Super-smooth cubic Powell–Sabin splines on three-directional triangulations: B-spline representation and subdivision

Abstract

Starting from a general B-spline representation for $C^1$ cubic Powell–Sabin splines on arbitrary triangulations, we focus on the construction of a B-spline representation for a particular subspace defined on three-directional triangulations with $C^2$ super-smoothness over each of the macro-triangles. We analyze the properties of the basis and point out the relation with simplex splines. Furthermore, we provide explicit expressions for the B-spline coefficients of any element of the spline space, and derive subdivision rules under dyadic refinement. Finally, we show simple conditions ensuring global $C^2$ smoothness on the domain.

Introduction

Bivariate polynomial splines on triangulations are usually investigated through the prisms of polyhedral splines or macro-elements. Polyhedral splines can be described as two-dimensional shadows of multi-dimensional polytopes, the most common examples of which are box and simplex splines; see, e.g., [1], [2]. Macro-elements are splines characterized by very particular interpolation problems that determine spline surfaces in a local sense, typically over every triangle of the triangulation separately; see, e.g., [3]. In theory, both frameworks are applicable on general triangulations. However, when one restricts to structured triangulations, box splines offer a neat environment for exploiting the geometric symmetries of the domain partition, while the macro-element approach is by design the tool of choice in the unstructured setting. This leaves us with two loosely connected worlds and forces us into compromises when dealing with partially structured and partially unstructured triangulations, which often arise in practice.

In this paper we study a space of super-smooth $C^1$ cubic splines on a three-directional triangulation, which after an additional refinement becomes six-directional. The space is introduced by considering first the full $C^1$ cubic Powell–Sabin macro-element space and its B-spline basis recently developed in [4], [5] for general triangulations. Those B-splines are then recombined and reduced by taking advantage of the symmetries of a three-directional triangulation in such a way that the spanning space consists of splines that are $C^2$ on every triangle of the triangulation. This result has several beneficial consequences. The introduced space has lower dimension and consists of smoother splines compared to the full $C^1$ spline space but, importantly, retains the full approximation order. Hence, it surpasses the approximation qualities of the commonly used cubic space on a three-directional triangulation spanned by the translates of the standard $C^1$ cubic box splines, which only ensure quadratic precision. Also, since the construction is completely in line with the general construction tailored for unstructured meshes, we foresee that the reduced B-splines can be used in combination with the general B-splines on triangulations that are only partially structured.

The adapted B-splines are, up to affine transformations, of two types and can be geometrically associated with the vertices, triangles and boundary edges of the triangulation. They are linearly independent, have local supports and form a convex partition of unity, i.e., they possess the basic properties that one expects from a B-spline basis. Besides providing an explicit relation to the B-splines on general triangulations, we describe them in terms of the Bernstein–Bézier representation, and we also interpret them as simplex splines, which endows us with a clearer insight into their super-smoothness properties. Moreover, we relate them to particular linearly dependent box splines extensively investigated in [6], [7], [8], [9], [10], which can be obtained by enforcing overall $C^2$ smoothness constraints. This offers several viewpoints for possible further investigation.

For the splines examined herein, the degrees of freedom related to a single triangle in the interior of the triangulation are three values per vertex (that can be specified by value and derivative interpolation) and one extra value corresponding to the barycenter of the triangle. This resembles the standard interpolation problem for cubic polynomials on a triangle, and the cubic spline construction can be seen as a blending scheme for ensuring $C^1$ smoothness across the edges by sacrificing the polynomial form for a macro-element structure inside the triangles. The relation is further amplified by the coefficients in the B-spline representation that we derive in terms of blossom values and which happen to agree with the expressions for certain Bézier ordinates of a cubic polynomial on a triangle. On the basis of these observations, we establish a control net for the spline surface and prove that the B-spline representation is stable with respect to perturbations of the B-spline coefficients.

The presented $C^1$ spline space, with $C^2$ super-smoothness over any triangle of the triangulation, is refinable under dyadic refinement of the triangles into four smaller triangles (obtained by the addition of the edge midpoints to the vertices of the triangulation). We develop a set of subdivision rules that provide the coefficients of the B-spline representation on the finer triangulation expressed as convex combinations of the B-spline coefficients on the coarser triangulation. These can be seen as a generalization of the dyadic subdivision rules for uniform quadratic Powell–Sabin splines [11], [12]. The convexity guarantees computational stability and makes the subdivision geometrically intuitive, but cannot be taken for granted. For example, the full $C^1$ cubic Powell–Sabin B-spline representation without additional super-smoothness (developed in [4]) does not admit convex subdivision rules even though it is dyadically refinable on three-directional triangulations. Equipped with the refinement strategy, we propose a triangular mesh for spline approximation, which is a simplification of the control net and is more suitable for surface visualization.

In the literature one can find several other cubic spline representations on triangulations that are closely related to the present construction. Different $C^1$ cubic Powell–Sabin B-splines with certain super-smoothness have been proposed and analyzed in [13], [14]. However, those spline spaces are not refinable, and so no subdivision rules are available. For $C^1$ cubic Clough–Tocher splines, only partially satisfactory B-spline representations have been developed: a global B-spline basis of a reduced space [15] and a local simplex spline basis [16]. A local simplex spline basis has also been developed for $C^2$ cubic splines defined on a Powell–Sabin 12-split [17]. Furthermore, $C^1$ cubic half-box splines have been explored for deriving a triangular subdivision scheme [18].

Reducing the number of degrees of freedom (i.e., using a smaller set of data to specify all the parameters in the spline representation) by local imposition of super-smoothness is a common technique in the context of splines on triangulations; see, e.g., [3]. In [5] three such recipes are presented for $C^1$ cubic Powell–Sabin splines; they are completely general in the sense that they can be implemented with any local cubic polynomial approximation scheme. The construction of a B-spline basis for a reduced space of Powell–Sabin splines with full approximation order traces back to [19, Section 4.3] for degree five and was generalized in [20, Section 6] to higher degree. Even though the latter type of reduction of degrees of freedom was not based on imposition of additional smoothness, it was observed in [21] that locally higher smoothness is achieved when considering three-directional triangulations.

The remainder of the paper is organized as follows. Section 2 briefly reviews the Bernstein–Bézier framework for representation of bivariate polynomials on triangles and, based on a recombination of the $C^1$ cubic Powell–Sabin B-splines, introduces a super-smooth spline space on a six-directional triangulation. Section 3 is devoted to the B-spline representation of the super-smooth splines and to the examination of its properties. In Section 4 dyadic subdivision rules are derived for this B-spline representation. Section 5 relates the super-smooth B-splines to simplex splines and (multi-)box splines, and establishes sufficient conditions for ensuring overall $C^2$ smoothness of the spline surface. The paper concludes with some final remarks in Section 6.