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

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 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 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].

Figure 1

Lattice points 𝓐 and 𝓐͠.

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) Δ_𝓐 of 𝓐. When the lattice polygon Δ_𝓐 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, ⋯, a_n} ⊂ ℝ with a₀ ≤ a₁ ≤ ⋯ ≤ a_n−1 ≤ a_n, and Δ_𝓐 = [a₀, a_n]. Obviously, the endpoints of Δ_𝓐 are points a₀ and a_n and we assume a₀ < a_n. Set h₀(t) = k₀(t − a₀) and h₁(t) = k₁(a_n − t), where k₀, k₁ are positive real numbers such that h₀(t) ≥ 0, h₁(t) ≥ 0, t ∈ Δ_𝓐. Then, basis functions indexed by 𝓐 can be constructed as follow.

Definition 1

Let 𝓐 = {a₀, a₁, ⋯, a_n} ⊂ ℝ with a₀ ≤ a₁ ≤ ⋯ ≤ a_n−1 ≤ a_n and a₀ < a_n. Then, for any point a_i 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, (1)

where coefficient c_ai > 0 and a_i is called knot.

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

Tai(t)=ωaiβai(t)∑i=0nωaiβai(t),    t∈ΔA, (2)

where ω_ai > 0 is called weight.

Remark 1

The GT-Bernstein basis {β_ai(t)} defined by equation (1) depends on the selection of the coefficients k₀ and 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 {β_ai(t)} is independent of k₀ and k₁ in 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 β_ai(t) on Δ_𝓐 = [0, 1] are shown in Figure 2. The changes of basis function β_a2(t) while coefficient c_a2 varying 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 β_a2(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.

Figure 2

GT-Bernstein basis.

Figure 3

Effect of coefficient changing on GT-Bernstein basis.

Figure 4

Effect of knot changing on GT-Bernstein basis.

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

Theorem 1

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

1. Nonnegativity. T_ai(t) ≥ 0, t ∈ Δ_𝓐, i = 0, 1, ⋯, n.

2. Partition of the unity. ∑i=0nTai(t)≡1.

3. 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].

4. Endpoints property. At the endpoints of [a₀, a_n], we have

   Tai(a0)=1, i=0,0, i≠0,     Tai(an)=1, i=n,0, i≠n.

5. Degeneration property. The GT-Bernstein basis {β_ai(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 a₀ ≤ t₀ < t₁ < ⋯ < t_n ≤ a_n to be an any increasing sequence. Then the collocation matrix of {βai(t)}i=0n at t₀ < t₁ < ⋯ < t_n

Mβa0,⋯,βant0,⋯,tn=(βaj(ti))j=0,1,⋯,ni=0,1,⋯,n (3)

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 Δ_𝓐 for arbitrary positive coefficients, we present a method to choose coefficients by Theorem 2, which makes the basis {β_ai(t)} has partition of the unity on a given increasing sequence a₀ ≤ t₀ < t₁ < ⋯ < t_n ≤ a_n.

Given an increasing sequence a₀ ≤ t₀ < t₁ < ⋯ < t_n ≤ a_n, 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. (4)

If we write C = (c_a0, ⋯, c_an)^T and 1 = (1, ⋯, 1)^T, then we obtain

Mβa0,⋯,βant0,⋯,tnC=1. (5)

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, ⋯, a_n} ⊂ ℝ², Let Δ_𝓐 be the convex hull of 𝓐. The lines defined by edges ϕ_i of Δ_𝓐 are h_i(u, v) = ξ_i u + η_iv + ρ_i, where 〈ξ_i, η_i〉 is the normal vector of ϕ_i towards inside of Δ_𝓐 such that h_i(u, v) ≥ 0, (u, v) ∈ Δ_𝓐, i = 1, ⋯, r. We construct the generalized toric-Bernstein basis functions as follows.

Definition 2

Let 𝓐 = {a₀, a₁, ⋯, a_n} ⊂ ℝ² be a finite collection of real points, and set Δ_𝓐 to be the convex hull of 𝓐. Then, for any point a_i in 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, (6)

where c_ai > 0 is the coefficient and a_i is called knot.

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

Tai(u,v)=ωaiβai(u,v)∑i=0nωaiβai(u,v),  (u,v)∈ΔA. (7)

where ω_ai > 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 c_ai ≡ 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 ω_ai = 1, then the rational forms of these three basis functions on Δ_𝓐 are shown in Figure 6.

Figure 5

GT-Bernstein basis functions.

Figure 6

Rational GT-Bernstein basis functions.

Suppose edges ϕ_i(i = 1, ⋯, r) of the convex hull Δ_𝓐 are ordered counterclockwise and let V_i be vertex of Δ_𝓐 where two edges ϕ_i and ϕ_i+1 meet, (i = 1, ⋯, r). The indices will be treated in a cyclic fashion: for instance, ϕ₀ = ϕ_r, ϕ_r+1 = ϕ₁ and so on. Denote by ϕ̂_i = ϕ_i ∩ 𝓐 the intersection of 𝓐 and ϕ_i. Note that {Vi}i=1r and ϕ̂_i 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 {T_ai(u, v)} directly.

