Jing-Gai Li and Chun-Gang Zhu

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

Open Access
De Gruyter Open Access | Published online: March 2, 2020

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.

MSC 2010: 65D17; 68U07

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].

Figure 1 
Lattice points 𝓐 and 𝓐͠.

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) 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

β a i ( t ) = c a i h 0 ( t ) h 0 ( a i ) h 1 ( t ) h 1 ( a i ) , t Δ A , (1)

where coefficient cai > 0 and ai is called knot.

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

T a i ( t ) = ω a i β a i ( t ) i = 0 n ω a i β a i ( t ) , t Δ A , (2)

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 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 {βai(t)} is independent of k0 and k1 in Section 3.

Example 1

Let a 0 = 0 , a 1 = 2 4 , a 2 = 1 2 , a 3 = 2 2 , a 4 = 1 a n d c a 0 = 1 2 , c a 1 = 1 , c a 2 = 3 2 , c a 3 = 7 10 , c a 4 = 9 10 . By (1), we have

β a 0 ( t ) = 1 2 ( 1 t ) , β a 1 ( t ) = t 2 4 ( 1 t ) 1 2 4 , β a 2 ( t ) = 3 2 t 1 2 ( 1 t ) 1 2 , β a 3 ( t ) = 7 10 t 2 2 ( 1 t ) 1 2 2 , β a 4 ( t ) = 9 10 t ,

and the basis functions βai(t) on Δ𝓐 = [0, 1] are shown in Figure 2. The changes of basis function βa2(t) while coefficient ca2 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 βa2(t) when its corresponding knot changes are shown in Figure 4 (the knots corresponding to curves from left to right are 2 5 , 1 2 , 5 3 respectively), which means that the knot mainly affect the positions of the maximum point of the basis function.

Figure 2 
GT-Bernstein basis.

Figure 2

GT-Bernstein basis.

Figure 3 
Effect of coefficient changing on GT-Bernstein basis.

Figure 3

Effect of coefficient changing on GT-Bernstein basis.

Figure 4 
Effect of knot 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 {Tai(t)} can be obtained directly as follows.

Theorem 1

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

  1. Nonnegativity. Tai(t) ≥ 0, tΔ𝓐, i = 0, 1, ⋯, n.

  2. Partition of the unity. i = 0 n T a i ( t ) 1.

  3. Normalized totally positive (NTP). The rational GT-Bernstein basis { T a i ( t ) } i = 0 n is a NTP basis. This property is proved recently by Yu et al. [32].

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

    T a i ( a 0 ) = 1 , i = 0 , 0 , i 0 , T a i ( a n ) = 1 , i = n , 0 , i n .

  5. Degeneration property. The GT-Bernstein basis {βai(t)} degenerates to the classical Bernstein basis for 𝓐 = {0, 1, ⋯, n} or 𝓐 = {0, 1 n , ⋯, 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, 1 n , ⋯, 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 { β a i ( t ) } i = 0 n at t0 < t1 < ⋯ < tn

M β a 0 , , β a n t 0 , , t n = ( β a j ( t i ) ) j = 0 , 1 , , n i = 0 , 1 , , n (3)

is a strictly totally positive matrix.

Since the basis { β a i ( t ) } i = 0 n 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 = 0 n β a j ( t i ) = c a 0 h 0 ( t i ) h 0 ( a 0 ) h 1 ( t i ) h 1 ( a 0 ) + + c a n h 0 ( t i ) h 0 ( a n ) h 1 ( t i ) h 1 ( a n ) = 1 , i = 0 , , n . (4)

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

M β a 0 , , β a n t 0 , , t n C = 1 . (5)

It’s clear that the basis { β a i ( t ) } i = 0 n 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} ⊂ ℝ2 be a finite collection of real points, and set Δ𝓐 to be the convex hull of 𝓐. Then, for any point ai in 𝓐, we define the bivariate generalized toric-Bernstein (GT-Bernstein) basis function as

β a i ( u , v ) = c a i h 1 ( u , v ) h 1 ( a i ) h r ( u , v ) h r ( a i ) , ( u , v ) Δ A , (6)

where cai > 0 is the coefficient and ai is called knot.

