vector; repeated insertion at an existing knot is not part of this algorithm's input contract.