Applied Mathematics and Computation

We deal with the problem of constructing, representing, and manipulating 𝐶 3 quartic splines on a given arbitrary triangulation  , where every triangle of  is equipped with the quartic Wang–Shi macro-structure. The resulting 𝐶 3 quartic spline space has a stable dimension and any function in the space can be locally built via Hermite interpolation on each of the macro-triangles separately, without any geometrical restriction on  . We provide a simplex spline basis for the space of 𝐶 3 quartics deﬁned on a single macro-triangle which behaves like a B-spline basis within the triangle and like a Bernstein basis for imposing smoothness across the edges of the triangle. The basis functions form a nonnegative partition of unity, inherit recurrence relations and diﬀerentiation formulas from the simplex spline construction, and enjoy a Marsden-like identity.


Introduction
Splines, in the classical sense of the term, are piecewise functions consisting of polynomial pieces glued together in a certain smooth way, usually by imposing equality of derivatives up to a given order.Besides their theoretical interest, splines find application in a wide range of contexts such as geometric modeling, signal processing, data analysis, visualization, and numerical simulation, just to mention a few.For many of these applications, a high smooth join between the different pieces is beneficial or even required [6,13].
In the univariate case, splines of maximal smoothness, i.e., piecewise polynomials of degree  with  −1 joins, are the most popular splines.In fact, smoother splines give the same approximation order as less smooth splines of the same degree but involve fewer degrees of freedom and have less tendency to oscillate [14,24].
When moving to the bivariate setting and considering polygonal partitions, e.g., triangulations, maximal smoothness is still very appealing but becomes an arduous task to achieve.Bivariate spline spaces with too low degree compared to the smoothness are exposed to several shortcomings: they may lack a stable dimension, optimal approximation power, and stable locally supported bases.In addition, the practical wish of constructing any function of the spline space locally on each of the elements of the partition T. Lyche, C. Manni and H. Speleers ( ) may require a significant gap between the degree () and the smoothness ().For instance, on a triangulation a degree  ≥ 4 + 1 is necessary to admit such a local construction [3,13,33].
The above drawbacks can be mitigated by considering a so-called macro-structure, where the partition  is further refined in a specific manner (often referred to as splits).In case  is a triangulation, the most famous examples are the Clough-Tocher (CT) split [3,4,13,23] and the Powell-Sabin (PS) 6 and 12 splits [1,13,21,23,25].They subdivide each triangle of  into 3, 6, and 12 subtriangles, respectively.Nevertheless, no spline spaces of maximal smoothness  =  − 1 can be constructed over general triangulations with the above mentioned splits for degree  > 2; see, e.g., [13].
In [32], Wang and Shi introduced a family of degree-dependent splitting schemes to refine any triangle Δ of a given triangulation.The split of degree  is obtained by uniformly distributing  +1 points on each edge of Δ and by taking the complete graph connecting these boundary points.For  = 1 we have no split and for  = 2 the Wang-Shi (WS) split reduces to the PS-12 split.Contrarily to the well-known splits mentioned above, when  increases, the family of WS splits generates a very large number of polygonal pieces in each Δ.For cubics we get a set of 75 polygons which includes triangles, quadrilaterals, and pentagons, while for quartics the split consists of 250 polygonal regions.Thanks to this very articulated geometry, the cubic/quartic WS splits allow us to locally construct  2 cubic/ 3 quartic spline spaces on general triangulations.Unfortunately, in practice, the complexity of the geometry hampers a piecewise treatment -in terms of a local polynomial basis -of spline functions on WS splits and discourages the use of such interesting spaces.
To overcome this issue, a simplex spline basis for the local space of  2 cubics on the cubic WS split of any Δ in  has been presented in [15].Such a basis behaves like a B-spline basis within each triangle of  and like a Bernstein basis for imposing smoothness across the edges of  .More precisely, the basis functions form a nonnegative partition of unity, inherit recurrence relations and differentiation formulas from the simplex spline structure, and enjoy a Marsden-like identity for the representation of cubic polynomials.Moreover, they admit simple conditions for  2 joins to neighboring triangles in  and a control net can be set up that mimics the shape of the spline function.Other simplex spline bases for local spline spaces over a macro-triangle have been considered for the PS-12 split in [5,19] and for the CT split in [16,18].The multivariate Alfeld split has been addressed in [17].
In this paper, we consider the quartic WS split for constructing  3 quartic splines on a general triangulation  .Given the complex geometry of the split, it is imperative to produce a basis for the local spline space over each (refined) triangle Δ in  that intrinsically avoids in its construction to deal with separate polynomial representations on each of the 250 polygonal subelements of Δ. Taking into account the complete graph structure of the WS split, a basis formed by (scaled) simplex splines emerges again as the natural solution to the problem.We construct such a basis for the local space of  3 quartics on the quartic WS split of Δ and attain properties similar to the cubic case.The local representation in terms of the simplex spline basis makes that the complex geometry of the WS split is transparent to the user, offering a pathway for effective use of the related spline space.
We remark that for all the above mentioned splits, neither the Bernstein basis nor the simplex spline basis provides a global basis for the full spline space on (the refinement of)  .Global B-spline bases have been developed for  1 PS spline spaces on triangulations [7,10,11,30], for PS spline spaces with higher smoothness [8,27,29], and for  1 CT spline spaces [9,28].A general framework to obtain a global basis is the technique of minimal determining sets described for the local Bernstein basis in [13].The latter technique has been exploited in [15] to produce a global basis starting from local simplex spline bases for  2 cubic splines over triangulations refined according to the cubic WS split.Although a similar approach could be used in the quartic case as well, it seems more convenient to work directly with the local representations provided by the local simplex spline basis, rather than with the global basis for the full spline space.
The remainder of this paper is divided into four sections.In Section 2, we describe the family of WS splits and we summarize the definition and some properties of simplex splines.In Section 3, we present a local simplex spline basis for the refinement of a single triangle and discuss some of its properties.An interesting subspace containing the polynomials of degree four is also provided.The full space of  3 quartic splines is discussed in Section 4 paying special attention to smoothness conditions across the edges of the given triangulation.We end with some concluding remarks in Section 5. Finally, the appendix aggregates data related to the presented simplex spline bases that were used in the proofs and might be useful for practical purposes as well.
Throughout the paper, we use small boldface letters for vectors.Function spaces are denoted by symbols like .In particular, ℙ  stands for the space of bivariate polynomials with real coefficients of total degree ≤ .The partial derivatives in  and  are denoted by   and   , respectively.Given a vector , the associated directional derivative is denoted by   .The directional derivative in the direction of the vector from point  1 to  2 is denoted by   1  2 .