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

T a i ( u , v ) = ω a i β a i ( u , v ) i = 0 n ω a i β a i ( u , v ) , ( u , v ) Δ A . (7)

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 , 6 5 ) , ( 8 7 , 8 7 ) , ( 2 , 1 ) , ( 0 , 0 ) , ( 6 5 , 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 ) 2 v 2 , β ( 1 , 2 ) ( u , v ) = ( 2 u ) v 2 u , β ( 0 , 6 5 ) ( u , v ) = ( 2 v ) 4 5 ( 3 u v ) 9 5 ( 2 u ) 2 v 6 5 , β ( 8 7 , 8 7 ) ( u , v ) = ( 2 v ) 6 7 ( 3 u v ) 5 7 ( 2 u ) 6 7 v 8 7 u 8 7 , β ( 2 , 1 ) ( u , v ) = ( 2 v ) v u 2 , β ( 0 , 0 ) ( u , v ) = ( 2 v ) 2 ( 3 u v ) 3 ( 2 u ) 2 , β ( 6 5 , 0 ) ( u , v ) = ( 2 v ) 2 ( 3 u v ) 9 5 ( 2 u ) 4 5 u 6 5 , β ( 2 , 0 ) ( u , v ) = ( 2 v ) 2 ( 3 u v ) u 2 .

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 Δ𝓐 are shown in Figure 6.

Figure 5 
GT-Bernstein basis functions.

Figure 5

GT-Bernstein basis functions.

Figure 6 
Rational GT-Bernstein basis functions.

Figure 6

Rational GT-Bernstein basis functions.

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 { V i } i = 1 r 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:

  1. Nonnegativity. Tai(u, v) ≥ 0, (u, v) ∈ Δ𝓐, i = 0, 1, ⋯, n.

  2. Partition of the unity. i = 0 n T a i ( u , v ) 1.

  3. Boundary property. When (u, v) is constrained on the edge ϕj of Δ𝓐, all basis functions βai(u, v) and Tai(u, v) with indices ai ∈ 𝓐 ∖ ϕ̂j vanish, that is:

    β a i ( u , v ) = 0 , a i A ϕ j ^ , ( u , v ) ϕ j , β a i ( u , v ) 0 , a i ϕ j ^ , T a i ( u , v ) = 0 , a i A ϕ j ^ , ( u , v ) ϕ j . T a i ( u , v ) 0 , a i ϕ j ^ , (8)

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

    T a i ( V i ) = 1 , a i = V i , T a i ( V i ) = 0 , a i V i . (9)

  5. 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

P A , ω , B ( t ) = i = 0 n b a i T a i ( t ) = i = 0 n b a i ω a i β a i ( t ) i = 0 n ω a i β a i ( t ) , t Δ A (10)

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 k0 and 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 k0 and 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 = { a 0 = 0 , a 1 = 2 4 , a 2 = 1 2 , a 3 = 2 2 , a 4 = 1 } as show in Example 1, weights ωa0 = 1, ωa1 = 10, ωa2 = 20, ωa3 = 6, ωa4 = 5 and control points ba0 = (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

P A , ω , B ( t ) = i = 0 4 b a i T a i ( t ) , t [ 0 , 1 ] ,

and the curve is shown in Figure 7.

Figure 7 
Quadratic GT-Bézier curve.

Figure 7

Quadratic GT-Bézier curve.

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:

  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 P𝓐,ω,𝓑(a0) = ba0, P𝓐,ω,𝓑(an) = ban.

  3. Progressive iteration approximation (PIA) property. The GT-Bézier curve has PIA property from the result in [31] because its basis { T a i ( t ) } i = 0 n is a NTP basis.

  4. Degeneration property. If ai = i (or ai = i n ), (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].

  5. Endpoints tangent vectors. For

    k 0 = 1 a 1 a 0 , k 1 = 1 a n a n 1 ,

    the tangent vectors at the end points of GT-Bézier curve are

    P A , ω , B ( a 0 ) = c a 1 k 0 k 1 k 1 k 0 ( a n a 0 ) k 1 k 0 ω a 1 ( b a 1 b a 0 ) c a 0 ω a 0 , P A , ω , B ( a n ) = c a n 1 k 1 k 0 k 0 k 1 ( a n a 0 ) k 0 k 1 ω a n 1 ( b a n b a n 1 ) c a n ω a n . (11)

    We can see the tangent vectors at the end points of curve P𝓐,ω,𝓑(t) are parallel to b a 0 b a 1 and b a n 1 b a n 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.

