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Using Geometric Symmetries to Achieve Super-Smoothness for Cubic Powell-Sabin Splines

Jan Grošelj, Hendrik Speleers

Abstract

In this paper, we investigate $C^2$ super-smoothness of the full $C^1$ cubic spline space on a Powell-Sabin refined triangulation, for which a B-spline basis can be constructed. Blossoming is used to identify the $C^2$ smoothness conditions between the functionals of the dual basis. Some of these conditions can be enforced without difficulty on general triangulations. Others are more involved but greatly simplify if the triangulation and its corresponding Powell-Sabin refinement possess certain symmetries. Furthermore, it is shown how the $C^2$ smoothness constraints can be integrated into the spline representation by reducing the set of basis functions. As an application of the super-smooth basis functions, a reduced spline space is introduced that maintains the cubic precision of the full $C^1$ spline space.

Using Geometric Symmetries to Achieve Super-Smoothness for Cubic Powell-Sabin Splines

Abstract

In this paper, we investigate super-smoothness of the full cubic spline space on a Powell-Sabin refined triangulation, for which a B-spline basis can be constructed. Blossoming is used to identify the smoothness conditions between the functionals of the dual basis. Some of these conditions can be enforced without difficulty on general triangulations. Others are more involved but greatly simplify if the triangulation and its corresponding Powell-Sabin refinement possess certain symmetries. Furthermore, it is shown how the smoothness constraints can be integrated into the spline representation by reducing the set of basis functions. As an application of the super-smooth basis functions, a reduced spline space is introduced that maintains the cubic precision of the full spline space.

Paper Structure

This paper contains 19 sections, 22 theorems, 50 equations, 9 figures.

Table of Contents

  1. Introduction
  2. $C^1$ Spline Space
  3. Powell--Sabin Refinement
  4. Blossoming of a Cubic Polynomial
  5. Basis Functions
  6. $C^{2}$ Super-Smoothness
  7. $C^{2}$ Smoothness Across $[v_{ij}, v_{ijk}]$
  8. $C^{2}$ Smoothness Across $[v_i, v_{ijk}]$
  9. Super-Smooth Basis Functions
  10. Basis Functions Associated with Vertices
  11. Basis Functions Associated with Interior Edges
  12. Basis Functions Associated with Boundary Edges
  13. Basis Functions Associated with Triangles
  14. A Reduced Spline Space
  15. Basis Functions
  16. ...and 4 more sections

Key Result

Lemma 1

Suppose that $t_1, t_2 \subset {\mathbb R}^2$ are two nondegenerate and nonoverlapping triangles with a common edge $e$. Let $S: t_1 \cup t_2 \rightarrow {\mathbb R}$ be a $C^{0}$ smooth spline determined by polynomials $P_1, P_2 \in {\mathbb P}_3$. Let $p_1, p_2 \in {\mathbb R}^2$ be any pair of different points on the line containing $e$.

Figures (9)

  • Figure 1: An excerpt from a triangulation and its Powell--Sabin refinement.
  • Figure 2: Triangles (in gray) associated with vertices of a triangulation. The figure also depicts the points that each of the triangles must contain.
  • Figure 3: The basis functions $B_{i,1}^v$, $B_{i,2}^v$, $B_{i,3}^v$ (from left to right) associated with the vertex $v_i$ and the points $q_{i,1}^v$, $q_{i,2}^v$, $q_{i,3}^v$ shown in Figure \ref{['fig:ps3conf']}.
  • Figure 4: The basis functions $B_{i,j,k}^t$, $B_{j,i,k}^t$ (in the upper row) and $B_{i,j,k'}^t$, $B_{j,i,k'}^t$ (in the bottom row) associated with the triangles $t_{i,j,k}$, $t_{j,i,k}$, $t_{i,j,k'}$, $t_{j,i,k'}$ shown in Figure \ref{['fig:pstri']}.
  • Figure 5: The basis functions $B_{j,k,i}^t$, $B_{k,j,i}^t$ (in the upper row) and $B_{j,k}^e$, $B_{k,j}^e$ (in the bottom row) associated with the triangles $t_{j,k,i}$, $t_{k,j,i}$ and the edges $e_{j,k}$, $e_{k,j}$ shown in Figure \ref{['fig:pstri']}.
  • ...and 4 more figures

Theorems & Definitions (47)

  • Lemma 1
  • proof
  • Theorem 2
  • Remark 3
  • Proposition 4
  • proof
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 37 more