Preliminaries
In this section, we collect some preliminary material about the Wang-Shi splits and simplex splines of interest in the rest of the paper.

The Wang-Shi splits
Given three noncollinear points  1 ,  2 ,  3 in ℝ 2 , the triangle Δ ∶= ⟨ 1 ,  2 ,  3 ⟩ with vertices  1 ,  2 ,  3 will serve as our macrotriangle.Given a degree  ∈ ℕ, we divide each edge of Δ into  equal segments, respectively, resulting into 3 boundary points.Then, we refine Δ into a number of subelements delineated by the complete graph connecting those boundary points which consists of the three edges of the triangle and of 3( − 1) interior lines.The resulting partition is called the WS  split of Δ as it was originally proposed by Wang and Shi [32].We denote the obtained mesh structure by Δ WS  , and the set of polygons in Δ WS  by   .Note  that we are dealing with a cross-cut partition of Δ because we are drawing only lines connecting boundary points.All the possible intersections of these lines are called vertices of Δ WS  .The cases  = 2, 3, 4 are shown in Fig. 1.For  = 1 we have  1 = {Δ}, while for  = 2 we obtain the well-known PS-12 split [21].For  > 2 not all elements of   are triangles and the complexity of the mesh grows quickly.For  = 3 we have 58 vertices and 75 regions, while for  = 4 the number of vertices increases up to 178 and we get 250 regions; see Figs. 2 and 3.
We consider the space The dimension of  −1  (Δ WS  ) can be computed using the general dimension formula for spline spaces over cross-cut partitions from [2, Theorem 3.1].In particular, under the assumption that at most  + 1 cross-cuts intersect at an interior vertex of Δ WS  , we have Theorem 2].Thus, for  = 4, we get because it can be directly checked that at most 4 cross-cuts intersect at any interior vertex of Δ WS 4 ; see Fig. 2.

