2020/07/23 by Michael DiPasquale, DiPasquale, Michael, Nelly Villamizar +1
Computer Science · Engineering · #13D02 #41A15 #65D07 #Advanced Numerical Analysis Techniques #Commutative Algebra (math.AC) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2007.12274
openalex publication_date 2020/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive a formula which is a lower bound on the dimension of trivariate splines on a tetrahedral partition which are continuously differentiable of order r in large enough degree. While this formula may fail to be a lower bound on the dimension of the spline space in low degree, we illustrate in several examples considered by Alfeld and Schumaker that our formula may give the exact dimension of the spline space in large enough degree if vertex positions are generic. In contrast, for splines continuously differentiable of order r>1, every lower bound in the literature diverges (often significantly) in large degree from the dimension of the spline space in these examples. We derive the bound using commutative and homological algebra.