Curve and surface construction based on the generalized toric-Bernstein basis functions

Abstract The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.


Introduction
In Computer Aided Geometric Design (CAGD) and Computed Aided Design (CAD), Bézier curves/surfaces play the central role [1,2].They were rstly described by French engineer Pierre Bézier to design automobile bodies in 1962 [3].From the viewpoint of algebraic geometry, Bézier curves/surfaces are projections of real toric varieties from higher-dimensional space.In 1992, Probably Warren [4] was the rst who noticed that the real toric variety can be applied in CAGD.In 2002, Krasauskas [5] proposed a kind of rational multisided surface, namely toric surface, which is conncened with a nite set of integer lattice points based on the toric ideals and toric varieties.The toric surface is degenerated into rational Bézier curve if the lattice points set is constrained to a one-dimensional integer points.What's more, the tensor product Bézier surfaces and Bézier triangles are also special cases of the toric surface.García-Puente et al. [6] indicated that limiting surface of toric patch is the regular control surface when all weights tend to in nity, which is called the toric degeneration.Since toric surface patches are a muliti-sided generalization of Bézier surfaces, compared with the Bézier scheme, they require fewer surface patches in applications such as data tting, blending surfaces, hole-lling and so on, and the overall smoothness is better.
In recent years, the parametric curves/surfaces construction based on di erent basis functions have been studied by many scholars.Zhang [7,8] investigated curves in the space span{ , t, cos t, sin t}.Chen and Wang et al. [9,10] de ned the C-Bézier curve and the C-B spline curve (NUAT B-spline curve) by extending the space of mixed algebra and trigonometric polynomial.Oruç and Phillips [11] de ned the q-Bézier curve based on the q-Bernstein operator which was constructed by Phillips [12].Han et al. [13] presented the generalizations of Bézier curves and the tensor product surfaces.These curves and surfaces are based on the Lupaş q-analogue of Bernstein operator.Cai et al. [14] presented a new generalization of λ-Bernstein operators based on q-integers and established a statistical approximation theorem.Hu et al. [15] presented a novel shape-adjustable generalized Bézier curve with multiple shape parameters and discussed its applications to surface modeling in engineering.Hu and Wu [16] presented a kind of generalized quartic H-Bézier basis functions with four shape parameters.And the expression and some properties of the corresponding curves were discussed.Schaback [17] gave an introduction to certain techniques for the construction of surfaces from scattered data, which emphasis is putting on interpolation methods using compactly supported radial basis functions.Goldman and Simeonov [18] studied the properties of quantum Bernstein bases and quantum Bézier curves by introducing a new variant of the blossom.Zhou and Cai [19] constructed a triangular Meyer-König-Zeller surface based on bivariate Meyer-König-Zeller operator.Zhou et al. [20] constructed two kinds of bivariate S − λ basis functions, tensor product and triangular S − λ basis functions by means of the technique of generating functions and transformation factors.Moreover, the corresponding two kinds of S − λ surfaces were studied.Salvi and Várady [21] described a new patch that extends the concept of generalized Bézier patches [22] to concave polygonal domains.
Most of the basis functions constructing curves and surfaces de ned above are represented in nonnegative integer power forms.At present, some researchers have put many e orts on the construction of curves and surfaces based on basis functions with rational or irrational number powers .The multiquadric (MQ) function is a radial basis function (RBF) with the rational number power form, which is widely used in numerical analysis and scienti c computing [17].In 2015, Zhu et al. [23] extended the Bernstein basis functions and then constructed αβ-Bernstein-like basis with two exponential shape parameters α and β with real number degrees.
García-Puente and Sottile [24] showed that tuning a pentagonal toric patch by lattice points A (see Figure 1(b)) instead of A (see Figure 1(a)) to achieve linear precision, where A contains three non-integer points.In 2008, Craciun et al. [25] studied the theory of toric varieties de ned by generally real lattice sets, which were applied in algebraic statistics known as toric model [26] and studied the geometric properties of toric surfaces.In 2015, Postinghel et al. [27] presented the degenerations of real irrational toric varieties de ned by generally real number set.Pir and Sottile studied the theory of irrational toric varieties in [28].Li et al. preliminarily studied the T-Bézier curve constructed by the real points in [29].In this paper, inspired by above methods, we de ne a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions.The degree of GT-Bernstein basis functions is an arbitrary real number.Then, the corresponding generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches.Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots.We generalize the lattice points in the de nition of toric surface patches to the real points in this paper, and this leads to construct a wider range of shapes for applications.We may construct GT-Bézier surfaces with linear precision for barycentric coordinate construction (as Figure 1 shows), we can apply the limiting properties of weights for shape deformation, computer animation and costume designing as [30] did, and we also can construct multisided surface patch from a given points by progressive iteration approximation (PIA) by method in [31].
The rest of this paper is organized as follows.In Section 2, the generalized toric-Bernstein basis functions are de ned and the properties of the basis are studied.And then a class of generalized Bézier curve is constructed in Section 3, which is the generalization of the classical rational Bézier curve.In Section 4, we construct a new kind of multisided parametric surface by bivariate generalized toric-Bernstein basis functions.At last, we conclude the whole paper and point out the future work in Section 5.

