On the dimension of spline spaces on planar T-subdivisions

We analyze the space Sr,r ′ m,m (T ) of bivariate functions that are piecewise polynomial of bidegree (m,m) and class C ′ over the planar T-subdivision T . We give a new formula for the dimension of this space by exploiting homological techniques. We relate this dimension to the number of nodes on the maximal interior segments of the subdivision, give combinatorial lower and upper bounds on the dimension of these spline spaces for general hierarchical T-subdivisions. We show that these bounds are exact, for high enough degrees or if the subdivision is enough regular. Finally, we analyse cases of small degrees and regularities. Introduction Standard parametrisations of surfaces in Computer Aided Geometric Design are based on tensor product bspline functions, defined from a grid of nodes over a rectangular domain. These representations are easy to control but their refinement has some drawback. Inserting a node in one direction of the parameter domain implies the insertion of several control points in the other directions. If for instance, regions along the diagonal of the parameter domain should be refined, this create a fine grid at some region where it is not needed. To avoid this problem, while extending the standard tensor product representation of CAGD, representations associated to subdivisions with T-junctions instead of a grid, have been analyzed. Such a T-subdivision is a partition of an axis-aligned box Ω (e.g. the unit square) into smaller axis-aligned boxes, called the cells of the subdivision. The first type of T-splines introduced in [17, 16], are defined by blending functions which are product of univariate bspline basis function associated to some nodes of the subdivision. They are piecewise polynomial functions, but the pieces were these functions are polynomial do not match with the cells of the T-subdivision. There is no proof that these piecewise polynomial functions are linearly independent. Indeed, [2] shows that in some cases, these blending T-spline functions are not linear independent. Moreover, there is no characterisation of the vector space span by these functions. For this reason, the partition of unity properties useful in CAGD are not available directly in this space. The spline functions have to be combined into piecewise rational functions, so that these piecewise rational functions sum to 1. However, this construction tends to complexify the practical use of such T-splines. Being able to describe a basis of the vector space of piecewise polynomials on a T-subdivision is an important but non-trivial issue. It yields a construction of piecewise polynomial functions on the T-subdivision which form a partition of unity so that the use of piecewise rational functions is not needed. It has also a direct impact in approximation problems such as surface reconstruction or isogeometric analysis, where controlling the space of functions used to approximate a solution is critical. In CAGD, it also provides more degrees of freedom to control a shape. That is why further 1 in ria -0 05 33 18 7, v er si on 1 5 N ov 2 01 0 works have been developed to understand better the space of piecewise polynomial functions on a T-subdivision. In [4], [6], a second family of splines on hierarchical T-subdivisions have been studied to tackle these issues. These splines are piecewise polynomial and form a basis of the space of piecewise polynomial functions on a hierarchical T-subdivision. They are called PHT-splines (Polynomial Hierarchical Tsplines). Dimension formulae of the spline space on such a subdivision have been proposed when the degree is high enough compared to the regularity [4], [10], [12] and in some cases for biquadratic C piecewise polynomial functions [5]. The construction of a basis is described for bicubic C spline spaces in terms of the coefficients of the polynomials in the Bernstein basis attached to a cell. When a cell is subdivided into 4 subcells, the Bernstein coefficients of the basis functions of the old level are modified and independent functions are introduced, using Bernstein bases on the cells at the new level. In this paper, we analyse the space S ′ m,m′(T ) of bivariate functions that are piecewise polynomial of bidegree (m,m) and class C ′ over a planar T-subdivision T . We give a new formula for its dimension. By extending homological techniques developed in [1] and [15], we relate this dimension to the number of nodes on the maximal interior segments of the subdivision. We show that for m > 2r+1 and m > 2r + 1, the dimension depends directly on the number of faces, interior edges and interior points, providing a new proof of the dimension