Simplex splines
Simplex splines are a very elegant extension of univariate B-splines to the multivariate setting.Like B-splines, they are defined in terms of knots whose number and position completely determine their properties.In this section, we define simplex splines and we shortly summarize some properties of interest for the paper.For further properties and proofs, we refer the reader to, e.g., [20,22].
The simplex spline  Ξ can be defined geometrically by vol  () .
For  = 0 we have and the value of  Ξ on the boundary of ⟨Ξ⟩ has to be dealt with separately.We refer the reader to [26] for a convention to decide in which region each edge and vertex belongs.The simplex spline  Ξ is a nonnegative piecewise polynomial of total degree , supported on ⟨Ξ⟩ and has unit integral.Moreover,  Ξ enjoys the following fundamental recurrence formulas, sometimes referred to as the  of simplex splines, which generalize the well-known analogs for B-splines.
If  = 1 then  Ξ is the univariate B-spline of degree  with knots Ξ, normalized to have its integral equal to one.In the bivariate case,  = 2, the lines in the complete graph of Ξ are called knot lines and provide a partition of ⟨Ξ⟩ into polygonal elements.Denoting by #Ξ the cardinality of Ξ, the simplex spline  Ξ is a polynomial of degree  = #Ξ − 3 in each region of this partition.Across a knot line we have  Ξ ∈  +1− , where  is the number of knots on that knot line, including multiplicities.

𝟒 (𝚫 𝐖𝐒 𝟒 )
In this section, we focus on the space  3  4 (Δ WS 4 ); see (1) with  = 4.We provide a basis for the space consisting of scaled simplex splines and consider some properties of such a basis.With a slight abuse of notation we also refer to the basis functions as simplex splines.The simplex spline basis intrinsically takes care of the complex geometry of the WS 4 split by construction.Therefore, it provides a tool to practically deal with this quartic space.

A simplex spline basis
For a given triangle Δ = ⟨ 1 ,  2 ,  3 ⟩, the WS 4 split is shown in the right plot of Fig. 1.From (2) we know that the dimension of  3  4 (Δ WS 4 ) is 51.In order to construct a basis for this space, we first specify 12 points along the boundary of the triangle (see Fig. 4): the three vertices  1 ,  2 ,  3 , the points and the midpoints of the edges Note that these points are part of the WS 4 split.We then consider the quartic simplex splines  1 , … ,  51 as schematically illustrated in Fig. 5, where each simplex spline has seven (including multiplicity) knots chosen among the 12 points above.For instance,  4 is defined by the sequence The B-recurrence relation may be used to compute each of them.We define the following set of 51 (scaled) simplex splines: with the scaling factors given by Proof.Let  be one of the functions   .We first prove that  belongs to the space  3 4 (Δ WS 4 ).Since  has seven knots, it is a piecewise quartic polynomial.Moreover, since the knots of  are a subset of the knots shown in Fig. 4, the knot lines of  are a subset of the knot lines in the complete graph; see Fig. 1 (right).We also see that each interior knot line contains at most two knots, so  has  3 where The computed values are shown in Tables 3 12), ( 13), and ( 16).Right: ( 14) and ( 15).
The proof of Theorem 1 implies that we can formulate a Hermite interpolation problem to characterize any spline in  3 4 (Δ WS 4 ); see also [31,Theorem 7.28].
A schematic visualization of the Hermite degrees of freedom specified in Corollary 2 is shown in Fig. 6 using graphical symbols that are common in finite element literature; see, e.g., [3].

Domain points
For the scaled simplex spline basis (5), we now compute special points in Δ that are often used in geometric modeling to give a geometric interpretation to the representation of any element of  3 4 (Δ WS 4 ) in terms of the considered spline basis.To this end, we solve the system for the two functions  1 (, ) ∶=  and  2 (, ) ∶= .The points are called the domain points of the basis (5).Together with the partition of unity, the domain points provide an explicit representation of any affine function with respect to the basis (5).The barycentric coordinates with respect to the triangle Δ of the domain points (17), multiplied by the common factor 336, are given by These points are depicted in Fig. 7.When representing a spline  ∈  3 4 (Δ WS 4 ) in the basis (5), the corresponding control points are defined as ( *  ,   ),  = 1, … , 51 which are usually connected into a control net.Unfortunately, the domain points in (18) are not all distinct; there are three pairs of coincident points, namely { * 40 ,  * 41 }, { * 42 ,  * 43 }, and { * 44 ,  * 45 }.This makes their use of limited interest in geometric modeling.

