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arXiv · 1412.1125

Smooth Bezier Surfaces over Arbitrary Quadrilateral Meshes

Michel Bercovier·Tanya Matskewich
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Abstract

We solve the following problem: given a polynomial of order $n$ and the corresponding $B\'ezier$ tensor product patches over an unstructured regular quadrilateral mesh of any valence, find a solution to the $G^{1}1$ or $C^{1}1$ approximation (resp. interpolation) problem ! Constraints defining regularity conditions across patches have to be satisfied. The resulting number of free degrees of freedom must be such that for instance the interpolation problem has a solution. This is similar to studying the minimal determining set (MDS) for a $C^{1}$ continuity construction. The givenunstructured quadrilateral mesh can include a cubic boundary curve. The final surface approximation or PDE solution is obtained by energy methods. We completely solve the problem and show that there is always a solution for $n\ge 5$ and under some mesh restrictions for $n=4$. From a practical point of view, the present paper provides a way to build first order smooth interpolation/approximation and solutions to partial differential equations for arbitrary structures of quadrilateral meshes.

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Michel Bercovier, Tanya Matskewich. 2014-12-02. Smooth Bezier Surfaces over Arbitrary Quadrilateral Meshes. https://doi.org/10.1007/978-3-319-63841-6

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