Constructing $C^1$ limit surfaces from unstructured splines via averaging and refinement

Abstract

In this paper we present a construction for unstructured splines over quadrilateral meshes by iterative averaging and refinement. We represent the spline as a multi-patch B-spline, where the degrees of freedom are those B-spline coefficients on the quadrilateral patches that are not associated with interior edges and vertices of the mesh, i.e., their corresponding Greville points lie inside the patches. In every averaging step, we replace the remaining B-spline coefficients associated with interior edges and vertices by suitable averages of neighboring degrees of freedom. In the refinement step we apply regular splits to all patches by knot insertion. This process results in a subdivision scheme that, for degree $p=2$, is similar to the almost-$C^1$ spline construction from (Takacs, Toshniwal. CMAME, 2023) and behaves similar to Doo-Sabin subdivision, cf. (Doo, Sabin. CAD, 1978), and that can be defined for arbitrary degrees and regularities inside the patches. We derive two families of spline constructions, based on simple and coplanar averaging, respectively, and analyze their spectral properties when interpreted as subdivision schemes. Using this interpretation, we show that they are $C^1$ in the limit. Moreover, the coplanar averaging scheme produces splines that are $C^1$ at all vertices for every level of refinement, whereas the simple averaging is $C^1$ only in the limit. For both constructions, we have control over the subdominant eigenvalue, which has multiplicity two and can range between $\frac{1}{4}$ and $1$, with $\frac{1}{2}$ often being the desired option. The resulting basis functions form a partition of unity. Moreover, they form a non-negative partition of unity for suitably selected averaging parameters.