IJPAM: Volume 71, No. 1 (2011)

AN APPROXIMATING-INTERPOLATORY
SUBDIVISION SCHEME

Yacine Boumzaid$^1$, Sandrine Lanquetin$^2$
Marc Neveu$^3$, Francois Destelle$^4$
$^{1,2,3,4}$Laboratoire LE2I UMR CNRS
5158 Aile des Sciences de l'Ingenieur
BP 47870, 21078, Dijon Cedex, FRANCE


Abstract. In the last decade, study and construction of quad/triangle subdivision schemes have attracted attention.

The quad/triangle subdivision starts with a control mesh consisting of both quads and triangles and produces finer and finer meshes with quads and triangles (Figure 1). Designers often want to model certain regions with quad meshes and others with triangle meshes to get better visual quality of subdivision surfaces. Smoothness analysis tools exist for regular quad/triangle vertices. Moreover $C^{1}$ and $C^{2}$ quad/triangle schemes (for regular vertices) have been constructed. But to our knowledge, there are no quad/triangle schemes that unifies approximating and interpolatory subdivision schemes.

In this paper we introduce a new subdivision operator that unifies triangular and quadrilateral subdivision schemes. Our new scheme is a generalization of the well known Catmull- Clark and Butterfly subdivision algorithms. We show that in the regular case along the quad/triangle boundary where vertices are shared by two adjacent quads and three adjacent triangles our scheme is $C^{2}$ everywhere except for ordinary Butterfly where our scheme is $C^{1}$.

Received: May 11, 2011

AMS Subject Classification: 65-XX, 68-XX

Key Words and Phrases: subdivision, polynomial generation, quad/triangle subdivision, quasi-interpolants

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 71
Issue: 1