A Marsden-like identity
We now provide a Marsden-like identity to represent any quartic polynomial with respect to the scaled simplex spline basis (5).The results are obtained by direct computation.Theorem 3. Given the triangle Δ = ⟨ 1 ,  2 ,  3 ⟩, we first introduce the points T. Lyche, C. Manni and H. Speleers
and recall the points  , and   defined in (3) and (4), respectively.We have the identity where the functions   are called dual polynomials.The first 48 dual polynomials can be written in the form ) , ) , ) .

A proper subspace
As observed in Section 3.2, the domain points for the basis functions in (5) are not all distinct and this makes their use of limited interest in geometric modeling.We now provide a proper subspace of the space  3  4 (Δ WS 4 ), still containing ℙ 4 , which can be equipped with a simplex spline basis whose elements enjoy distinct domain points.
We consider the subspace spanned by the simplex splines identified by the knot sequences in Fig. 8 and scaled to form a partition of unity.We denote these functions by It can be directly checked that These points are depicted in Fig. 9 (left) and a possible control net with triangular and hexagonal connectivity in Fig. 9 (right).Note that, contrarily to the domain points associated with the basis (5), the points in ( 23) are all distinct and therefore allow us to define a proper control net.The distribution of the domain points in (23) makes the considered subspace particularly attractive from the geometric modeling point of view.The following result ensures that the subspace also contains all quartic polynomials and so can provide full approximation power.

𝑪 𝟑 quartic splines on general triangulations
In the previous section, we have provided a simplex spline basis for the space  3 4 (Δ WS 4 ) of  3 quartic splines on the WS 4 split of a given triangle Δ.In this section, we discuss how the WS 4 split can be used as refinement strategy to locally construct  3 quartic splines on general triangulations.Let  be a triangulation of a polygonal domain Ω and let  WS 4 denote its refinement obtained by taking the WS 4 split of each triangle in  .

𝐶 3 quartic splines on  WS 4
Let us consider the space of  3 quartic splines on  WS 4 , i.e., The unisolvency of the local Hermite interpolation problem stated in Corollary 2 implies that the dimension of the global space  3  4 ( WS 4 ) only depends on combinatorial properties of the triangulation.Indeed, for a given triangle Δ and any of its edges, say  1  2 , the interpolation data ( 11)-( 16) associated with points on the considered edge (namely  1 ,  3,1 ,  3 ,  3,2 , and  2 ) completely characterize the values of the spline and its derivatives up to order three along the entire edge.More precisely, along this edge, any element of  3  4 (Δ WS 4 ) is a univariate  3 quartic spline with break points at  1 ,  3,1 ,  3 ,  3,2 , and  2 .Therefore, it is uniquely determined by its values and derivatives (in the direction of the edge) up to order three at the two vertices  1 and  2 .Similarly, T. Lyche, C. Manni and H. Speleers along the edge, the first, second, and third derivatives (in any direction) of the spline are univariate  2 cubic,  1 quadratic, and  0 linear splines, respectively, with the same break points.The considered Hermite data also uniquely specify their values.Hence, we directly deduce that dim( 3  4 ( WS 4 )) = 10  + 6  + 3  , (24) where   ,   ,   are the number of vertices, edges, and triangles of  , respectively; see also [31,Theorem 7.28].
Any spline function of  3 4 ( WS 4 ) can be locally constructed on each (macro-)triangle Δ of  via the Hermite data (used in Corollary 2), and the corresponding spline piece on Δ can be represented in the form (19). Conversely, any function, which is represented locally in the form (19) on each Δ of  , is  3 smooth over each Δ of  but not necessary  3 smooth across the edges of  .For this reason we refer to the basis in (5) as local simplex spline basis for the space  3  4 ( WS 4 ).Such a local basis plays the same role as Bernstein polynomials for spline spaces on triangulations that are not refined.
A global basis for the space  3 4 ( WS 4 ) can be constructed by using the technique of minimal determining sets [13], following the same line of arguments as in [15,Section 4.2].However, paraphrasing [13, Section 5.8], such a global basis has mainly a theoretical interest.For computation in  3  4 ( WS 4 ), it is more convenient to work directly with the local representations provided by the basis in (5), rather than with the global basis for the full spline space.The smoothness relations between the local representations across the edges of the triangulation  are a crucial ingredient in the construction of minimal determining sets.In the next two sections, we will derive conditions on the local spline coefficients in (19) that ensure global smoothness across the edges of  .

