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

Jing-Gai Li
and Chun-Gang Zhu

## 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.

## 1 Introduction

In Computer Aided Geometric Design (CAGD) and Computed Aided Design (CAD), Bézier curves/surfaces play the central role [1, 2]. They were firstly 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 first 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 finite 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 infinity, 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 fitting, blending surfaces, hole-filling and so on, and the overall smoothness is better.

In recent years, the parametric curves/surfaces construction based on different basis functions have been studied by many scholars. Zhang [7, 8] investigated curves in the space span{1, t, cos t, sin t}. Chen and Wang et al. [9, 10] defined 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] defined 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 Sλ basis functions, tensor product and triangular Sλ basis functions by means of the technique of generating functions and transformation factors. Moreover, the corresponding two kinds of Sλ 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 defined above are represented in non-negative integer power forms. At present, some researchers have put many efforts 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 scientific 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 𝓐͠ (see Figure 1(b)) instead of 𝓐 (see Figure 1(a)) to achieve linear precision, where 𝓐͠ contains three non-integer points. In 2008, Craciun et al. [25] studied the theory of toric varieties defined 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 defined 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 define 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 definition 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 defined 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.

## 2 Generalized toric-Bernstein basis functions

It is well known that toric Bernstein basis functions depend on the finite set of integer lattice points 𝓐 and boundary functions of the convex hull(lattice polygon) Δ𝓐 of 𝓐. When the lattice polygon Δ𝓐 is a standard triangle or a rectangle, if we take appropriate coefficients, 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 finite set of real points, and give the definitions of generalized toric-Bernstein basis functions in one and two dimensions.

### 2.1 Univariate generalized toric-Bernstein basis functions

Consider 𝓐 = {a0, a1, ⋯, an} ⊂ ℝ with a0a1 ≤ ⋯ ≤ an−1an, and Δ𝓐 = [a0, an]. Obviously, the endpoints of Δ𝓐 are points a0 and an and we assume a0 < an. Set h0(t) = k0(ta0) and h1(t) = k1(ant), where k0, k1 are positive real numbers such that h0(t) ≥ 0, h1(t) ≥ 0, tΔ𝓐. Then, basis functions indexed by 𝓐 can be constructed as follow.

Definition 1

Let 𝓐 = {a0, a1, ⋯, an} ⊂ ℝ with a0a1 ≤ ⋯ ≤ an−1an and a0 < an. Then, for any point ai in 𝓐, we define the generalized toric-Bernstein (GT-Bernstein) basis functions as

$βai(t)=caih0(t)h0(ai)h1(t)h1(ai), t∈ΔA,$

where coefficient cai > 0 and ai is called knot.

The rational form of the GT-Bernstein basis βai(t) is

$Tai(t)=ωaiβai(t)∑i=0nωaiβai(t), t∈ΔA,$

where ωai > 0 is called weight.

Remark 1

The GT-Bernstein basis {βai(t)} defined by equation (1) depends on the selection of the coefficients k0and k1. Since any positive real numbers can be selected, if there is no special explanation, we set k0 = k1 = 1. We will show that the curve defined by {βai(t)} is independent of k0and k1in Section 3.

Example 1

Let$a0=0,a1=24,a2=12,a3=22,a4=1 and ca0=12,ca1=1,ca2=32,ca3=710,ca4=910.$By (1), we have

$βa0(t)=12(1−t), βa1(t)=t24(1−t)1−24,βa2(t)=32t12(1−t)12, βa3(t)=710t22(1−t)1−22, βa4(t)=910t,$

and the basis functions βai(t) on Δ𝓐 = [0, 1] are shown in Figure 2. The changes of basis function βa2(t) while coefficient ca2varying as shown in Figure 3 (the coefficients of curves from bottom to top are 0.1, 0.7, 1.5, 1.9 respectively), which shows that the coefficient mainly affects the function value of the basis function at each point. However, the changes of the basis function βa2(t) when its corresponding knot changes are shown in Figure 4 (the knots corresponding to curves from left to right are$25,12,53$respectively), which means that the knot mainly affect the positions of the maximum point of the basis function.

From Definition 1 and rational form (2), some properties of the basis functions {Tai(t)} can be obtained directly as follows.

Theorem 1

The rational GT-Bernstein basis functions defined in (2) have the following properties:

• Nonnegativity. Tai(t) ≥ 0, tΔ𝓐, i = 0, 1, ⋯, n.
• Partition of the unity. $∑i=0nTai(t)≡1.$
• Normalized totally positive (NTP). The rational GT-Bernstein basis${Tai(t)}i=0n$is a NTP basis. This property is proved recently by Yu et al. [32].
• Endpoints property. At the endpoints of [a0, an], we have
$Tai(a0)=1, i=0,0, i≠0, Tai(an)=1, i=n,0, i≠n.$
• Degeneration property. The GT-Bernstein basis {βai(t)} degenerates to the classical Bernstein basis for 𝓐 = {0, 1, ⋯, n} or 𝓐 = {0, $1n$, ⋯, 1} after proper parameter transformation, and to toric-Bernstein basis for 𝓐 ⊂ ℤ. Therefore, the rational GT-Bernstein basis degenerates to rational Bernstein basis for 𝓐 = {0, 1, ⋯, n} or 𝓐 = {0, $1n$, ⋯, 1} after proper parameter transformation.

Yu et al. [32] presented the following result for GT-Bernstein basis.

Theorem 2

Suppose k0 = k1 = k and set a0t0 < t1 < ⋯ < tnan to be an any increasing sequence. Then the collocation matrix of${βai(t)}i=0n$at t0 < t1 < ⋯ < tn

$Mβa0,⋯,βant0,⋯,tn=(βaj(ti))j=0,1,⋯,ni=0,1,⋯,n$

is a strictly totally positive matrix.

Since the basis ${βai(t)}i=0n$ defined by equation (1) may do not hold the property of partition of the unity on Δ𝓐 for arbitrary positive coefficients, we present a method to choose coefficients by Theorem 2, which makes the basis {βai(t)} has partition of the unity on a given increasing sequence a0t0 < t1 < ⋯ < tnan.