Generalized toric-Bernstein basis functions
It is well known that toric Bernstein basis functions depend on the nite set of integer lattice points A and boundary functions of the convex hull(lattice polygon) ∆ A of A. When the lattice polygon ∆ A is a standard triangle or a rectangle, if we take appropriate coe cients, the toric Bernstein basis functions degenerate into the classical Bernstein basis functions after the parameter transformation.So they are the generalizations of the classical Bernstein basis functions.In this section, we generalize the toric Bernstein basis functions to nite set of real points, and give the de nitions of generalized toric-Bernstein basis functions in one and two dimensions.where k , k are positive real numbers such that h (t) ≥ , h (t) ≥ , t ∈ ∆ A .Then, basis functions indexed by A can be constructed as follow.

De nition 1. Let
an and a < an.Then, for any point a i in A, we de ne the generalized toric-Bernstein (GT-Bernstein) basis functions as where coe cient ca i > and a i is called knot.
The rational form of the GT-Bernstein basis βa i (t) is where ωa i > is called weight.
Remark 1.The GT-Bernstein basis {βa i (t)} de ned by equation (1) depends on the selection of the coe cients k and k .Since any positive real numbers can be selected, if there is no special explanation, we set k = k = .
We will show that the curve de ned by {βa i (t)} is independent of k and k in Section 3.
and the basis functions βa i (t) on ∆ A = [ , ] are shown in Figure 2. The changes of basis function βa (t) while coe cient ca varying as shown in Figure 3 (the coe cients of curves from bottom to top are ., ., ., .respectively), which shows that the coe cient mainly a ects the function value of the basis function at each point.However, the changes of the basis function βa (t) when its corresponding knot changes are shown in Figure 4 (the knots corresponding to curves from left to right are √ , , √ respectively), which means that the knot mainly a ect the positions of the maximum point of the basis function.From De nition 1 and rational form (2), some properties of the basis functions {Ta i (t)} can be obtained directly as follows.
Theorem 1.The rational GT-Bernstein basis functions de ned in (2) have the following properties:

c) Normalized totally positive (NTP). The rational GT-Bernstein basis {Ta i (t)} n
i= is a NTP basis.This property is proved recently by Yu et al. [32].

after proper parameter transformation, and to toric-Bernstein basis for A ⊂ Z. Therefore, the rational GT-Bernstein basis degenerates to rational Bernstein basis for
Yu et al. [32] presented the following result for GT-Bernstein basis.
Theorem 2. Suppose k = k = k and set a ≤ t < t < • • • < tn ≤ an to be an any increasing sequence.Then the collocation matrix of is a strictly totally positive matrix.
Since the basis {βa i (t)} n i= de ned by equation (1) may do not hold the property of partition of the unity on ∆ A for arbitrary positive coe cients, we present a method to choose coe cients by Theorem 2, which makes the basis {βa i (t)} has partition of the unity on a given increasing sequence a ≤ t < t < • • • < tn ≤ an.
Given an increasing sequence a ≤ t < t < • • • < tn ≤ an, we have the following system of equations: It's clear that the basis {βa i (t)} n i= satis es the conditions of Theorem 2, then the matrix M is a strictly totally positive matrix and system of equations ( 5) has a unique solution.For the bivariate generalized toric-Bernstein basis in Section 2.2, the method for selection of the coe cients is similar to the univariate case.

Consider a nite set of real points
We construct the generalized toric-Bernstein basis functions as follows.