Global 𝐶 2 smoothness conditions
In this section, we seek conditions on the local spline coefficients in (19) to guarantee  0 ,  1 , and  2 smoothness across a common edge of two adjacent triangles of  .In Section 4.3, we will address  3 smoothness.To this end, we start by defining specific operators, similar to the ones in ( 7)-( 9):  1 , … ,  6 are related to first derivatives at points on the edge We apply them to all the basis functions in (5); the resulting values are collected in Tables 6-8 in the appendix.
Next, we consider  1 smoothness across the common edge.It suffices to prove that along the edge  1  2 the functions   1  3   and   1  3   agree.These functions are univariate  2 cubic splines with (interior) knots at the points  3,1 ,  3 , and  3,2 .Therefore, each of them is uniquely determined by its value and its first and second derivative at the two endpoints of the edge and by the value at the midpoint  3 .From (28) we know that By employing the values in Table 6 we get With the aid of the  0 smoothness conditions, equating the above expressions results in the first condition of (30).With a similar line of arguments, we can show the remaining six conditions of (30).

T. Lyche, C. Manni and H. Speleers
Finally, we outline the proof for  2 smoothness.Assume   and   join  1 across the common edge  1  2 .To prove  2 smoothness across the same edge, it suffices to prove that along  1  2 the functions  2  1  3   and  2  1  3   agree.Along the edge  1  2 , the above functions are univariate  1 quadratic splines with (interior) knots at the points  3,1 ,  3 , and  3,2 .Therefore, each of them is uniquely determined by its value and its first derivative at the two endpoints of the edge and by the value at the points  3,1 and  3,2 .From (28) we know that By employing the values in Table 6 in the appendix, a direct computation gives By equating these expressions and taking into account the  1 smoothness conditions, we get the first equation in (31).With a similar line of arguments, we can show the remaining five relations in (31).□ Some remarks are in order.

𝑖
and  *   ,  = 1, … , 51 in two adjacent triangles Δ  and Δ  satisfy conditions ( 29)-( 31) across the common edge because they are the coefficients related to the two local simplex spline bases of the smooth functions  1 (, ) ∶=  and  2 (, ) ∶= .This allows us to give a nice geometric interpretation to the above mentioned smoothness conditions in terms of control points in a full analogy with the interpretation of smoothness conditions for the Bernstein-Bézier representation.For  0 smoothness, only the eight pairs of control points associated with the common edge are involved and the elements of each pair must coincide, i.e., For  1 smoothness, there are seven sets of four control points along the common edge that must be all pairwise coplanar, i.e.,    6 ); see Fig. 10 for an illustration.Finally, also the  2 smoothness conditions show a similarity with the  2 join of two adjacent triangular Bernstein-Bézier patches; see [12,Example 2].There are six sets of seven control points that must belong (separately) to a quadratic surface.For the sake of brevity we omit the equations that can be immediately obtained from (31).Remark 7. Since the smoothness conditions provided in Theorem 5 do not involve the functions {  ,  = 40, … , 51}, the same relations also hold true for the subspace Ŝ3 4 (Δ WS 4 ) defined in Section 3.4.

