##
**Smoothness of stationary subdivision on irregular meshes.**
*(English)*
Zbl 0965.65024

The goal of this comprehensive work is to establish characterizations for tangent plane continuity and \(C^k\)-continuity of subdivision surfaces at extraordinary vertices. The author shows that any subdivision surface in \(\mathbb R^3\) can be regarded as a projection of a unique universal surface in a higher-dimensional space. The key idea consists in reducing the analysis of smoothness of the division scheme to that of the universal surface. This allows to separate the algebraic and the geometric aspects of the problem. In doing so, a more general theory of smoothness of subdivision surfaces near extraordinary vertices is constructed.

Namely, subdivision surfaces \(f: |K|\to \mathbb R^3\), defined on a simplicial complex \(K\) are considered. As in the regular setting, \(f\) can be decomposed into a sum of basis functions. Let \(\psi\) be the vector of basis functions that contribute to \(f\) on the neighborhood \(U\) of an extraordinary vertex, consisting of the triangles of the complex adjacent to the vertex. Then \(f\) can be written in vector form as \(f=(p,\psi)\), where \(p\) are the control points in \(\mathbb R^3\). This equation allows to regard \(f\) as a projection of a higher-dimensional surface \(\psi: U \to \mathbb R^p\), called the universal surface.

It is proved that under general conditions subdivision surfaces are tangent plane continuous (resp., \(C^k\)-continuous) if and only if the universal surface has the same property. In all the derivations it is assumed that the subdivision scheme produces at least \(C^1\)-continuous (resp., \(C^k\)-continuous) limit function, and the configurations of control points are not degenerate.

Namely, subdivision surfaces \(f: |K|\to \mathbb R^3\), defined on a simplicial complex \(K\) are considered. As in the regular setting, \(f\) can be decomposed into a sum of basis functions. Let \(\psi\) be the vector of basis functions that contribute to \(f\) on the neighborhood \(U\) of an extraordinary vertex, consisting of the triangles of the complex adjacent to the vertex. Then \(f\) can be written in vector form as \(f=(p,\psi)\), where \(p\) are the control points in \(\mathbb R^3\). This equation allows to regard \(f\) as a projection of a higher-dimensional surface \(\psi: U \to \mathbb R^p\), called the universal surface.

It is proved that under general conditions subdivision surfaces are tangent plane continuous (resp., \(C^k\)-continuous) if and only if the universal surface has the same property. In all the derivations it is assumed that the subdivision scheme produces at least \(C^1\)-continuous (resp., \(C^k\)-continuous) limit function, and the configurations of control points are not degenerate.

Reviewer: Andrei Martínez Finkelshtein (Almeria)

### MSC:

65D17 | Computer-aided design (modeling of curves and surfaces) |

65D10 | Numerical smoothing, curve fitting |