Nonuniform interpolatory subdivision schemes with improved smoothness

Dyn, Nira
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

Hormann , Kai
ORCID
Facoltà di scienze informatiche, Università della Svizzera italiana, Svizzera

Mancinelli, Claudio
ORCID
DIBRIS, Università degli Studi di Genova, Genova, Italy
Published in:
 Computer aided geometric design.  2022, vol. 94, p. 102083
English
Subdivision schemes are used to generate smooth curves or surfaces by iteratively refining an initial control polygon or mesh. We focus on univariate, linear, binary subdivision schemes, where the vertices of the refined polygon are computed as linear combinations of the current neighbouring vertices. In the classical stationary setting, there are just two such subdivision rules, which are used throughout all subdivision steps to construct the new vertices with even and odd indices, respectively. These schemes are well understood and many tools have been developed for deriving their properties, including the smoothness of the limit curves. For nonstationary schemes, the subdivision rules are not fixed and can be different in each subdivision step. Nonuniform schemes are even more general, as they allow the subdivision rules to be different for every new vertex that is generated by the scheme. The properties of nonstationary and nonuniform schemes are usually derived by relating the scheme to a corresponding stationary scheme and then exploiting the fact that the properties of the stationary scheme carry over under certain proximity conditions. In particular, this approach can be used to show that the limit curves of a nonstationary or nonuniform scheme are as smooth as those of a corresponding stationary scheme. In this paper we show that nonuniform subdivision schemes have the potential to generate limit curves that are smoother than those of stationary schemes with the same support size of the subdivision rule. For that, we derive interpolatory 2point and 4point schemes that generate and limit curves, respectively. These values of smoothness exceed the smoothness of classical interpolating schemes with the same support size by one.

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