Global 𝐶 3 smoothness conditions
The basis functions   ,  = 40, … , 48 and  = 49, 50, 51, can be associated with the degrees of freedom corresponding to third derivatives related to the edges of Δ (i.e., those used by the linear operators   ,  = 40, … , 48) and to function values at interior points (i.e., those used by the linear operators   ,  = 49, 50, 51), however, they do not allow for a complete separation of these degrees of freedom.This can be understood by looking at the structure of the Hermite collocation matrix which is block triangular but not triangular in the part containing the last 11 rows/columns; see Tables 3-5 in the appendix.Unfortunately, this prevents the possibility of obtaining elegant localized conditions on the local spline coefficients in (19) for  3 smoothness across the edges of  .
Inspired by the approach in [15, Section 3.4], we consider an alternative local basis for  3 4 (Δ WS 4 ) to achieve more practical  3 smoothness conditions.We denote this basis by T. Lyche, C. Manni and H. Speleers The functions in (32) are computed as where  is the conversion matrix specified in Table 2.

Proof. Since the conversion matrix in Table
Regarding the property of vanishing third derivatives, let us focus on the first case a).From the smoothness property of simplex splines it follows that the functions  35 ,  36 ,  40 ,  41 ,  42 ,  43 ,  46 ,  49 are  2 smooth across the edge  1  2 (see Fig. 5), thus they vanish with all their derivatives up to order two along the edge.By construction, this property carries over to the functions B35 , B36 , B41 , B42 , B49 .Moreover, along the same edge, the third derivative in any direction of these functions is a univariate  0 linear spline with break points at  1 ,  3,1 ,  3 ,  3,2 ,  2 .Now, let us consider the function B35 .Taking into account the supports of  35 ,  40 , and  41 , a direct inspection of the values in Table 2 and Tables 4,5 in the appendix shows that Therefore,  3  3,1  3 B35 vanishes along  1  2 .Since we already know that all the derivatives of B35 vanish up to order two along  1  2 , this implies that all third derivatives (in any direction) of B35 also vanish along  1  2 .With the same line of arguments, the stated results for the other functions and the other derivative cases can be obtained as well.□ When representing spline functions on two adjacent macro-triangles sharing a common edge, say  1  2 , in terms of the new basis (32), the  2 smoothness conditions remain the same as given in Theorem 5. Indeed, any derivative up to the second order in any direction along  1  2 agrees for   and B ,  = 1, … , 51.On the other hand, the representation in the basis (32) allows us to derive simpler  3 smoothness conditions.

T. Lyche, C. Manni and H. Speleers
and (37) Proof.Assume   and   join  2 across the common edge  1  2 .To prove  3 smoothness across the same edge, it suffices to prove that along  1  2 the functions  3 agree.Along the edge  1  2 , the above functions are univariate  0 linear splines with (interior) knots at the points  3,1 ,  3 , and  3,2 .Therefore, each of them is uniquely determined by its value at the two endpoints of the edge and by its value at the points  3,1 ,  3 , and  3,2 .From (28) we know that By using the values in Tables 6-8 in the appendix together with those in Table 2, a direct computation provides the expressions of By equating these expressions and taking into account the  2 smoothness conditions, we obtain the conditions ( 33)-(37).□ Remark 10.Using the conversion between the bases ( 5) and (32), see Table 2, we can immediately rewrite the  3 smoothness conditions provided in Theorem 9 solely in terms of coefficients of the basis (5).

Remark 11.
A global basis for the space  3 4 ( WS 4 ) can be constructed by using the technique of minimal determining sets [13], following the same line of arguments as in [15,Section 4.2].Even though both the local bases ( 5) and ( 32) may be used to this aim, the second one is preferable.Indeed, when using the basis (32), the locality of the smoothness conditions provided in Theorems 5 and 9 implies the local support of the global basis functions.