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 curve P𝓐,ω,𝓑(t) of degree n defined in (10) is exactly the GT-Bézier curve P͠𝓐͠,ω͠,𝓑͠(t) of degree n − 1, defined as

lim a k a k + 1 P A , ω , B ( t ) = P ~ A ~ , ω ~ , B ~ ( t ) = i = 0 k 1 ω a i b a i β a i ( t ) + ω ~ a k + 1 b ~ a k + 1 β a k + 1 ( t ) + i = k + 2 n ω a i b a i β a i ( t ) i = 0 k 1 ω a i β a i ( t ) + ω ~ a k + 1 β a k + 1 ( t ) + i = k + 2 n ω a i β a i ( t ) , (12)

where

ω ~ a k + 1 = ω a k + ω a k + 1 , b ~ a k + 1 = ω a k ω a k + ω a k + 1 b a k + ω a k + 1 ω a k + ω a k + 1 b a k + 1 ,

A ~ = { a 0 , , a k 1 , a k + 1 , , a n } , B ~ = { b a 0 , , b a k 1 , b ~ a k + 1 , b a k + 2 , , b a n } , ω ~ = { ω a 0 , , ω a k 1 , ω ~ a k + 1 , ω a k + 2 , , ω a n } .

Proof

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

lim a k a k + 1 β a k ( t ) = lim a k a k + 1 ( t a 0 ) a k a 0 ( a n t ) a n a k = ( t a 0 ) a k + 1 a 0 ( a n t ) a n a k + 1 = β a k + 1 ( t ) .

Thus,

lim a k a k + 1 i = 0 n b a i T a i ( t ) = P ~ A ~ , ω ~ , B ~ ( t ) = lim a k a k + 1 i = 0 n b a i ω a i β a i ( t ) i = 0 n ω a i β a i ( t ) = i k , k + 1 ω a i b a i β a i ( t ) + ω a k b a k β a k + 1 ( t ) + ω a k + 1 b a k + 1 β a k + 1 ( t ) i k , k + 1 ω a i β a i ( t ) + ω a k β a k + 1 ( t ) + ω a k + 1 β a k + 1 ( t ) = i k , k + 1 ω a i b a i β a i ( t ) + ( ω a k b a k + ω a k + 1 b a k + 1 ) β a k + 1 ( t ) i k , k + 1 ω a i β a i ( t ) + ( ω a k + ω a k + 1 ) β a k + 1 ( t ) .

Let ω ~ a k + 1 = ω a k + ω a k + 1 , b ~ a k + 1 = ω a k ω a k + ω a k + 1 b a k + ω a k + 1 ω a k + ω a k + 1 b a k + 1 , we can obtain

lim a k a k + 1 P A , ω , B ( t ) = P ~ A ~ , ω ~ , B ~ ( t ) = i k , k + 1 ω a i b a i β a i ( t ) + ω ~ a k + 1 b ~ a k + 1 β a k + 1 ( t ) i k , k + 1 ω a i β a i ( t ) + ω ~ a k + 1 β a k + 1 ( t ) ,

where

A ~ = { a 0 , , a k 1 , a k + 1 , , a n } , B ~ = { b a 0 , , b a k 1 , b ~ a k + 1 , b a k + 2 , , b a n } , ω ~ = { ω a 0 , , ω a k 1 , ω ~ a k + 1 , ω a k + 2 , , ω a n } .

This leads to prove the result. □

Example 4

Consider the curve P𝓐,ω,𝓑(t) defined as in Example 3. Let knots 𝓐 = {a0 = 0, a1 = 2 4 , a2 = 1 2 , a3 = 2 2 , a4 = 1}, weights ωa0 = 1, ωa1 = 10, ωa2 = 20, ωa3 = 6, ωa4 = 5, control points ba0 = (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 a1 approaches a2, then the changes of the GT-Bézier curve are shown in Figure 8. We can see that the limit curve lima1a2 P𝓐,ω,𝓑(t) coincides with the target curve P͠𝓐͠,ω͠,𝓑͠(t), which verifies the Theorem 4.