Given an increasing sequence a0t0 < t1 < ⋯ < tnan, we have the following system of equations:

$∑j=0nβaj(ti)=ca0h0(ti)h0(a0)h1(ti)h1(a0)+⋯+canh0(ti)h0(an)h1(ti)h1(an)=1, i=0,⋯,n.$

If we write C = (ca0, ⋯, can)T and 1 = (1, ⋯, 1)T, then we obtain

$Mβa0,⋯,βant0,⋯,tnC=1.$

It’s clear that the basis ${βai(t)}i=0n$ satisfies 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 coefficients is similar to the univariate case.

### 2.2 Bivariate generalized toric-Bernstein basis functions

Consider a finite set of real points 𝓐 = {a0, a1, ⋯, an} ⊂ ℝ2, Let Δ𝓐 be the convex hull of 𝓐. The lines defined by edges ϕi of Δ𝓐 are hi(u, v) = ξi u + ηiv + ρi, where 〈ξi, ηi〉 is the normal vector of ϕi towards inside of Δ𝓐 such that hi(u, v) ≥ 0, (u, v) ∈ Δ𝓐, i = 1, ⋯, r. We construct the generalized toric-Bernstein basis functions as follows.

Definition 2

Let 𝓐 = {a0, a1, ⋯, an} ⊂ ℝ2be a finite collection of real points, and set Δ𝓐to be the convex hull of 𝓐. Then, for any pointaiin 𝓐, we define the bivariate generalized toric-Bernstein (GT-Bernstein) basis function as

$βai(u,v)=caih1(u,v)h1(ai)⋯hr(u,v)hr(ai), (u,v)∈ΔA,$

where cai > 0 is the coefficient andaiis called knot.

The rational form of the GT-Bernstein basis function βai(u, v) is

$Tai(u,v)=ωaiβai(u,v)∑i=0nωaiβai(u,v), (u,v)∈ΔA.$

where ωai > 0 is called weight.

Remark 2

In (6), the basis function depends on the choice of coefficients, and the coefficients can vary from case to case. If there is no special explanation, we set cai ≡ 1.

Example 2

Let $A~={(0,2),(1,2),(0,65),(87,87),(2,1),(0,0),(65,0),(2,0)}$(see Figure 1(b)). By (6), the GT-Bernstein basis defined by 𝓐͠ are given as belows

$β(0,2)(u,v)=(3−u−v)(2−u)2v2,β(1,2)(u,v)=(2−u)v2u,β(0,65)(u,v)=(2−v)45(3−u−v)95(2−u)2v65,β(87,87)(u,v)=(2−v)67(3−u−v)57(2−u)67v87u87,β(2,1)(u,v)=(2−v)vu2,β(0,0)(u,v)=(2−v)2(3−u−v)3(2−u)2,β(65,0)(u,v)=(2−v)2(3−u−v)95(2−u)45u65,β(2,0)(u,v)=(2−v)2(3−u−v)u2.$

Three of the basis functions are shown in Figure 5. We further set each weight ωai = 1, then the rational forms of these three basis functions on Δ𝓐are shown in Figure 6.

Suppose edges ϕi(i = 1, ⋯, r) of the convex hull Δ𝓐 are ordered counterclockwise and let Vi be vertex of Δ𝓐 where two edges ϕi and ϕi+1 meet, (i = 1, ⋯, r). The indices will be treated in a cyclic fashion: for instance, ϕ0 = ϕr, ϕr+1 = ϕ1 and so on. Denote by ϕ̂i = ϕi ∩ 𝓐 the intersection of 𝓐 and ϕi. Note that ${Vi}i=1r$ and ϕ̂i are subsets of 𝓐 respectively, i = 1, ⋯, r.

From Definition 2 and rational form (7), we can obtain the following properties of the basis functions {Tai(u, v)} directly.

Theorem 3

The rational forms of the GT-Bernstein basis functions defined in (7) have the following properties:

• Nonnegativity. Tai(u, v) ≥ 0, (u, v) ∈ Δ𝓐, i = 0, 1, ⋯, n.
• Partition of the unity. $∑i=0nTai(u,v)≡1.$
• Boundary property. When (u, v) is constrained on the edge ϕj of Δ𝓐, all basis functions βai(u, v) and Tai(u, v) with indicesai ∈ 𝓐 ∖ ϕ̂j vanish, that is:
$βai(u,v)=0, ai∈A∖ϕj^,(u,v)∈ϕj,βai(u,v)≠0, ai∈ϕj^,Tai(u,v)=0, ai∈A∖ϕj^,(u,v)∈ϕj.Tai(u,v)≠0, ai∈ϕj^,$
• Corner points property. At the vertices of Δ𝓐, we have
$Tai(Vi)=1, ai=Vi,Tai(Vi)=0, ai≠Vi.$
• Degeneration property. For 𝓐 = {a0, a1, ⋯, an} ⊂ ℤ2, the basis defined by (6) degenerates into toric-Bernstein basis defined in [5]. In particular, the GT-Bernstein basis degenerates to the bivariate triangular Bernstein basis for 𝓐 = {(i, j) ∈ ℤ2 | i + jk, i ≥ 0, j ≥ 0}, and to the tensor product Bernstein basis for 𝓐 = {(i, j) ∈ ℤ2 | 0 ≤ im, 0 ≤ jn}, if coefficients selected properly.

## 3 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 affine invariance. In the same way, the basis functions defined by (2) can be used to define a new class of rational curves.

Definition 3

Given real points set 𝓐 = {a0, a1, ⋯, an}, control points 𝓑 = {bai | ai ∈ 𝓐} ⊂ ℝ3, and weights ω = {ωai > 0 | ai ∈ 𝓐}, the rational parametric curve

$PA,ω,B(t)=∑i=0nbaiTai(t)=∑i=0nbaiωaiβai(t)∑i=0nωaiβai(t), t∈ΔA$

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 defined by equation (1) depends on the selection of the coefficients k0and k1, 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 defined 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 defined by the 𝓐 = {a0, a1, ⋯, an}. Given point set 𝓐, for different coefficients k0and k1, after unitizing the corresponding toric variety and eliminating the constant in the projective space, the toric varieties are identical, then the GT-Bézier curve defined by point set 𝓐 is also the same. For more Details refer to [6, 27].