Conclusion
We have addressed the 51 dimensional space of  3 quartic splines on the WS 4 split of a given triangle.For this space we have constructed a (scaled) simplex spline basis and we have provided a unisolvent Hermite interpolation problem which allows us to locally characterize any function belonging to the space of  3 quartic splines on the WS 4 refinement of a given triangulation.These results are a quartic extension of those obtained in [5] and [15] for the PS-12 split and the WS 3 split, respectively, i.e., for the quadratic and the cubic members of the Wang-Shi split family.
As in the quadratic and cubic cases, the basis is a B-spline basis within a single macro-triangle but globally it behaves like a Bernstein basis across macro-triangles.We have proved that the basis has many desirable B-spline properties including that it forms a nonnegative partition of unity and enjoys a Marsden-like identity.The restriction of each basis function onto a boundary edge of the macro-triangle reduces to a classical univariate  3 quartic B-spline.
The basis can be equipped with control points and a control net.The control net is unusual in the sense that it incorporates three pairs of coincident domain points.A proper 48 dimensional subspace containing quartic polynomials has been identified for which a classical control net (based on different domain points) with triangular and hexagonal connectivity can be naturally constructed.Furthermore, explicit conditions for smoothness up to the third order across a common edge of two macro-triangles have been derived in terms of the respective control points.We have also proposed an alternative basis for the space  3 4 (Δ WS 4 ) that simplifies the imposition of  3 smoothness conditions across edges.
The technique of minimal determining sets can be used to build (stable) locally supported global basis functions for the space  3  4 ( WS 4 ); see Remark 11.In addition, the availability of such a global basis allows us to show the optimal approximation power of the space using a similar line of arguments as in [15,Section 4.2].
For computation with splines belonging to  3 4 ( WS 4 ), it is convenient to work directly with the local representations provided by the simplex spline basis (5) or the alternative basis (32).The evaluation of the simplex spline basis functions can be achieved by applying the recurrence relation (B-recurrence) of simplex splines; see Section 2.2.However, it might be more convenient to use a lookup-table approach as proposed in [15, Section 5.1] for the  2 cubic case.The explicit expressions of the simplex spline basis elements can be stored in a suitable table and then accessed by a search algorithm based on boolean vectors.The alternative basis functions in (32) can be immediately deduced from the previous ones by means of the conversion matrix in Table 2.
The local simplex spline basis makes the treatment of the complex geometry of the WS 4 split more affordable because it intrinsically avoids to consider separate polynomial representations on each of the 250 polygonal subelements of the split.This transparency of the WS 4 split to the user offers a pathway for effective use of the interesting space of  3 quartic splines on the WS 4 refinement of a given triangulation.Generalizations of the presented construction to higher degrees  ≥ 5 would be desirable.Extensions to splines in three or more variables would be of great interest as well.However, neither appears straightforward.3 Values of   (  ) for  = 1, … , 51,  = 1, … , 18, where   is defined in (7), (8)

Fig. 5 .
Fig. 5. Sequences of knots for a set of simplex spline basis functions for  3 4 (Δ WS 4 ).Each black disc shows the position of a knot and the number inside indicates its multiplicity.

Fig. 8 .
Fig. 8. Sequences of knots for a set of simplex spline basis functions for a subspace of  3 4 (Δ WS 4 ).Each black disc shows the position of a knot and the number inside indicates its multiplicity.

Fig. 10 .
Fig. 10.A  1 spline surface on two adjacent domain triangles.The 7 pairs of triangles in the control nets that must be coplanar according to the smoothness conditions are colored.

Fig. 17
Fig. 17.The simplex spline basis function  31 and its support.

.
Fig. 17.The simplex spline basis function  31 and its support.

𝒑 1 , 𝒑 1 , 𝒑 1 , 𝒑 1 , 𝒒 3,1 , 𝒒 3 , 𝒒 3,2 , 𝒑 2 , 𝒑 2 , 𝒑 2 , 𝒑 2 , 𝒑 2 }. Theorem 1. The
Here |Δ| stands for the area of Δ.Note that the scaling factors sum up to |Δ|.There are eleven different types of simplex splines in .For each type, a representative   is depicted in Figs.11-21 in the appendix; the remaining ones can be obtained by symmetry.On any edge of Δ, there are eight nonzero basis functions.Their restrictions to that edge are nothing but the set of univariate  3 quartic B-splines defined on a uniform open-knot sequence with three interior knots.For instance, for the edge  1  2 , they correspond to the univariate quartic B-splines on the knot sequence specified by { 1 , simplex splines { 1 , … ,  51 } in (5) form a nonnegative partition of unity basis for the space  3 4 (Δ WS 4 ).
(32)be the spline bases as in(32)on Δ  and Δ  , respectively.Then, the spline functions Theorem 9. Consider the same assumptions as in Theorem 5. Let { B  ,  = 1, … , 51} and { B  ,  = 1, … ,  join  3 across the common edge if and only if they join  2 and in addition , and (9).