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

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, ⋯, 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 curve P𝓐,ω,𝓑(t) of degree n defined in (10) is exactly the GT-Bézier curve of degree nk + 1 as

lim a q , , a q + k 2 a q + k 1 P A , ω , B ( t ) = P ~ A ~ , ω ~ , B ~ ( t ) = i = 0 q 1 ω a i b a i β a i ( t ) + ω ~ a q + k 1 b ~ a q + k 1 β a q + k 1 ( t ) + i = q + k n ω a i b a i β a i ( t ) i = 0 q 1 ω a i β a i ( t ) + ω ~ a q + k 1 β a q + k 1 ( t ) + i = q + k n ω a i β a i ( t ) ,

where

ω ~ a q + k 1 = ω a q + ω a q + 1 + + ω a q + k 1 , b ~ a q + k 1 = ω a q ω ~ a q + k 1 b a q + + ω a q + k 1 ω ~ a q + k 1 b a q + k 1 , A ~ = { a 0 , , a q 1 , a q + k 1 , , a n } , B ~ = { b a 0 , , b a q 1 , b ~ a q + k 1 , b a q + k , , b a n } , ω ~ = { ω a 0 , , ω a q 1 , ω ~ a q + k 1 , ω a q + k , , ω a n } .

Example 5

Consider the curve P𝓐,ω,𝓑(t) defined as in Example 3. If a1a2 and 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 = 1 2 , a4 = 1}, control points B ~ = { b a 0 = ( 0 , 0 ) , b ~ a 2 = ( 66.2 36 , 62 36 ) , b a 4 = ( 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.

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

Figure 9

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

  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

    lim ω a i + P A , ω , B ( t ) = b a 0 t = a 0 , b a i t ( a 0 , a n ) , b a n t = a n .

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 → +∞.

Figure 10 
Limit of GT-Bézier curve with ωa1 → +∞.

Figure 10

Limit of GT-Bézier curve 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

P A , ω , B ( S λ ) = F S λ P F , 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

P A , ω λ ( x ) , B ( t ) = i = 0 n x λ ( a i ) ω a i b a i β a i ( t ) i = 0 n x λ ( a i ) ω a i β a i ( t ) , t Δ A . (13)

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 curve P𝓐,ωλ(x),𝓑 as x → ∞ is the regular control curve induced by regular decomposition 𝓢λ, that is

lim x P A , ω λ ( x ) , B = P A , ω , B ( S λ ) .

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 { β a i a i A } X A ω X A , ω B P A , ω , B .

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

A { β a i a i A } X A ω λ ( x ) X A , ω λ ( 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

lim x X A , ω λ ( x ) = F S λ X F , ω | F .

Then add control points 𝓑, we have

F S λ X F , ω | F B F S λ P F , ω | F , B | F = P A , ω , B ( S λ ) .

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 , 2 4 , 1 2 , 2 2 , 1 } , and the lifted values of 𝓐 by a lifting function λ be (2, 1, 5, 9 − 4 2 , 1). This induces a regular decomposition of 𝓐 as

{ 0 , 1 2 } , { 1 2 , 2 2 , 1 } .

The lifted point 2 4 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.

Figure 11 
Toric degeneration of GT-Bézier curve.

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 , 2 4 } , { 2 4 , 1 2 } , { 1 2 , 2 2 } , { 2 2 , 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.

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 , 2 4 } , { 2 4 , 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 d i = p i q i ( p i , q i 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 curve P𝓐,ω,𝓑(t) is variation diminishing, which means that the number of intersections of any straight line with the GT-Bézier curve P𝓐,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 + m1 t + ⋯ + 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

Z t > 0 [ m 0 + m 1 t + + m n t n ] V [ m 0 , m 1 , , m n ] .

