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

In [1], Figure 2 and the text description at the beginning of page 374 should be replaced with the ones given below. The illustrative example in Fig. 2 left (respectively, right) shows the degree-5 (respectively, degree-7) fundamental function with compact support [−3, 3] (respectively, [−4, 4]), that is obtained when selecting as a degree-2 (respectively, degree-3) polynomial B-spline and P as a degree-3 (respectively, degree-4) polynomial interpolating a subset of four (respectively, five) consecutive points (xk, δk, j ), k ≥ . Since the assumption of Proposition 1 is fulfilled, 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 j is 6 (respectively, 8). Finally, according to Proposition 3, since is C1 (respectively, C2) then j is C2 (respectively, C3). In [1], the following changes should be considered when reading pages 375–379:

In [1], Figure 2 and the text description at the beginning of page 374 should be replaced with the ones given below.
In [1], the following changes should be considered when reading pages 375-379: • in the caption of Table 1 replace R 1 with R 0 ; • in Definition 5 replace k-ring neighbourhood with (k + 1)-ring neighbourhood; • on page 375, two lines below eq.( 3), replace R n+1 with R n ; The original article can be found online at https://doi.org/10.1007/s11565-022-00423-8.• on page 376, line 7, replace R n+1 with R n ; • on page 376, line 16, replace R 2 with R 1 ; • in Proposition 4, second line, replace R n+1 with R 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 #L 1 = 6N + 1 with N ∈ {3, 4, 5, 6} denoting the valence of x , we should require that dim( d ) ≥ max{6N } + 1.In our experiments we have worked with polynomials P ,1 of total degree 6 ≤ d ≤ 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 x of valence N ≤ 6, we have computed the coefficients of the degree-d polynomial P ,1 by solving a weighted least-squares fitting problem with big weights assigned to the 6N + 1 interpolation points with parameter values in R 1 .Examples of bivariate polynomials that interpolate the 6N + 1 points (x k , δ k, ), x k ∈ R 1 and approximate in the least-squares sense the 6N points (x k , 0), x k ∈ R 2 \R 1 when the valence of x is N ∈ {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.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Fig. 2 Fig. 4
Fig.2Left: 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

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