De nition 2.
Let A = {a , a , • • • , an} ⊂ R be a nite collection of real points, and set ∆ A to be the convex hull of A. Then, for any point a i in A, we de ne the bivariate generalized toric-Bernstein (GT-Bernstein) basis function as where ca i > is the coe cient and a i is called knot.
The rational form of the GT-Bernstein basis function βa i (u, v) is where ωa i > is called weight.
Remark 2. In (6), the basis function depends on the choice of coe cients, and the coe cients can vary from case to case.If there is no special explanation, we set ca i ≡ .
Example 2. Let A = {( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )} (see Figure 1(b)).By (6), the GT-Bernstein basis de ned by A are given as belows Three of the basis functions are shown in Figure 5.We further set each weight ωa i = , then the rational forms of these three basis functions on ∆ A are shown in Figure 6.(a) From De nition 2 and rational form (7), we can obtain the following properties of the basis functions {Ta i (u, v)} directly.Theorem 3. The rational forms of the GT-Bernstein basis functions de ned in (7) have the following properties: ) is constrained on the edge ϕ j of ∆ A , all basis functions βa i (u, v) and Ta i (u, v) with indices a i ∈ A \ φj vanish, that is:

(d) Corner points property. At the vertices of ∆ A , we have
(e) Degeneration property.For A = {a , a , • • • , an} ⊂ Z , the basis de ned by (6) degenerates into toric-Bernstein basis de ned in [5].In particular, the GT-Bernstein basis degenerates to the bivariate triangular Bernstein basis for A = {(i, j) ∈ Z | i + j ≤ k, i ≥ , j ≥ }, and to the tensor product Bernstein basis for

Generalized toric-Bézier curves
For given control points and weights, we can use the Bernstein basis functions to construct the classical rational Bézier curve.The classical rational Bézier curve has many good properties, such as convex hull property, boundary property, and a ne invariance.In the same way, the basis functions de ned by ( 2) can be used to de ne a new class of rational curves.

De nition 3. Given real points set
, and weights ω = {ωa i > | a i ∈ A}, the rational parametric curve is called the generalized toric-Bézier curve (GT-Bézier curve for short) of degree n.The n-edge polyline polygon is obtained by sequentially connecting two adjacent control points of B with a straight line segment, is called control polygon.

Remark 3. Although the GT-Bernstein basis de ned by equation (1) depends on the selection of the coe cients
k and k , the GT-Bézier curve is independent on the choice of these two parameters.It can be known from the results in [27], the GT-Bézier curve de ned by the equation ( 10) is obtained by the projection (the projection is related to the weights and the control points) of the high-dimensional real projective toric variety de ned by the A = {a , a , • • • , an}.Given point set A, for di erent coe cients k and k , after unitizing the corresponding toric variety and eliminating the constant in the projective space, the toric varieties are identical, then the GT-Bézier curve de ned by point set A is also the same.For more Details refer to [6,27].The degree of n of curve in De nition 3 is just the number of forms in the curve, one less than the number of knots of A, not exactly the polynomial degree of curve in general sense.If A ⊂ Z, then this degree is exactly the polynomial degree of curve .Example 1 , weights ωa = , ωa = , ωa = , ωa = , ωa = and control points ba = ( , ), ba = ( ., .), ba = ( , ), ba = ( ., .), ba = ( , ).Suppose ca i = (i = , • • • , ), then the quadratic GT-Bézier curve is

Example 3. Let
and the curve is shown in Figure 7.
From the properties of the GT-Bernstein basis functions associated with A = {a , a , • • • , an} ⊂ R, some properties of the GT-Bézier curve can be obtained as follows: (a) A ne invariance and convex hull property.Since the basis (2) have the properties of nonnegativity and partition of the unity, these show that the corresponding GT-Bézier curve (10) has a ne invariance and convex hull property.(c) Progressive iteration approximation (PIA) property.The GT-Bézier curve has PIA property from the result in [31] because its basis {Ta i (t)} n i= is a NTP basis.
then the GT-Bézier curve (10) degenerates into the classical rational Bézier curve after reparameterization and coe cients selected properly.For A = {a , a , • • • , an} ⊂ Z, the GT-Bézier curve (10) is the toric Bézier curve de ned in [33], which is exactly the one-dimensional form of the toric surface de ned in [5].(f) Multiple knot property.When a knot in A tends to its adjacent knot, the following results describe the limit property of the GT-Bézier curve, which also show the resulting GT-Bézier curve de ned by A with multiple knots.

