# $G^1$-smooth splines on quad meshes with 4-split macro-patch elements

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA | UoA - National and Kapodistrian University of Athens = University of Athens
Abstract : We analyze the space of differentiable functions on a quad-mesh $\cM$, which are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across shared edges, that satisfy conditions which ensure that the space of differentiable functions is ample on a quad-mesh of arbitrary topology. These transition maps define a finite dimensional vector space of $G^{1}$ spline functions of bi-degree $\le (k,k)$ on each quadrangular face of $\cM$. We determine the dimension of this space of $G^{1}$ spline functions for $k$ big enough and provide explicit constructions of basis functions attached respectively to vertices, edges and faces. This construction requires the analysis of the module of syzygies of univariate b-spline functions with b-spline function coefficients. New results on their generators and dimensions are provided. Examples of bases of $G^{1}$ splines of small degree for simple topological surfaces are detailed and illustrated by parametric surface constructions.
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https://hal.archives-ouvertes.fr/hal-01491676
Contributor : Ahmed Blidia <>
Submitted on : Friday, March 17, 2017 - 11:24:25 AM
Last modification on : Tuesday, January 14, 2020 - 1:36:09 PM
Document(s) archivé(s) le : Sunday, June 18, 2017 - 1:10:04 PM

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### Citation

Ahmed Blidia, Bernard Mourrain, Nelly Villamizar. $G^1$-smooth splines on quad meshes with 4-split macro-patch elements. Computer Aided Geometric Design, Elsevier, 2017, 52–53, pp.106-125. ⟨10.1016/j.cagd.2017.03.003⟩. ⟨hal-01491676⟩

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