Correction to: Fundamental functions for local interpolation of quadrilateral meshes with extraordinary vertices

In the original publication of the article, some figures and their descriptions are incorrect. This erratum corrects the same and adjusts the meaning of the notation $R^k_{\ell }$ in Definition 5.

In [1], Figure 2 and the text description at the beginning of page 374 should be replaced with the ones given below.

Fig. 2

Left: the $C^2$ fundamental function supported on $[-3,3]$ that is obtained by using our constructive approach with the quadratic B-splines as blending functions. Right: the $C^3$ fundamental function supported on $[-4,4]$ that is obtained by using our constructive approach with the cubic B-splines as blending functions. In both figures the interpolating polynomials blended by the B-splines (dashed graphs) are the dotted graphs

The illustrative example in Fig. 2left (respectively, right) shows the degree-5 (respectively, degree-7) fundamental function with compact support $[-3,3]$ (respectively, $[-4,4]$), that is obtained when selecting $\Phi _{\ell }$ as a degree-2 (respectively, degree-3) polynomial B-spline and ${\mathcal {P}}_{\ell }$ as a degree-3 (respectively, degree-4) polynomial interpolating a subset of four (respectively, five) consecutive points $(x_k, \delta _{k,j})$, $k \ge \ell$. Since the assumption of Proposition 1 is fulfilled, $\Psi _j$ is a fundamental function for interpolation. Moreover, according to Proposition 2, since $m=3$ and $w=3$ (respectively, $m=4$ and $w=4$), then the support width of $\Psi _j$ is 6 (respectively, 8). Finally, according to Proposition 3, since $\Phi _{\ell }$ is $C^1$ (respectively, $C^2$) then $\Psi _j$ is $C^{2}$ (respectively, $C^3$).

In [1], the following changes should be considered when reading pages 375–379:

Fig. 3

Degree-6,7,8 bivariate polynomials that interpolate the $6N_{\ell }+1$ points $({{\varvec{x}}}_k, \delta _{k,\ell })$, ${{\varvec{x}}}_k \in {{{\mathcal {R}}}}_{\ell }^1$ and approximate in the least-squares sense the $6N_{\ell }$ points $({{\varvec{x}}}_k, 0)$, ${{\varvec{x}}}_k \in {{{\mathcal {R}}}}_{\ell }^2 {\setminus } {{{\mathcal {R}}}}_{\ell }^1$ when the valence of ${{\varvec{x}}}_{\ell }$ is $N_{\ell } \in \{3, 5, 6\}$

Fig. 4

Left: fundamental function for local interpolation that is defined on a regular subregion of $\Omega$ and is $C^3$ everywhere. Right: globally $C^2$ fundamental function that is centered at a valence-4 vertex and contains a valence-5 vertex in its support

Fig. 5

First column: globally $C^2$ fundamental functions for local interpolation centered at an extraordinary vertex of valence 3, 5, 6

• in the caption of Table 1 replace ${{{\mathcal {R}}}}_{\ell }^1$ with ${{{\mathcal {R}}}}_{\ell }^0$;

• in Definition 5 replace k-ring neighbourhood with $(k+1)$-ring neighbourhood;

• on page 375, two lines below eq. (3), replace ${{{\mathcal {R}}}}_{\ell }^{n+1}$ with ${{{\mathcal {R}}}}_{\ell }^n$;

• on page 376, line 7, replace ${{{\mathcal {R}}}}_{\ell }^{n+1}$ with ${{{\mathcal {R}}}}_{\ell }^n$;

• on page 376, line 16, replace ${{{\mathcal {R}}}}_{\ell }^{2}$ with ${{{\mathcal {R}}}}_{\ell }^1$;

• in Proposition 4, second line, replace ${{{\mathcal {R}}}}_{\ell }^{n+1}$ with ${{{\mathcal {R}}}}_{\ell }^n$;

• on page 379, fifth line from the bottom, replace 2-ring with 3-ring.

Additionally, the second paragraph of [1, Section 5] should be substituted with the following one:

As to the polynomial interpolants, since when $n=1$ condition (5) must hold for $\#{\mathcal {L}}_\ell ^1=6N_\ell +1$ with $N_{\ell } \in \{3,4,5,6\}$ denoting the valence of ${{\varvec{x}}}_{\ell }$, we should require that $\dim (\Pi _d) \geqslant \max \{ 6N_\ell \} +1$. In our experiments we have worked with polynomials ${{{\mathcal {P}}}}_{\ell ,1}$ of total degree $6 \le d \le 8$, as it is a reasonably low degree that allows us to satisfy the condition (5) for all valences up to 6. Hence, for a vertex ${{\varvec{x}}}_\ell$ of valence $N_\ell \leqslant 6$, we have computed the coefficients of the degree-d polynomial ${{{\mathcal {P}}}}_{\ell ,1}$ by solving a weighted least-squares fitting problem with big weights assigned to the $6N_\ell +1$ interpolation points with parameter values in ${{{\mathcal {R}}}}_{\ell }^1$. Examples of bivariate polynomials that interpolate the $6N_{\ell }+1$ points $({{\varvec{x}}}_k, \delta _{k,\ell })$, ${{\varvec{x}}}_k \in {{{\mathcal {R}}}}_{\ell }^1$ and approximate in the least-squares sense the $6N_{\ell }$ points $({{\varvec{x}}}_k, 0)$, ${{\varvec{x}}}_k \in {{{\mathcal {R}}}}_{\ell }^2 {\setminus } {{{\mathcal {R}}}}_{\ell }^1$ when the valence of ${{\varvec{x}}}_{\ell }$ is $N_{\ell } \in \{3, 5, 6\}$, are shown in Fig. 3.

Finally, Figure 4, 5 and the first column of Figure 5 in [1] should be replaced with the ones given below.