Theorem 4. Suppose ca
When knot a k ( ≤ k < n) approaches to its adjacent knot a k+ , the limit of GT-Bézier curve P A,ω,B (t) of degree n de ned in (10)  , where This leads to prove the result.The following corollary generalizes Theorem 4, and gives the limit of GT-Bézier curve with multiple knots.The proof of the corollary is similar to Theorem 4 and will be omitted here.

Corollary 1. Suppose ca
approaches to the knot a q+k− , the limit of GT-Bézier curve P A,ω,B (t) of degree n de ned in (10) is exactly the GT-Bézier curve of degree n−k+ as i= ωa i ba i βa i (t)+ ωa q+k− ba q+k− βa q+k− (t)+ n i=q+k ωa i ba i βa i (t) q− i= ωa i βa i (t)+ ωa q+k− βa q+k− (t)+ n i=q+k ωa i βa i (t) , where  (g) Toric degeneration property.For each t ∈ ∆ A , we have the limiting property of GT-Bézier curve while a single weight of curve tends to in nity, that is lim And this property can be derived from weight property of rational Bézier curve directly.Figure 10 shows the limit curve of GT-Bézier curve de ned in Example 3 with ωa → +∞.Next, we consider the property of GT-Bézier curve if all the weights tend to in nity.Let λ : A → R be a lifting function to lift the points a i of A to (a i , λ(a i )) ∈ R .We denote P λ = conv{(a i , λ(a i )) | a i ∈ A} the convex hull of the lifted points.Each edge of the convex hull P λ has a and this leads to end the proof.
From Theorem 6, we have the following property.
(a) Toric surface (b) GT-Bézier Surface From the properties of the GT-Bernstein basis functions, we have the following properties of the GT-Bézier surface.
(a) A ne invariance and convex hull property.Since the basis functions (7) possess of nonnegativity and partition of unity, the corresponding GT-Bézier surface ( 14) has a ne invariance and convex hull property.
(b) Degeneration property.When A={a ,a ,• • •,an}⊂Z , the GT-Bézier surface associated of A degenerates to the toric surface de ned in [5] by the property of basis (7).In particular, the rational Bézier triangle de ned by A = {(i, j) ∈ Z | i + j ≤ k, i ≥ , j ≥ }, and the rational tensor product Bézier surface de ned by A = {(i, j) ∈ Z | ≤ i ≤ m, ≤ j ≥ n} are special cases of the GT-Bézier surface.
(c) Corner points interpolation property.This property follows directly from the property at the corner points property of the basis (7), that is Let the angle between the edge ϕ and the u axis be α.Then tan α = − ξ η , and All basis functions βa i (u, v) with indices a i ∈ A \ φ vanishes if (u, v) ∈ ϕ, hence P φ,ω| φ ,B| φ depends only on weights and control points indexed by a i ∈ φ. If . By geometric relationship, we have u j = u + l j cos α.
For the edge equation h k (u, v) = for the edge ϕ k of ∆ A , we evaluate h k (u, v) at point a j , Thus the basis de ned on the edge ϕ can be expressed as Here the rst r factors h k (u, v) (ρ k − η k η ρ+ ηξ k −ξη k η u ) do not depend on j and can be canceled in the de nition of GT-Bézier surface.
φ ωa j ba j ca j s l j a j ∈ φ ωa j ca j s l j = a j ∈ φ ωa j ba j ca j t l j (σ − t) σ−l j a j ∈ φ ωa j ca j t l j (σ − t) σ−l j .
We choose a natural parameter τ on the edge, u = u + τσ cos α( < τ < ), to prove that this reparametrization is 1- Hence the reparametrization τ −→ t is monotonic.Also it is easy to check that it preserves endpoints.Therefore it is 1-1 and ends the proof.
(e) Multiple knot property.When a knot of A tends to its adjacent knot, the following theorem describes the limit property of the GT-Bézier surface, and demonstrates the construction of GT-Bézier surface by A with multiple knots.Theorem 8. Suppose ca i = (i = , • • • , n).When the knot a k ( ≤ k < n) approaches to aq ( ≤ q < n, and q ≠ k) along line a k aq with the convex hull ∆ A unchanging, the limit of GT-Bézier surface P A,ω,B (u, v) de ned in ( 14) is exactly the GT-Bézier surface P A, ω, B (u, v), de ned as where ωa q = ωa k + ωa q , ba q = ωa k ωa k + ωa q ba k + ωa q ωa k + ωa q ba Proof.When a k tends to aq, we have lim Thus, lim = i≠ k,q ωa i ba i βa i (u,v)+ωa k ba k βa q (u,v)+ωa q ba q βa q (u,v) i≠ k,q ωa i βa i (u, v)+ωa k βa q (u, v)+ωa q βa q (u, v) = i≠ k,q ωa i ba i βa i (u, v)+(ωa k ba k +ωa q ba q )βa q (u, v) i≠ k,q ωa i βa i (u, v)+(ωa k + ωa q )βa q (u, v) .
Let ωa q = ωa k + ωa q , ba q = ωa k ωa k +ωa q ba k + ωa q ωa k +ωa q ba q , we can obtain lim a k →aq P A,ω,B (u, v)= P A, ω, B (u, v)= i≠ k,q ωa i ba i βa i (u, v) + ωa q ba q βa q (u, v) i≠ k,q ωa i βa i (u, v) + ωa q βa q (u, v) ,  Theorem 8 shows that the limit surface of the GT-Bézier surface when single knot approaches to another with the convex hull ∆ A unchanging.For the limit of multiple knots with the convex hull ∆ A unchanging, we only need to treat it by Theorem 8 repeatedly.
(f) Toric degeneration property.Similarly, let λ : A → R be a lifting function to lift the points a i of A to (a i , λ(a i )) ∈ R .We denote P λ = conv{(a i , λ(a i )) | a i ∈ A} the convex hull of the lifted points.Each face of the convex hull P λ has a normal vector pointing to the outer side.We call it the upper face of P λ if the last coordinate of the normal vector is positive.If we project these upper faces back vertically into R , they can cover ∆ A and form a regular subdivision Γ λ of ∆ A induced by λ (see [6]).
We group together the points of A that are in the same subset of the Γ λ and on the same upper face of the P λ .Then we get a decomposition of A, which is called regular decomposition S λ of A induced by λ.For each subset F of S λ , we can use the weights ω| F = {ωa i | a i ∈ F} and the control points B| F = {ba i | a i ∈ F} to de ne a new GT-Bézier surface P F,ω| F ,B| F on ∆ F = conv{a i ∈ F} by De nition 4.