The degree of n of curve in Definition 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 3

Let$A={a0=0,a1=24,a2=12,a3=22,a4=1}$as show in Example 1, weights ωa0 = 1, ωa1 = 10, ωa2 = 20, ωa3 = 6, ωa4 = 5 and control pointsba0 = (0, 0), ba1 = (0.4, 1.3), ba2 = (2, 2), ba3 = (3.7, 1.5), ba4 = (4, 0). Suppose cai = 1(i = 0, ⋯, 4), then the quadratic GT-Bézier curve is

$PA,ω,B(t)=∑i=04baiTai(t), t∈[0,1],$

and the curve is shown in Figure 7.

From the properties of the GT-Bernstein basis functions associated with 𝓐 = {a0, a1, ⋯, an} ⊂ ℝ, some properties of the GT-Bézier curve can be obtained as follows:

• Affine 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 affine invariance and convex hull property.
• Endpoints interpolation property. This property follows directly from the endpoints property of the basis (2), that is P𝓐,ω,𝓑(a0) = ba0, P𝓐,ω,𝓑(an) = ban.
• Progressive iteration approximation (PIA) property. The GT-Bézier curve has PIA property from the result in [31] because its basis ${Tai(t)}i=0n$ is a NTP basis.
• Degeneration property. If ai = i (or ai = $in$), (i = 0, 1, ⋯, n) and k0 = k1 = n, then the GT-Bézier curve (10) degenerates into the classical rational Bézier curve after reparameterization and coefficients selected properly. For 𝓐 = {a0, a1, ⋯, an} ⊂ ℤ, the GT-Bézier curve (10) is the toric Bézier curve defined in [33], which is exactly the one-dimensional form of the toric surface defined in [5].
• Endpoints tangent vectors. For
$k0=1a1−a0,k1=1an−an−1,$
the tangent vectors at the end points of GT-Bézier curve are
$PA,ω,B′(a0)=ca1k0k1−k1k0(an−a0)−k1k0ωa1(ba1−ba0)ca0ωa0,PA,ω,B′(an)=can−1k1k0−k0k1(an−a0)−k0k1ωan−1(ban−ban−1)canωan.$
We can see the tangent vectors at the end points of curve P𝓐,ω,𝓑(t) are parallel to $ba0ba1→ and ban−1ban→$ respectively. And this property can be used to construct G1 continuous piecewise GT-Bézier curve.
• 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 defined by 𝓐 with multiple knots.

Theorem 4

Suppose cai = 1(i = 0, ⋯, n). When knot ak (0 ≤ k < n) approaches to its adjacent knot ak+1, the limit of GT-Bézier curveP𝓐,ω,𝓑(t) of degree n defined in (10) is exactly the GT-Bézier curveP͠𝓐͠,ω͠,𝓑͠(t) of degree n − 1, defined as

$limak→ak+1PA,ω,B(t)=P~A~,ω~,B~(t)=∑i=0k−1ωaibaiβai(t)+ω~ak+1b~ak+1βak+1(t)+∑i=k+2nωaibaiβai(t)∑i=0k−1ωaiβai(t)+ω~ak+1βak+1(t)+∑i=k+2nωaiβai(t),$

where

$ω~ak+1=ωak+ωak+1,b~ak+1=ωakωak+ωak+1bak+ωak+1ωak+ωak+1bak+1,$

$A~={a0,⋯,ak−1,ak+1,⋯,an},B~={ba0,⋯,bak−1,b~ak+1,bak+2,⋯,ban},ω~={ωa0,⋯,ωak−1,ω~ak+1,ωak+2,⋯,ωan}.$

Proof

When ak (0 ≤ k < n) tends to ak+1, we have

$limak→ak+1βak(t)=limak→ak+1(t−a0)ak−a0(an−t)an−ak=(t−a0)ak+1−a0(an−t)an−ak+1=βak+1(t).$

Thus,

$limak→ak+1∑i=0nbaiTai(t)=P~A~,ω~,B~(t)=limak→ak+1∑i=0nbaiωaiβai(t)∑i=0nωaiβai(t)=∑i≠k,k+1ωaibaiβai(t)+ωakbakβak+1(t)+ωak+1bak+1βak+1(t)∑i≠k,k+1ωaiβai(t)+ωakβak+1(t)+ωak+1βak+1(t)=∑i≠k,k+1ωaibaiβai(t)+(ωakbak+ωak+1bak+1)βak+1(t)∑i≠k,k+1ωaiβai(t)+(ωak+ωak+1)βak+1(t).$

Let $ω~ak+1=ωak+ωak+1,b~ak+1=ωakωak+ωak+1bak+ωak+1ωak+ωak+1bak+1,$ we can obtain

$limak→ak+1PA,ω,B(t)=P~A~,ω~,B~(t)=∑i≠k,k+1ωaibaiβai(t)+ω~ak+1b~ak+1βak+1(t)∑i≠k,k+1ωaiβai(t)+ω~ak+1βak+1(t),$

where

$A~={a0,⋯,ak−1,ak+1,⋯,an},B~={ba0,⋯,bak−1,b~ak+1,bak+2,⋯,ban}, ω~={ωa0,⋯,ωak−1,ω~ak+1,ωak+2,⋯,ωan}.$

This leads to prove the result. □

Example 4

Consider the curveP𝓐,ω,𝓑(t) defined as in Example 3. Let knots 𝓐 = {a0 = 0, a1 = $24$, a2 = $12$, a3 = $22$, a4 = 1}, weights ωa0 = 1, ωa1 = 10, ωa2 = 20, ωa3 = 6, ωa4 = 5, control pointsba0 = (0, 0), ba1 = (0.4, 1.3), ba2 = (2, 2), ba3 = (3.7, 1.5), ba4 = (4, 0) and cai = 1 (i = 0, ⋯, 4). If a1approaches a2, then the changes of the GT-Bézier curve are shown in Figure 8. We can see that the limit curve lima1a2P𝓐,ω,𝓑(t) coincides with the target curveP͠𝓐͠,ω͠,𝓑͠(t), which verifies the Theorem 4.

