Algorithm.

Spline construction at an irregular node.

Input: B-spline-like control points $c_{ij}^k$, i, j ∈ {1, 2}, k = 0, …, n − 1 (see Fig. 2a) of quadrilaterals surrounding an n ≠ 4-valent node.

Output: A C1 spline complex consisting of 4n polynomial pieces bk,αβ that is singularly parameterized at the central point (Fig. 2b). Conversion to BB form: set $a_{ij}^k:=c_{ij}^k$ for i, j ∈ {1, 2} and then compute the remaining coefficients $a_{ij}^k$, i, j ∈ {0, 1, 2, 3} of the bicubic BB-patches ak(u, v), k = 1, …, n, as averages of the $c_{ij}^k$ so that the ak join C1 except at the irregular point $a_{00}^k$ that we set to the average of the surrounding control points $a_{00}^k:={\sum }_{k=1}^n{c}_{11}^k/n$. Subdivide the patches ak to get sub-patches hk,ij := Sak, i, j ∈ {1, 2}. When i +j > 2 then bk,ij := hk,ij. Only the subpatches hk,11 that include the central point still require work. Abbreviating $h_{\alpha \beta }:=h_{\alpha \beta }^{k,11}$, we apply the projection P (cf. [Rei97, Sect.6]): $$\left(\begin{array}{c}{b}_{11}\\ b_{21}\\ b_{12}\end{array}\right):=\mathbf{P}\left(\begin{array}{c}{h}_{11}\\ h_{21}\\ h_{12}\end{array}\right):=\left(\begin{array}{ccc}{\mathbf{P}}_1& {\mathbf{P}}_2& {\mathbf{P}}_3\\ {\mathbf{P}}_4& {\mathbf{P}}_5& {\mathbf{P}}_6\\ {\mathbf{P}}_7& {\mathbf{P}}_8& {\mathbf{P}}_9\end{array}\right)\left(\begin{array}{c}{h}_{11}\\ h_{21}\\ h_{12}\end{array}\right)$$ (1) where ${\mathbf{P}}_{\nu }(j,k):=p_{\nu }^{j-k}$, for ν = 1, …, 9, and j, k = 0, …, n − 1, φn := 2π/n, ψ := arg((1 + iβsinφn)e−iφn/2), β := 1/10, $p_1^j=p_2^j=p_3^j=p_4^j=p_7^j=1/3n,p_5^j=p_9^j=(1+3cos(j{\phi }_n))/3n$, $p_6^j=(1+3cos(2\psi +j{\phi }_n))/3n,p_8^j=(1+3cos(2\psi -j{\phi }_n))/3n$. Average $b_{10}^k:=(b_{11}^k+b_{11}^{k+1}/2)$ for all edges and set $b_{00}^k:={\sum }_{k=1}^n{b}_{11}^k/n$ for all vertices of valence n.