Conclusions and future work
In this paper, we present novel generalized toric-Bézier (GT-Bézier) curves and surfaces and discuss their properties.Firstly, we de ne a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions.The degree of GT-Bernstein basis functions is an arbitrary real number.Secondly, the corresponding generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches.Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots.We indicate that the GT-Bézier curve and surface we presented partially preserve the properties of rational Bézier curves and surfaces.Finally, some representative examples verify the properties and results.
Our further work will be devoted to elevation algorithm and de Casteljau algorithm of GT-Bézier curves and surfaces.In addition, although the basis de ned by real knots limits the application in computation, it provides a wider of shapes for design.In this paper, we present the de nition and study the properties of curves and surfaces theoretically only.We will study the applications of GT-Bézier curves and surfaces in future, such as barycentric coordinate construction, shape deformation, computer animation, and surface construction by PIA method.

Figure 1 :
Figure 1: Lattice points A and A.
an, and ∆ A = [a , an].Obviously, the endpoints of ∆ A are points a and an and we assume a < an.Set h (t) = k (t − a ) and h (t) = k (an − t),

Figure 4 :
Figure 4: E ect of knot changing on GT-Bernstein basis.

( e )
Endpoints tangent vectors.For k = a − a , k = an − a n− , the tangent vectors at the end points of GT-Bézier curve are P A,ω,B (a )= ca k k n− (ba n −ba n− ) ca n ωa n .(11) We can see the tangent vectors at the end points of curve P A,ω,B (t) are parallel to − −−− → ba ba and − −−−− → ba n− ba n respectively.And this property can be used to construct G continuous piecewise GT-Bézier curve.

Figure 8 :
Figure 8: Limits of the quadratic GT-Bézier curve of single knot.

Example 5 .
Consider the curve P A,ω,B (t) de ned as in Example 3. If a → a and a → a , then the changes of the GT-Bézier curve are shown in Figure 9.The limit curve is constructed by knots A = {a = , a = , a = }, control points B = {ba = ( , ), ba = ( ., ), ba = ( , )} and weights ω = {ωa = , ωa =, ωa = }.We can see that the limit curve coincides with the target curve together, which veri es the result of Corollary 1.

Figure 9 :
Figure 9: Limits of the quadratic GT-Bézier curve with multiple knots.

Figure 14 :
Figure 14: Limit of GT-Bézier surface with a → a .