Theorem 3

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

1. Nonnegativity. T_ai(u, v) ≥ 0, (u, v) ∈ Δ_𝓐, i = 0, 1, ⋯, n.

2. Partition of the unity. ∑i=0nTai(u,v)≡1.

3. Boundary property. When (u, v) is constrained on the edge ϕ_j of Δ_𝓐, all basis functions β_ai(u, v) and T_ai(u, v) with indices a_i ∈ 𝓐 ∖ ϕ̂_j 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^, (8)

4. Corner points property. At the vertices of Δ_𝓐, we have

   Tai(Vi)=1, ai=Vi,Tai(Vi)=0, ai≠Vi. (9)

5. Degeneration property. For 𝓐 = {a₀, a₁, ⋯, a_n} ⊂ 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 + j ≤ k, i ≥ 0, j ≥ 0}, and to the tensor product Bernstein basis for 𝓐 = {(i, j) ∈ ℤ² | 0 ≤ i ≤ m, 0 ≤ j ≥ n}, 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.

Example 3

Let A={a0=0,a1=24,a2=12,a3=22,a4=1} as show in Example 1, weights ω_a0 = 1, ω_a1 = 10, ω_a2 = 20, ω_a3 = 6, ω_a4 = 5 and control points b_a0 = (0, 0), b_a1 = (0.4, 1.3), b_a2 = (2, 2), b_a3 = (3.7, 1.5), b_a4 = (4, 0). Suppose c_ai = 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.

Figure 7

Quadratic GT-Bézier curve.

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

1. 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.

2. Endpoints interpolation property. This property follows directly from the endpoints property of the basis (2), that is P_𝓐,ω,𝓑(a₀) = b_a0, P_𝓐,ω,𝓑(a_n) = b_an.

3. 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.

4. Degeneration property. If a_i = i (or a_i = 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, ⋯, a_n} ⊂ 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].

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. (11)

   We can see the tangent vectors at the end points of curve P_𝓐,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.

6. 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.

Example 4

Consider the curve P_𝓐,ω,𝓑(t) defined as in Example 3. Let knots 𝓐 = {a₀ = 0, a₁ = 24 , a₂ = 12 , a₃ = 22 , a₄ = 1}, weights ω_a0 = 1, ω_a1 = 10, ω_a2 = 20, ω_a3 = 6, ω_a4 = 5, control points b_a0 = (0, 0), b_a1 = (0.4, 1.3), b_a2 = (2, 2), b_a3 = (3.7, 1.5), b_a4 = (4, 0) and c_ai = 1 (i = 0, ⋯, 4). If a₁ approaches a₂, then the changes of the GT-Bézier curve are shown in Figure 8. We can see that the limit curve lim_a1→a2 P_𝓐,ω,𝓑(t) coincides with the target curve P͠_{𝓐͠,ω͠,𝓑͠}(t), which verifies the Theorem 4.

Figure 8

Limits of the quadratic GT-Bézier curve of single knot.

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, ⋯, a_k−1, a_k+1, ⋯, a_n}, control points 𝓑͠ = {b_a0, ⋯, b_ak−1, b͠_ak+1, b_ak+2, ⋯, b_an} and weights ω͠ = {ω_a0, ⋯, ω_ak−1, ω͠_ak+1, ω_ak+2, ⋯, ω_an} when a_k = a_k+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.

Example 5

Consider the curve P_𝓐,ω,𝓑(t) defined as in Example 3. If a₁ → a₂ and a₃ → a₂, 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 = {ω_a0 = 1, ω͠_a2 = 36, ω_a4 = 5}. We can see that the limit curve coincides with the target curve together, which verifies the result of Corollary 1.

Figure 9

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

1. Toric degeneration property. For each t ∈ Δ_𝓐, 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 ω_a1 → +∞.

Figure 10

Limit of GT-Bézier curve with ω_a1 → +∞.

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 a_i of 𝓐 to (a_i, λ(a_i)) ∈ ℝ². 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 normal vector pointing to the outer side. We call it the upper edges of P_λ if the last coordinate of the normal vector is positive. If we project these upper edges back vertically into ℝ, they can cover Δ_𝓐 and form a regular subdivision Γ_λ of Δ_𝓐 induced by λ [6].

We group together the points of 𝓐 that are in the same subset of the Γ_λ and on the same upper edge 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 = {ω_ai | a_i ∈ 𝓕} and the control points 𝓑|_𝓕 = {b_ai | a_i ∈ 𝓕} to define a new GT-Bézier curve P_{𝓕,ω|𝓕,𝓑|𝓕} on Δ_𝓕 = conv{a_i ∈ 𝓕 | a_i ∈ 𝓐} by Definition 3. The union of these curves

is called the regular control curve of P_𝓐,ω,𝓑 induced by regular decomposition 𝓢_λ.

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

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

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 P_λ, 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.

Figure 11

Toric degeneration of GT-Bézier curve.

If λ^′ takes 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.

Figure 12

Regular decompositions of 𝓐.

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