formula proved in [4], [10], [12] for a hierarchical Tsubdivision. The algebraic approach provides an homological interpretation of the so-called Smoothing Cofactor-Conformality method [19]. It allows us to generalize the dimension formulae obtained by this technique [12], [10]. In particular, it yields combinatorial lower and upper bounds on the dimension of these spline spaces for general T-subdivisions. We also show that these bounds are exact for T subdivisions which are enough regular. As a consequence, we give the dimension of the space of Locally Refined splines described in [7]. We do not consider the problem of constructing explicit bases for these spline spaces, which will be analyzed in a forthcoming paper [13]. In the first section, we recall the notations, the polynomial properties and the homological constructions which are needed in the following. Section 2 deals with the properties of T -subdivisions that we will use. In section 3, we detail the construction of the topological complex, describe its homology and proof the dimension formulae. In the last section, we analyse some examples for small degree and regularity. 1 Planar T-splines 1.1 Notations We denote by R = K[s, t] the space of polynomials in the variables s, t with coefficients in the field K of characteristic 0. Let Rm,m′ denote the space of polynomials of degree 6 m in s and degree 6 m ′ in t. For any line L of K with equation ∆L(s, t) = 0 and for r ∈ N, let IL,r = (∆L(s, t) ). For two such lines L,L which intersect at a point γ ∈ K and for m,m ∈ N, the ideal IL,r + IL′,r′ = (∆L(s, t) ) + (∆L′(s, t) r′+1) defines the point γ with multiplicity (r + 1)× (r + 1). Let T be a subdivision of a rectangular domain D0 ⊂ R 2 into a rectangular complex. • The faces of dimension 2 of T are rectangles. The set 2-faces of T is denoted by T2 and the number of elements in T2 is f2. • The faces of dimension 1 of T are edges which are either vertical or horizontal. The set 1-faces of T is denoted by T1. The set of interior faces, that is those which intersect the interior of Ω is denoted by T o 1 . The number of edges in T1 is f1. Let f h 1 (resp. f v 1 ) be the number of interior horizontal (resp. vertical) edges.The number of interior edges is f 1 = f h 1 + f v 1 . Let ∆τ (s, t) = 0 be the equation associated to line supporting τ . If an edge is vertical, we define δ(τ) = (r + 1, 0) and ∆ (r,r′) τ = ∆ τ . Otherwise δ(τ) = (0, r ′ + 1) and ∆ (r,r′) τ = ∆ ′+1 τ . We 2 in ria -0 05 33 18 7, v er si on 1 5 N ov 2 01 0 denote I ′ (τ) = (∆ (r,r′) τ ) and I r,r′ m,m′(τ) = I r,r′(τ) ∩Rm,m′ . Notice that two horizontal (resp. vertical) edges τ1, τ2 which intersect define the same ideal I ′(τ1) = I ′(τ2). • The faces of dimension 0 of T are the vertices of the subdivision. The set of vertices of T is denoted T0. The set of interior vertices is denoted T o 0 . Let f0 be the number of vertices of T0, f 0 be the number of interior vertices, f + 0 be the number of interior crossing vertices, f T 0 the number of interior T -vertices, f b 0 the number of boundary vertices, counting the 4 corner points. There are f b 0 − 4 boundary points which are T -vertices. A vertex γ ∈ T0 is the intersection of a vertical edge τv with an horizontal edge τh. Let I r,r′(γ) = I ′(τv)+I ′(τh) = (∆ (r,r′) τv ,∆ (r,r′) τh ). We denote by I r,r′ m,m′(γ) = I r,r′ m,m′(τv)+I r,r′ m,m′(τh). Notice that these definitions are independent of the choice (if any) of the vertical edge τv and horizontal edge τh which contain γ. In the following, we are going to analyse the following space. Definition 1.1 Let S ′ m,m′(T ) be the vector space of functions which are polynomials in Rm,m′ on each face σ ∈ T2 and of class C r in s and C ′ in t on Ω. We will say that f ∈ S ′ m,m′(T ) is C r,r′ regular. Our aim is to describe its dimension in terms of combinatorial quantities attached to T and an explicit basis. Given an element f ∈ S ′ m,m′(T ), we denote by f ε the function on K, which coincide with f on Ω and which is 0 outside. We are interested in the analysis of the following space. 1.2 Polynomial properties We recall here basic results on the dimension of the vector spaces involved in the analysis of S ′ m,m′(T ): Lemma 1.2 • dimRm,m′ = (m+ 1)× (m ′ + 1). • dimRm,m′/I r,r′ m,m′(τ) = { (m+ 1)× (r + 1) if τ is a horizontal edge (r + 1)× (m + 1) if τ is a vertical edge • dimRm,m′/I r,r′ m,m′(γ)) = (r + 1)× (r ′ + 1). An algebraic way to characterise the C ′ -regularity is given by the next lemma [1]: Lemma 1.3 Let τ ∈ T1 and let p1, p2 be two polynomials. Their derivatives coincide on τ up to order (r, r) iff p1 − p2 ∈ I r,r′(τ). In the following, we will use the apolar product defined on polynomials of degree 6 n of K[x] by: