Generalized Tsplines and VMCR Tmeshes
Abstract
The paper considers the extension of the Tspline approach to the Generalized Bsplines (GBsplines), a relevant class of nonpolynomial splines. The Generalized Tsplines (GTsplines) are based both on the framework of classical polynomial Tsplines and on the Trigonometric GTsplines (TGTsplines), a particular case of GTsplines. Our study of GTsplines introduces a class of Tmeshes (named VMCR Tmeshes) for which both the corresponding GTsplines and the corresponding polynomial Tsplines are linearly independent. A practical characterization can be given for a subclass of VMCR Tmeshes, which we refer to as weakly dualcompatible Tmeshes, which properly includes the class of dualcompatible (equivalently, analysissuitable) Tmeshes for an arbitrary (polynomial) order.
Keywords: Tspline, Tmesh, GBspline, analysissuitable, dualcompatible, linear independence.
1 Introduction
^{†}^{†}footnotetext: Email addresses: cesare.bracco@unito.it (C. Bracco).In the last years, the introduction of the socalled Tsplines and of the spline spaces defined over Tmeshes introduced significant advancements for the use of polynomial spline functions in the CAD and CAGD techniques. The main idea of this approach, in the basic case of surface modelling in , is to free the control points of the surface from the constraint to lie, topologically, on a rectangular grid whose edges intersect only at “cross junctions”, and allow instead partial lines of control points, which leads to the possibility to have “Tjunctions”between the edges of the grid. Such a framework gave some important improvements in CAD and CAGD methods: the possibility to locally refine the surfaces, a considerable reduction of the quantity of control points needed, the ability to easily avoid gaps between surfaces to be joined (see, e.g., [19] and [20]), just to name a few. All these advantages became even more important in the applications, such as the isogeometric approach for the analysis problems represented by partial differential equations (see, e.g., [7], [8] and [1]).
The Tspline idea has been applied mainly to polynomial splines, while we know that several types of nonpolynomial splines are used for certain applications because of their particular properties. For this reason, recently we proposed a generalization of the Tspline approach to the trigonometric GBsplines (see [3]), a particularly relevant class of nonpolynomial splines because of their adaptability and their application to the already mentioned isogeometric analysis (see, e.g., [10] and [12]). Roughly speaking, the GBsplines are a basis of spaces of piecewise functions, locally spanned both by polynomials and by nonpolynomial functions, which in the trigonometric case are and , with a given frequency . Note that these splines can be seen as particular cases of the piecewise Extended Chebyshevian splines (see, e.g., [13], [14] and [15]). GBsplines have been successfully used to construct tensorproduct surfaces (see, e.g., [12] and references therein) with control points on rectangular grids.
In this paper, we will first extend the results in [3] to any type of GBsplines of arbitrary biorder , so that we can take full advantage of the good features of GBsplines and Tsplines. In order to achieve this goal, we will start by presenting the univariate GBsplines and their properties, including a knot insertion formula with necessary conditions, which will be also essential in the study of the linear independence of the GTspline functions. Then, we will introduce the GTsplines, whose definition (and notations) is based both on the polynomial Tsplines (see, e.g., [4], [5] and [11]) and on the TGTsplines (see [3]). Similarly to the case of TGTsplines [3], we will show that there exists a relation between the GTsplines of biorder and the polynomial Tsplines of the same biorder. The study of their linear independence will lead us to the introduction of the class of VMCR Tmeshes (Void Matrix after Column Reduction Tmeshes), which guarantee the linear independence of the associated GTspline and Tspline blending functions of the same biorder. The basic concept behind VMCR Tmeshes involves the idea of column reduction (used in [11]), and implicitly helped to show in [3] that in the case of biorder the wellknown analysissuitable Tmeshes are also VMCR Tmeshes. In this paper we provide a simple characterization of a subclass of VMCR Tmeshes, which we refer to as weakly dualcompatible Tmeshes: we will prove that such class strictly includes the one of dualcompatible/analysissuitable Tmeshes (see, e.g., [5] and [11]) for any biorder . Finally, we will present an explicit example of weakly dualcompatible Tmeshes which is not dualcompatible/analysissuitable.
The paper is organized as follows. In Section 2 we recall the definition and the basic properties of the univariate GBsplines, and we deal with the conditions needed to get a knot insertion formula. In Section 3, after having recalled the definition of Tmesh, we introduce the GTsplines and we give some properties following directly from their definition. In Section 4 we study the linear independence of the GTspline blending functions and, more importantly, the classes of VMCR Tmeshes and of weakly dualcompatible Tmeshes. Finally, Section 5 contains some concluding remarks.
2 Univariate generalized Bsplines
2.1 Definition and main properties
Let , , and let be a nondecreasing knot sequence (knot vector); we associate to two vectors of functions and , where, for , belong to and are such that the space spanned by the derivatives
is a Chebyshev space, that is, any function belonging to it has at most one zero in . Let, for , be the multiplicity of in , that is, the cardinality of the set
Note that if . We assume that , for . We consider the generalized spline space spanned, in each interval , by for and by for . For this space we can define a basis of compactlysupported splines, which are called Generalized Bsplines (GBsplines).
The definition of such basis is usually given in a recursive fashion, which we briefly recall (see also [10] and [12]). Since we required that the space spanned by and , denoted by , is a Chebyshev space, it is not restrictive to choose, as generating functions of , two functions and such that
(1) 
Since is a Chebyshev space, for any and for any . We will call the selected functions and generating functions associated to . Then, following [10], [12], we can define a basis of compactlysupported spline functions for the generalized spline space in the following way: for
(2) 
while, for ,
(3) 
where
(4) 
Moreover, if , we set
The GBsplines have essentially the same properties of the classical polynomial splines.
Property 2.1.
The GBsplines satisfy the following properties.

Continuity: each is times continuously differentiable at the knot , where is the multiplicity of in the knot vector , with .

Positivity: for , and , .

Local support: if , and , .

Partition of unity: for and , .

Linear independence: for any are linearly independent.
Therefore, the GBsplines can be used, just like the polynomial Bsplines, to define a GBspline curve:
(5) 
where are the control points corresponding to . Because of the properties of positivity, local support and partition of unity, the control points in (5) play the same role of the control points in the polynomial Bspline curves.
2.2 Knot insertion formula
One of the main reasons to introduce the Tspline approach in the construction of spline surfaces is, as already mentioned, the possibility to apply local refinement techniques. Therefore, it is crucial to have a knot insertion formula in the univariate case. In the case of GBsplines, differently from the polynomial case (see Boehm’s seminal work [2]), we need to pay attention to the issue that a knot insertion requires two new additional functions and that the space refined by applying the knot insertion formula must contain the original space. The resulting knot insertion rule is stated below.
Theorem 2.2.
Let be a knot vector, the knot vector obtained by inserting a new knot , . Let , and , be the corresponding vectors of functions, where
(6) 
If we denote by and the GBsplines of order , respectively before and after the knot insertion, and by the multiplicity of in , then we obtain
(7) 
with, for ,
and, for ,
where and are the constants defined by (4) for and respectively, and and , and , and are the generating functions associated to , , , respectively, and such that .
3 Generalized Tsplines
In order to define the GTsplines, we need to briefly recall some definitions and notations about the Tmeshes, which are the same used for the classical polynomial case (see, e.g., [4] and [5]) and for the TGTsplines in [3].
Let and be two index vectors, where , are equal to or greater than and, for any real number , is the largest integer smaller than or equal to . Analogously, , , and are the associated vectors of functions.
An index Tmesh is a rectangular partition of the index domain such that the vertices have integer coordinates (see Figure 3(a)). In other words, is the collection of all the elements of such partition, which are called cells. Note that, since the elements are rectangular, Tjunctions are allowed but Ljunctions or Ijunctions are not. We call edge any segment, either horizontal or vertical, linking two vertices of the mesh. We denote the set of vertices by and by , and the sets containing only horizontal, only vertical and all the edges respectively. The valence of a vertex is the number of edges such that . Finally, we denote by the union of all the edges and vertices.
We define the active region and frame region (see Figure 3(b)) as
and
Definition 3.1.
A Tmesh is admissible for the biorder if includes the segments
and all vertices belonging to have valence 4. will denote the set of admissible Tmeshes for the biorder .
Definition 3.2.
A Tmesh belongs to if, for any couple of vertices both belonging to the boundary of a cell and such that (, resp.), the segment (, resp.) belongs to .
In other words, A Tmesh satisfying the definition 3.2 does not have any “facing”Tjunctions. While considering this additional requirement is not necessary now, we will need it later to guarantee the equivalence between analysissuitable and dualcompatible Tmeshes (see [5]).
The socalled anchors, which are basic to the construction of Tsplines, are defined as follows.
Definition 3.3.
Given Tmesh , the set of anchors is defined in the following way:

if both and are even, ;

if is odd and is even, ;

if is even and is odd, ;

if both and are odd, .
To each anchor we associate a global horizontal (vertical) index vector () and a local horizontal (vertical) index vector (), which is a subset of (). The construction of these vectors, which depends on the local topology of the Tmesh, is fundamental in the theory of Tsplines, and a formal presentation can be found for example in [5].
The Tmesh in parameter space is naturally defined as the partition of the domain obtained by considering the elements of the form
where . Let us introduce the notation
for any index vectors , . In this way, the global and local index vectors associated to each anchor naturally define corresponding global and local knot and functions vectors.
Then, we define, for each anchor, a bivariate Generalized Tspline (GTspline):
(8) 
where and are the univariate GBsplines in the variables and constructed on the horizontal and vertical local knot and function vectors associated to . Of course, the TGTsplines introduced in [3] are a particular case of the just defined GTsplines, obtained by setting and , where and are frequencies such that and for any .
Several properties holding for the polynomial Tsplines are also satisfied by the GTsplines.
Property 3.4.
The GTsplines enjoy the following properties, as direct consequence of their definition.

Continuity: each blending function , for any , is times continuously differentiable with respect to and times continuously differentiable with respect to at the point , where and are the multiplicities of and in the knot vectors and , respectively.

Positivity: for , and , .

Local support: if , then , and , .

Linear independence for tensorproduct case: If is a tensorproduct mesh, that is, all the vertices have valence 4, then the corresponding blending functions are linearly independent.

Partition of unity for tensorproduct case: If is a tensorproduct mesh, then the corresponding blending functions form a partition of unity.
We can use the blending functions (8) to construct a spline surface in the same way as in the polynomial case:
(9) 
where are given control points and are the weights.
Note that the constant function may not belong to , and then considering the rational form (9) allows to get the partition of unity property. It can be easily shown that the concepts of standard and semistandard Tsplines (see [19] and [20]) can be extended to this nonpolynomial setting; therefore, if we construct standard or semistandard GTsplines, we can avoid using the rational form, which allows us to combine the features of the Tspline approach and the reproduction properties of the GBsplines.
As in the GBsplines case, the use of GTsplines is particularly relevant to exactly represent certain shapes, which cannot be obtained with classical Tsplines. For example, helicalshaped domains such as helicoids, helicoidal springs and screws can be exactly reproduced by trigonometric GTsplines.
Example 1. Let us consider the helicoid section parametrized by
The helicoid section in Figure 4(a), where , , and , is exactly modeled by using suitable GTsplines of biorder (4,4), which span a space containing .
Example 2. Let us consider a helicoidal spring parametrized by
with . The helicoidal spring in Figure 4(b), where , , , and , is exactly modeled by using suitable GTsplines of biorder (4,4), which span a space containing .
Such shapes are used for example in boundary layer problems on helicalshaped domains (see, e.g., [16] and [17]), in modeling of dental implants (see, e.g., [6] and [22]) and in finite element methods on helicalspring models (see, e.g., [9] and [21]). Note that using VMCR Tmeshes defined in Section 4.2 guarantees the linear independence of the GTsplines and then makes them suitable to numerically solve the above mentioned problems.
4 Linear independence of the GTsplines and VMCR Tmeshes
4.1 GTsplines and tensorproduct splines
The linear independence of the Tsplines is a key point for at least one of their main applications, that is, isogeometric analysis (see, e.g., [1]). Therefore, the study of linear independence is basic for the theory of the just introduced GTsplines as well, which is the reason why we devote Section 4 to this topic. First we will generalize the results for TGTsplines in [3].
In general, the situation about linear independence of GTsplines may not coincide with the one of the classical polynomial Tsplines. For instance, it has been shown that there are examples where the arguments used to prove the linear dependence of the Tsplines do not hold in the case of the TGTsplines (see [3]). The linear independence of the GTsplines can be studied by examining the relation between them and the tensorproduct spline functions associated to the socalled underlying tensor product mesh. Note that the tensorproduct GBspline functions are linearly independent (see Property 3.4).
Definition 4.1.
Given a Tmesh , its underlying tensorproduct mesh is the Tmesh with the same index domain of and obtained by adding to vertices and edges such that all the vertices have valence and the knot and functions vectors , , , , and are unvaried.
Then, we can get the underlying tensorproduct mesh of a Tmesh by adding edges. Therefore, there is a linear relation between the two sets of GTsplines associated to and , since adding the edges needed to get from corresponds to inserting knots belonging to ( respectively) in the global knot vectors ( respectively) and the corresponding elements in the vectors of functions and ( and respectively) for , which can be handled by using the knot insertion formula. Then these knot insertions must satisfy the requirements of Theorem 2.2. More precisely, condition (6) must be satisfied when we insert new functions belonging to and ( and respectively), in the global function vectors and ( and respectively) of an anchor . By repeatedly applying the knot insertion formula (7), we get a relation of type
(10) 
where and are the sets of GTsplines associated to and , respectively. If we denote the sets of the anchors of and by and , (10) can be also written in the matrix form
(11) 
where , , and is an matrix , whose elements are obtained by relabeling the coefficients in (10). The linear independence of the GTspline blending functions is equivalent to being a fullrank matrix.
Theorem 4.2.
A necessary and sufficient condition for the GTspline blending functions to be linearly independent is that is full rank.
Proof. See the analogous Theorem in [3].
Given the same Tmesh , the same knot vectors and , and as a consequence the same anchors and the same global and local knot vectors, we denote by the polynomial Tspline blending functions of bidegree , and by the tensorproduct Bspline functions associated to the underlying tensorproduct mesh . We can obtain also in this case, by repeatedly applying Boehm’s knot insertion formula for the polynomial splines (see, e.g., [2]), the relation
(12) 
where , and is an matrix.
Theorem 4.3.
A necessary and sufficient condition for the Tspline blending functions to be linearly independent is that is full rank.
Proof. See, e.g., [11].
In the nonpolynomial and the polynomial cases, the linear independence of the respective blending functions is equivalent to the respective matrices and being full rank. We will prove that there is a strong connection between the two matrices. Since the elements of the two matrices and are obtained by a repeated application of the respective knot insertion formulae, we need to understand the relation between their knot insertion formulae, stated in the following Lemma.
Lemma 4.4.
Let be given a knot vector and another knot vector obtained by inserting a new knot between and . Moreover, let , and , be their respective vectors of functions, where and if or and if . Consider, for any order , the knot insertion formulas for the univariate GBsplines and for the univariate polynomial Bsplines
where by and we denote the GBsplines and the Bsplines obtained after the knot insertion, and the coefficients are obtained by using, respectively, (7) and the classical knot insertion formula (see, e.g., [2]). Then, for , we have
(13) 
Proof. The result, analogously to the trigonometric case considered in [3], follows from the analysis of the expressions of , .
This Lemma allows us to establish a connection between the matrices and . Combining Lemma 4.4 with the same arguments in the proof of Corollary 4.6 in [3], we prove below that the sparsity pattern of the two matrices coincide.
Theorem 4.5.
Let , and let and be the sets of the GTspline blending functions and of the Tspline blending functions associated to , respectively. Moreover, let and be the sets of the GTspline blending functions and of the Tspline blending functions associated to the underlying tensorproduct mesh . If we denote by and the matrices expressing the relation between the functions and , and between