Theorem 4 indicates that the GT-Bézier curve of degree n degenerates into the GT-Bézier curve of degree n − 1 with knots 𝓐͠ = {a0, ⋯, ak−1, ak+1, ⋯, an}, control points 𝓑͠ = {ba0, ⋯, bak−1, b͠ak+1, bak+2, ⋯, ban} and weights ω͠ = {ωa0, ⋯, ωak−1, ω͠ak+1, ωak+2, ⋯, ωan} when ak = ak+1. 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 cai = 1 (i = 0, ⋯, n). When knots aq, aq+1, ⋯, aq+k−2(0 ≤ q < n, 1 < kn + 1 − q) approaches to the knot aq+k−1, the limit of GT-Bézier curveP𝓐,ω,𝓑(t) of degree n defined in (10) is exactly the GT-Bézier curve of degree nk + 1 as

$limaq,⋯,aq+k−2→aq+k−1PA,ω,B(t)=P~A~,ω~,B~(t)=∑i=0q−1ωaibaiβai(t)+ω~aq+k−1b~aq+k−1βaq+k−1(t)+∑i=q+knωaibaiβai(t)∑i=0q−1ωaiβai(t)+ω~aq+k−1βaq+k−1(t)+∑i=q+knωaiβai(t),$

where

$ω~aq+k−1=ωaq+ωaq+1+⋯+ωaq+k−1,b~aq+k−1=ωaqω~aq+k−1baq+⋯+ωaq+k−1ω~aq+k−1baq+k−1,A~={a0,⋯,aq−1,aq+k−1,⋯,an},B~={ba0,⋯,baq−1,b~aq+k−1,baq+k,⋯,ban},ω~={ωa0,⋯,ωaq−1,ω~aq+k−1,ωaq+k,⋯,ωan}.$

Example 5

Consider the curveP𝓐,ω,𝓑(t) defined as in Example 3. If a1a2and a3a2, then the changes of the GT-Bézier curve are shown in Figure 9. The limit curve is constructed by knots 𝓐͠ = {a0 = 0, a2 = $12$, a4 = 1}, control points$B~={ba0=(0,0),b~a2=(66.236,6236),ba4=(4,0)}$and weights ω͠ = {ωa0 = 1, ω͠a2 = 36, ωa4 = 5}. We can see that the limit curve coincides with the target curve together, which verifies the result of Corollary 1.

1. (g)Toric degeneration property. For each tΔ𝓐, we have the limiting property of GT-Bézier curve while a single weight of curve tends to infinity, that is
$limωai→+∞PA,ω,B(t)=ba0t=a0,bait∈(a0,an),bant=an.$

And this property can be derived from weight property of rational Bézier curve directly. Figure 10 shows the limit curve of GT-Bézier curve defined in Example 3 with ωa1 → +∞.

Next, we consider the property of GT-Bézier curve if all the weights tend to infinity.

Let λ : 𝓐 → ℝ be a lifting function to lift the points ai of 𝓐 to (ai, λ(ai)) ∈ ℝ2. We denote Pλ = conv{(ai, λ(ai)) | ai ∈ 𝓐} the convex hull of the lifted points. Each edge of the convex hull Pλ has a normal vector pointing to the outer side. We call it the upper edges of Pλ if the last coordinate of the normal vector is positive. If we project these upper edges back vertically into ℝ, they can cover Δ𝓐 and form a regular subdivision Γλ of Δ𝓐 induced by λ [6].

We group together the points of 𝓐 that are in the same subset of the Γλ and on the same upper edge of the Pλ. Then we get a decomposition of 𝓐, which is called regular decomposition 𝓢λ of 𝓐 induced by λ. For each subset 𝓕 of 𝓢λ, we can use the weights ω|𝓕 = {ωai | ai ∈ 𝓕} and the control points 𝓑|𝓕 = {bai | ai ∈ 𝓕} to define a new GT-Bézier curve P𝓕,ω|𝓕,𝓑|𝓕 on Δ𝓕 = conv{ai ∈ 𝓕 | ai ∈ 𝓐} by Definition 3. The union of these curves

$PA,ω,B(Sλ)=⋃F∈SλPF,ω|F,B|F$

is called the regular control curve of P𝓐,ω,𝓑 induced by regular decomposition 𝓢λ.

We can use lifting function λ to get a set of weights with a parameter x, ωλ(x) := {xλ(ai)ωai | ai ∈ 𝓐}. These weights are used to define the map

$PA,ωλ(x),B(t)=∑i=0nxλ(ai)ωaibaiβai(t)∑i=0nxλ(ai)ωaiβai(t), t∈ΔA.$

The image of Δ𝓐 under this map is a GT-Bézier curve with a parameter x, denoted as P𝓐,ωλ(x),𝓑. We have the following result.

Theorem 5

The limit of the GT-Bézier curveP𝓐,ωλ(x),𝓑as x → ∞ is the regular control curve induced by regular decomposition 𝓢λ, that is

$limx→∞PA,ωλ(x),B=PA,ω,B(Sλ).$

Proof

According to the theory of real irrational toric varieties in [27], the GT-Bézier curve P𝓐,ω,𝓑 is obtained by the projection of the high-dimensional real projective toric variety formed by 𝓐. Then P𝓐,ω,𝓑 is projection after the composition of a sequence of mappings

$A⟶{βai∣ai∈A}XA⟶ωXA,ω⟶BPA,ω,B.$

For 𝓐 and weights ωλ(x) with parameter x, we can get a family of translated toric varieties X𝓐,ωλ(x)

$A⟶{βai∣ai∈A}XA⟶ωλ(x)XA,ωλ(x).$

When x → ∞, X𝓐,ωλ(x) limits to a union of irrational toric varieties in the Hausdorff distance, which are defined by the all of subset of 𝓢λ. That is

$limx→∞XA,ωλ(x)=⋃F∈SλXF,ω|F.$

Then add control points 𝓑, we have

$⋃F∈SλXF,ω|F⟶B⋃F∈SλPF,ω|F,B|F=PA,ω,B(Sλ).$

So the result holds. □

Theorem 5 shows that regular control curves are exactly the limits of the GT-Bézier curve when all the weights tend to infinity. Obviously the control polygon is the regular control curve of GT-Bézier curve. This property is also called toric degeneration of GT-Bézier curves.

Example 6

Let$A={0,24,12,22,1},$and the lifted values of 𝓐 by a lifting function λ be (2, 1, 5, 9 − $42$, 1). This induces a regular decomposition of 𝓐 as

${0,12} , {12,22,1}.$

The lifted point$24$doesn’t lie on any upper edge of the lifting polygon Pλ, then it doesn’t lie on any subset of the decomposition.

Figure 11(a) shows 𝓐, the lifted values of 𝓐 by λ, and the corresponding regular decomposition. Figure 11(b) and 11(c) show the toric degeneration of this GT-Bézier curve for x = 2, and x = 3. The GT-Bézier curve approaches its regular control curve as the parameter x becomes larger.

If λtakes the values of 𝓐 as {0, 2.5, 3, 2.5, 0}, then this induces a regular decomposition of 𝓐 as

${0,24} , {24,12} , {12,22} , {22,1}.$

The corresponding regular decomposition is shown in Figure 12(a) and the regular control curve is exactly the control polygon of the curve.

Moreover λtakes the values of 𝓐 as {1, 3, 1, 0, 1}, then the regular decomposition of 𝓐 is${0,24},{24,1}$(see Figure 12(b)) and the regular control curve is as shown in Figure 10.

1. (h)Variation diminishing (VD) property. Let di = aia0 (i = 1, ⋯, n) for 𝓐 = {a0, a1, ⋯, an} ⊂ ℝ with a0a1 ≤ ⋯ ≤ an−1an. If di (i = 1, ⋯, n) are rational numbers, then di can be expressed as $di=piqi(pi,qi∈N).$ Let q be the least common multiple of q1, q2, ⋯, qn, namely, q = [q1, q2, ⋯, qn], then q di ∈ ℕ ∖ {0}. At this point, we have the following theorem.

Theorem 6

If di = aia0 ∈ ℚ (i = 1, 2, ⋯, n), then the planar GT-Bézier curveP𝓐,ω,𝓑(t) is variation diminishing, which means that the number of intersections of any straight line with the GT-Bézier curveP𝓐,ω,𝓑(t) is no more than the number of intersections of the line with its control polygon.

Proof

In order to prove this theorem, we need to use the Cartesian notation rule, which presents the upper bound of the number of the positive roots of the polynomial. For any polynomial f(t) = m0 + m1t + ⋯ + mn tn, if we write Zt>0 [f(t)] to denote the number of positive roots of f(t) and denote V[m0, m1, ⋯, mn] as the number of strict sign changes of polynomial coefficients, then

$Zt>0[m0+m1t+⋯+mntn]≤V[m0,m1,⋯,mn].$

Let L denote any straight line, C denote the planar GT-Bézier curve defined by 𝓐, and write I(C, L) to denote the number of times L crosses C. Establish the Cartesian coordinate system with L as the abscissa axis. Because GT-Bézier curve is geometric invariant, we can let (xai, yai)(i = 0, 1, ⋯, n) represent the new coordinates of the control points. Let P denote the control polygon and I(P, L) denote the number of times L crosses P. We only need to prove that I(C, L) ≤ I(P, L).

We set a parameter transformation as $u=t−a0an−t,t∈(a0,an),$ so that u ∈ (0, +∞). Then by the Cartesian notation rule

$I(C,L)=Za0≤t≤an∑i=0nyaiTai(t)=Za0≤t≤an∑i=0nyaiωaicai(t−a0)t−ai(an−t)an−ai∑i=0nωaicai(t−a0)t−ai(an−t)an−ai=Z0

and this leads to end the proof. □

From Theorem 6, we have the following property.

(i) Convexity-preserving property. Suppose di = aia0 ∈ ℚ (i = 1, ⋯, n), then the planar GT-Bézier curve is convex if its control polygon is convex.

## 4 Generalized toric-Bézier surfaces

Definition 4

Let 𝓐 = {a0, a1, ⋯, an} ⊂ ℝ2be a finite set of real points. Given positive weights ω = {ωai | ai ∈ 𝓐} and control points 𝓑 = {bai | ai ∈ 𝓐}, the generalized toric-Bézier surface (GT-Bézier surface for short) is defined as

$PA,ω,B(u,v)=∑i=0nbaiTai(u,v)=∑i=0nbaiωaiβai(u,v)∑i=0nωaiβai(u,v), (u,v)∈ΔA.$

Example 7

Let 𝓐 = {(0, 2), (1, 2), (0, 1), (1, 1), (2, 1), (0, 0), (1, 0), (2, 0)} be the integer points in the pentagon as shown in Figure 1(a). Set control points 𝓑 = {(0, 2, 0), (1, 2, 4), (0, $65$, 2), ($87$, $87$, 5), (2, 1, 2), (0, 0, 0), ($65$, 0, 2), (2, 0, 0)} and weights ω = {2, 2, 5, 7, 2, 3, 5, 2}. Suppose cai = 1 (i = 0, ⋯, 7), then we can define a toric surface as shown in Figure 13(a). This toric surface does not have linear precision, but we can tune it to achieve linear precision. We set$A~={(0,2),(1,2),(0,65),(87,87),(2,1),(0,0),(65,0),(2,0)}$by moving the non-extreme points of 𝓐 within the pentagon (Figure 1(b)). The GT-Bézier surface constructed by 𝓐͠, ω and 𝓑 has linear precision, as shown in Figure 13(b). The theoretical proof can be found in [24].

From the properties of the GT-Bernstein basis functions, we have the following properties of the GT-Bézier surface.

• Affine invariance and convex hull property. Since the basis functions (7) possess of nonnegativity and partition of unity, the corresponding GT-Bézier surface (14) has affine invariance and convex hull property.
• Degeneration property. When 𝓐 = {a0, a1, ⋯, an} ⊂ ℤ2, the GT-Bézier surface associated of 𝓐 degenerates to the toric surface defined in [5] by the property of basis(7). In particular, the rational Bézier triangle defined by 𝓐 = {(i, j) ∈ ℤ2 | i + jk, i ≥ 0, j ≥ 0}, and the rational tensor product Bézier surface defined by 𝓐 = {(i, j) ∈ ℤ2 | 0 ≤ im, 0 ≤ jn} are special cases of the GT-Bézier surface.
• Corner points interpolation property. This property follows directly from the property at the corner points property of the basis (7), that is P𝓐,ω,𝓑(Vi) = bVi, i = 1, ⋯, r, where Vi ∈ 𝓐 are the vertices of Δ𝓐.
• Isoparametric curves property. The isoparametric curves P𝓐,ω,𝓑(u, v) and P𝓐,ω,𝓑(u, v) of a GT-Bézier surface are respectively the GT-Bézier curves.

Theorem 7

Each boundary of the GT-Bézier surface is a GT-Bézier curvePϕ̂i,ω|ϕ̂i,𝓑|ϕ̂i, which defined by control pointsbaiand weights ωaibyaiϕ̂i of corresponding edges ϕiΔ𝓐, where i = 1, ⋯, r.

Proof

Consider the restriction Pϕ̂,ω|ϕ̂,𝓑|ϕ̂ of the GT-Bézier surface at the fixed edge ϕ = ϕi of Δ𝓐. Denote V0 = (u0, v0) = Vi−1, V1 = (u1, v1) = Vi, and hi(u, v) = h(u, v) = ξ u + η v + ρ is the equation of ϕ for simplicity. Let the angle between the edge ϕ and the u axis be α. Then tan $α=−ξη,$ and

$cos⁡α=ηξ2+η2.$

Let $σ=|V0V1|=(u1−u0)2+(v1−v0)2.$ All basis functions βai(u, v) with indices ai ∈ 𝓐 ∖ ϕ̂ vanishes if (u, v) ∈ ϕ, hence Pϕ̂,ω|ϕ̂,𝓑|ϕ̂ depends only on weights and control points indexed by aiϕ̂. If aj = (uj, vj) ∈ ϕ̂, then h(aj) = ξ uj + η vj + ρ = 0, $vj=−ρ+ξujη.$ Let $lj=|ajV0|=(uj−u0)2+(vj−v0)2.$ By geometric relationship, we have

$uj=u0+ljcos⁡α.$

For the edge equation hk(u, v) = 0 for the edge ϕk of Δ𝓐, we evaluate hk(u, v) at point aj,

$hk(aj)=ξkuj+ηkvj+ρk=ηξk−ξηkηuj+ρk−ηkηρ=ρk−ηkηρ+ηξk−ξηkηu0+(ηξk−ξηk)cos⁡αηlj.$

Thus the basis defined on the edge ϕ can be expressed as

$βai(u,v)=cai∏k=1rhk(u,v)(ρk−ηkηρ+ηξk−ξηkηu0)(h1(u,v)(ηξ1−ξη1)cos⁡αη⋯hr(u,v)(ηξr−ξηr)cos⁡αη)lj.$

Here the first r factors $hk(u,v)(ρk−ηkηρ+ηξk−ξηkηu0)$ do not depend on j and can be canceled in the definition of GT-Bézier surface.

When (u, v) ∈ ϕ, then $h(u,v)=0, v=−ρ+ξuη.$ So when (u, v) ∈ ϕ, hk(u, v) is univariate function of u, written hk(u). If we set new variables

$s=h1(u)(ηξ1−ξη1)cos⁡αη⋯hr(u)(ηξr−ξηr)cos⁡αη, t=σs1+s,$

we obtain

$Pϕ^,ω|ϕ^,B|ϕ^(u)=∑aj∈ϕ^ωajbajcajslj∑aj∈ϕ^ωajcajslj=∑aj∈ϕ^ωajbajcajtlj(σ−t)σ−lj∑aj∈ϕ^ωajcajtlj(σ−t)σ−lj.$

We choose a natural parameter τ on the edge, u = u0 + τσ cosα(0 < τ < 1), to prove that this reparametrization is 1-1, and calculate derivatives

$dsdτ=ddτ(h1(u)(ηξ1−ξη1)cos⁡αη)⋯hr(u)(ηξr−ξηr)cos⁡αη+…+h1(u)(ηξ1−ξη1)cos⁡αη⋯ddτ(hr(u)(ηξr−ξηr)cos⁡αη)=σ∏k=1rhk(u)(ηξk−ξηk)cos⁡αη∑j=1r(ηξj−ξηjηcos⁡α)2hj(u)>0, 0<τ<1,$

and

$dtdτ=ddτ(σs1+s)=σ(1+s)2dsdτ>0.$

Hence the reparametrization τt 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 cai = 1(i = 0, ⋯, n). When the knotak (0 ≤ k < n) approaches toaq (0 ≤ q < n, and qk) along lineakaqwith the convex hull Δ𝓐unchanging, the limit of GT-Bézier surfaceP𝓐,ω,𝓑(u, v) defined in (14) is exactly the GT-Bézier surfaceP͠𝓐͠, ω͠, 𝓑͠(u, v), defined as

$limak→aqPA,ω,B(u,v)=P~A~,ω~,B~(u,v)=∑i≠k,qωaibaiβai(u,v)+ω~aqb~aqβaq(u,v)∑i≠k,qωaiβai(u,v)+ω~aqβaq(u,v),$

where

$ω~aq=ωak+ωaq,b~aq=ωakωak+ωaqbak+ωaqωak+ωaqbaq,ω~={ωa0,⋯,ωak−1,ωak+1,⋯,ω~aq,⋯,ωan},A~={a0,⋯,ak−1,ak+1,⋯,an},B~={ba0,⋯,bak−1,bak+1,⋯,b~aq,⋯,ban}.$

Proof

When ak tends to aq, we have

$limak→aqβak(u,v)=limak→aqh1(u,v)h1(ak)⋯hr(u,v)hr(ak)=h1(u,v)h1(aq)⋯hr(u,v)hr(aq)=βaq(u,v).$

Thus,

$limak→aq∑i=0nbaiTai(u,v)=P~A~,ω~,B~(u,v)=limak→aq∑i=0nbaiωaiβai(u,v)∑i=0nωaiβai(u,v)=∑i≠k,qωaibaiβai(u,v)+ωakbakβaq(u,v)+ωaqbaqβaq(u,v)∑i≠k,qωaiβai(u,v)+ωakβaq(u,v)+ωaqβaq(u,v)=∑i≠k,qωaibaiβai(u,v)+(ωakbak+ωaqbaq)βaq(u,v)∑i≠k,qωaiβai(u,v)+(ωak+ωaq)βaq(u,v).$

Let $ω~aq=ωak+ωaq, b~aq=ωakωak+ωaqbak+ωaqωak+ωaqbaq,$ we can obtain

$limak→aqPA,ω,B(u,v)=P~A~,ω~,B~(u,v)=∑i≠k,qωaibaiβai(u,v)+ω~aqb~aqβaq(u,v)∑i≠k,qωaiβai(u,v)+ω~aqβaq(u,v),$

where 𝓐͠ = {a0, ⋯, ak−1, ak+1, ⋯, an}, 𝓑͠ = {ba0, ⋯, bak−1, bak+1, ⋯, b͠aq, ⋯, ban} and ω͠ = {ωa0, ⋯, ωak−1, ωak+1, ⋯, ω͠aq, ⋯, ωan}. □

Example 8

Consider the GT-Bézier surface defined in Example 7. Let 𝓐 = {(0, 2), (1, 2), $(0,65),(87,87),$ (2, 1), (0, 0), ($65$, 0), (2, 0)}, control points 𝓑 = {(0, 2, 0), (1, 2, 4), (0, $65$, 2), ($87$, $87$, 5), (2, 1, 2), (0, 0, 0), ($65$, 0, 2), (2, 0, 0)}, weights ω = {2, 2, 5, 7, 2, 3, 5, 2} and cai = 1 (i = 0, ⋯, 7). Ifa3 = ($87$, $87$) approachesa1 = (1, 2), then the changes of the GT-Bézier surface are shown in Figure 14.

Since the shape of the convex hull Δ𝓐and control points 𝓑 are unchanging during the process ofa3tending toa1, the original curved surface is stretched like an elastic film by the boundary property of the GT-Bézier surface. Untila3 = a1, the resulting surface is defined by 𝓐͠ = {(0, 2), (1, 2), (0, $65$), (2, 1), (0, 0), ($65$, 0), (2, 0)}, control points 𝓑͠ = {(0, 2, 0), $(109,129,439),(0,62,2),$ (2, 1, 2), (0, 0, 0), ($65$, 0, 2), (2, 0, 0)}, weights ω͠ = {2, 9, 5, 2, 3, 5, 2}.

Theorem 8 shows that the limit surface of the GT-Bézier surface when single knot approaches to another with the convex hull Δ𝓐 unchanging. For the limit of multiple knots with the convex hull Δ𝓐 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 ai of 𝓐 to (ai, λ(ai)) ∈ ℝ3. We denote Pλ = conv{(ai, λ(ai)) | ai ∈ 𝓐} the convex hull of the lifted points. Each face of the convex hull Pλ has a normal vector pointing to the outer side. We call it the upper face of Pλ if the last coordinate of the normal vector is positive. If we project these upper faces back vertically into ℝ2, they can cover Δ𝓐 and form a regular subdivision Γλ of Δ𝓐 induced by λ (see [6]).

We group together the points of 𝓐 that are in the same subset of the Γλ and on the same upper face of the Pλ. Then we get a decomposition of 𝓐, which is called regular decomposition 𝓢λ of 𝓐 induced by λ. For each subset 𝓕 of 𝓢λ, we can use the weights ω|𝓕 = {ωai | ai ∈ 𝓕} and the control points 𝓑|𝓕 = {bai | ai ∈ 𝓕} to define a new GT-Bézier surface P𝓕,ω|𝓕,𝓑|𝓕 on Δ𝓕 = conv{ai ∈ 𝓕} by Definition 4. The union of these patches

$PA,ω,B(Sλ)=⋃F∈SλPF,ω|F,B|F$

is called the regular control surface of P𝓐,ω,𝓑 induced by regular decomposition 𝓢λ.

We can use lifting function λ to get a set of weights with a parameter x, ωλ(x) := {xλ(ai)ωai | ai ∈ 𝓐}. These weights are used to define the map

$PA,ωλ(x),B(u,v)=∑i=0nxλ(ai)ωaibaiβai(u,v)∑i=0nxλ(ai)ωaiβai(u,v), (u,v)∈ΔA.$

The image of Δ𝓐 under this map is a GT-Bézier surface with a parameter x, denoted as P𝓐,ωλ(x),𝓑. We have the following result.

Theorem 9

The limit of the GT-Bézier surfaceP𝓐,ωλ(x),𝓑as x → ∞ is the regular control surface induced by the regular decomposition 𝓢λ, that is

$limx→∞PA,ωλ(x),B=PA,ω,B(Sλ).$

Proof

The proof of the theorem is similar to Theorem 5 and will be omitted here. □

Theorem 9 describes the conclusion that the limit surface of the GT-Bézier surface is its regular control surface, and explains the geometric meaning of the limit surface of the GT-Bézier surface when all the weights tend to infinity. And this property is called toric degeneration of GT-Bézier surfaces.

Example 9

Given point set 𝓐͠ is shown in Figure 1(b), and the lifted values of 𝓐͠ by λ are shown in Figure 15(a). The upper hull and the subdivision of Δ𝓐͠by λ are shown in Figure 15(b), and the regular decomposition 𝓢λis shown in Figure 15(c). Let control points 𝓑 = {(0, 2, 0), (1, 2, 4), (0, $65$, 2), ($87$, $87$, 18), (2, 1, 2), (0, 0, 0), ($65$, 0, 2), (2, 0, 0)} and weights ω = {2, 2, 5, 7, 2, 3, 5, 2} corresponding to 𝓐͠. The toric degeneration process of this GT-Bézier surface is shown in Figure 16. This figure also shows the GT-Bézier surfaces for the parameters x = 5, x = 100 and x = 600 respectively. As the parameter x becomes larger, the GT-Bézier surface approaches its regular control surface in Figure 16(d) (consists of surface patches defined by three triangles and two quadrilaterals).

## 5 Conclusions and future work

In this paper, we present novel generalized toric-Bézier (GT-Bézier) curves and surfaces and discuss their properties. Firstly, we define 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 defined by real knots limits the application in computation, it provides a wider of shapes for design. In this paper, we present the definition 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.

Acknowledgements

The authors appreciate the valuable comments and suggestions from the anonymous reviewers, which improve the clarity of the paper. This work is partly supported by the National Natural Science Foundation of China (Nos. 11671068, 11801053).

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G. Hu and J. L. Wu, Generalized quartic H-Bézier curves: Construction and application to developable surfaces, Adv. Eng. Softw. 138 (2019), 102723, .

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G. R. Zhou and Q. B. Cai, Triangular surface patch based on bivariate Meyer-König-Zeller operator, Open Math. 17 (2019), no. 1, 282–296, .

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G. R. Zhou, X. M. Zeng, and F. L. Fan, Bivariate S-λ bases and S-λ surface patches, Comput. Aided Geom. Design. 31 (2014), 674–688, .

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P. Salvi and T. Várady, Multi-sided Bézier surfaces over concave polygonal domains, Comput Graph. 74 (2018), 56–65, .

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T. Várady, P. Salvi, and G. Karikó, A multi-sided Bézier patch with a simple control structure, Comput Graph. Forum 35 (2016), no. 2, 307–317, .

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Y. P. Zhu, X. L. Han, and S. J. Liu, Curve construction based on four αβ-Bernstein-like basis functions, J. Comput. Appl. Math. 273 (2015), 160–181, .

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Y. Zhang and C. G. Zhu, Degenerations of NURBS curves while all of weights approaching infinity, Jpn. J. Ind. Appl. Math. 35 (2018), no. 2, 787–816, .

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C. G. Zhu and X. Y. Zhao, Self-intersections of rational Bézier curves, Graph. Models 76 (2014), no. 5, 312–320, .

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• Export Citation

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G. Hu, J. L. Wu, and X. Q. Qin, A novel extension of the Bézier mode and its applications to surface modeling, Adv. Eng. Softw. 125 (2018), 27–54, .

• Crossref
• Export Citation
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G. Hu and J. L. Wu, Generalized quartic H-Bézier curves: Construction and application to developable surfaces, Adv. Eng. Softw. 138 (2019), 102723, .

• Crossref
• Export Citation
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R. Schaback, Creating surfaces from scattered data using radial basis functions, in: M. Dæhlen, T. Lyche, and L. L. Schumaker (Eds.), Mathematical Methods for Curves and Surfaces, Vanderbilt University Press, Nashville, TN, 38 (1995), no. 4, 477–496.

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R. Goldman and P. Simeonov, Quantum Bernstein bases and quantum Bézier curves, J. Comput. Appl. Math. 288 (2015), 284–303, .

• Crossref
• Export Citation
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G. R. Zhou and Q. B. Cai, Triangular surface patch based on bivariate Meyer-König-Zeller operator, Open Math. 17 (2019), no. 1, 282–296, .

• Crossref
• Export Citation
• [20]

G. R. Zhou, X. M. Zeng, and F. L. Fan, Bivariate S-λ bases and S-λ surface patches, Comput. Aided Geom. Design. 31 (2014), 674–688, .

• Crossref
• Export Citation
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P. Salvi and T. Várady, Multi-sided Bézier surfaces over concave polygonal domains, Comput Graph. 74 (2018), 56–65, .

• Crossref
• Export Citation
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T. Várady, P. Salvi, and G. Karikó, A multi-sided Bézier patch with a simple control structure, Comput Graph. Forum 35 (2016), no. 2, 307–317, .

• Crossref
• Export Citation
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Y. P. Zhu, X. L. Han, and S. J. Liu, Curve construction based on four αβ-Bernstein-like basis functions, J. Comput. Appl. Math. 273 (2015), 160–181, .

• Crossref
• Export Citation
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L. D. García-Puente and F. Sottile, Linear precision for parametric patches, Adv. Comput. Math. 33 (2010), no. 2, 191–214, .

• Crossref
• Export Citation
• [25]

G. Craciun, L. D. García-Puente, and F. Sottile, Some geometrical aspects of control points for toric patches, in: M. Dæhlen, et al., (Eds.), Mathematical Methods for Curves and Surfaces, Springer, Berlin, Heidelberg, LNCS 5862 (2008), 111–135, .

• Crossref
• Export Citation
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L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cambridge University Press, Cambridge, 2005, 1183–1202.

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E. Postinghel, F. Sottile, and N. Villamizar, Degenerations of real irrational toric varieties, J. Lond. Math. Soc. 92 (2015), no. 2, 223–241, .

• Crossref
• Export Citation
• [28]

A. F. Pir and F. Sottile, Irrational Toric Varieties, arXiv:1807.05919.

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J. G. Li, Q. Y. Chen, J. Q. Han, Q. L. Huang, and C. G. Zhu, A new kind of parametric curves by special basis function, Comput. Sci. 45 (2018), no. 3, 46–50, .

• Crossref
• Export Citation
• [30]

Y. Zhang and C. G. Zhu, Degenerations of NURBS curves while all of weights approaching infinity, Jpn. J. Ind. Appl. Math. 35 (2018), no. 2, 787–816, .

• Crossref
• Export Citation
• [31]

H. W. Lin, H. J. Bao, and G. J. Wang, Totally positive bases and progressive iteration approximation, Comput. Math. Appl. 50 (2005), no. 3, 575–586, .

• Crossref
• Export Citation
• [32]

Y. Y. Yu, H. Ma, and C. G. Zhu, Total positivity of a kind of generalized toric-Bernstein basis, Linear Algebra Appl. 579 (2019), 449–462, .

• Crossref
• Export Citation
• [33]

C. G. Zhu and X. Y. Zhao, Self-intersections of rational Bézier curves, Graph. Models 76 (2014), no. 5, 312–320, .

• Crossref
• Export Citation
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### Open Mathematics

Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant and original works in all areas of mathematics. The journal publishes both research papers and comprehensive and timely survey articles. Open Math aims at presenting high-impact and relevant research on topics across the full span of mathematics.