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 a 0 a n t , t ( a 0 , a n ) , so that u ∈ (0, +∞). Then by the Cartesian notation rule

I ( C , L ) = Z a 0 t a n i = 0 n y a i T a i ( t ) = Z a 0 t a n i = 0 n y a i ω a i c a i ( t a 0 ) t a i ( a n t ) a n a i i = 0 n ω a i c a i ( t a 0 ) t a i ( a n t ) a n a i = Z 0 < u < + i = 0 n y a i ω a i c a i u a i a 0 i = 0 n ω a i c a i u a i a 0 = Z 0 < u < + i = 0 n y a i ω a i c a i u q d i i = 0 n ω a i c a i u q d i = Z 0 < u < + i = 0 n y a i ω a i c a i u q d i i = 0 n ω a i c a i u q d i = Z 0 < u < + i = 0 n y a i u q d i V y a 0 , y a 1 , , y a n = I ( P , L ) ,

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} ⊂ ℝ2 be 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

P A , ω , B ( u , v ) = i = 0 n b a i T a i ( u , v ) = i = 0 n b a i ω a i β a i ( u , v ) i = 0 n ω a i β a i ( u , v ) , ( u , v ) Δ A . (14)

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, 6 5 , 2), ( 8 7 , 8 7 , 5), (2, 1, 2), (0, 0, 0), ( 6 5 , 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 , 6 5 ) , ( 8 7 , 8 7 ) , ( 2 , 1 ) , ( 0 , 0 ) , ( 6 5 , 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].

Figure 13 
Toric Surface and GT-Bézier Surface.

Figure 13

Toric Surface and GT-Bézier Surface.

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

  1. 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.

  2. 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.

  3. 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.

  4. 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 curve Pϕ̂i,ω|ϕ̂i,𝓑|ϕ̂i, which defined by control points bai and weights ωai by aiϕ̂i 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 = ξ η , and

cos α = η ξ 2 + η 2 .

Let σ = | V 0 V 1 | = ( u 1 u 0 ) 2 + ( v 1 v 0 ) 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, v j = ρ + ξ u j η . Let l j = | a j V 0 | = ( u j u 0 ) 2 + ( v j v 0 ) 2 . By geometric relationship, we have

u j = u 0 + l j cos α .

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

h k ( a j ) = ξ k u j + η k v j + ρ k = η ξ k ξ η k η u j + ρ k η k η ρ = ρ k η k η ρ + η ξ k ξ η k η u 0 + ( η ξ k ξ η k ) cos α η l j .

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

β a i ( u , v ) = c a i k = 1 r h k ( u , v ) ( ρ k η k η ρ + η ξ k ξ η k η u 0 ) ( h 1 ( u , v ) ( η ξ 1 ξ η 1 ) cos α η h r ( u , v ) ( η ξ r ξ η r ) cos α η ) l j .

Here the first r factors h k ( u , v ) ( ρ k η k η ρ + η ξ k ξ η k η u 0 ) 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 + ξ u η . So when (u, v) ∈ ϕ, hk(u, v) is univariate function of u, written hk(u). If we set new variables

s = h 1 ( u ) ( η ξ 1 ξ η 1 ) cos α η h r ( u ) ( η ξ r ξ η r ) cos α η , t = σ s 1 + s ,

we obtain

P ϕ ^ , ω | ϕ ^ , B | ϕ ^ ( u ) = a j ϕ ^ ω a j b a j c a j s l j a j ϕ ^ ω a j c a j s l j = a j ϕ ^ ω a j b a j c a j t l j ( σ t ) σ l j a j ϕ ^ ω a j c a j t l j ( σ t ) σ l j .

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

d s d τ = d d τ ( h 1 ( u ) ( η ξ 1 ξ η 1 ) cos α η ) h r ( u ) ( η ξ r ξ η r ) cos α η + + h 1 ( u ) ( η ξ 1 ξ η 1 ) cos α η d d τ ( h r ( u ) ( η ξ r ξ η r ) cos α η ) = σ k = 1 r h k ( u ) ( η ξ k ξ η k ) cos α η j = 1 r ( η ξ j ξ η j η cos α ) 2 h j ( u ) > 0 , 0 < τ < 1 ,

and

d t d τ = d d τ ( σ s 1 + s ) = σ ( 1 + s ) 2 d s d τ > 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 knot ak (0 ≤ k < n) approaches to aq (0 ≤ q < n, and qk) along line ak aq with the convex hull Δ𝓐 unchanging, the limit of GT-Bézier surface P𝓐,ω,𝓑(u, v) defined in (14) is exactly the GT-Bézier surface P͠𝓐͠, ω͠, 𝓑͠(u, v), defined as

lim a k a q P A , ω , B ( u , v ) = P ~ A ~ , ω ~ , B ~ ( u , v ) = i k , q ω a i b a i β a i ( u , v ) + ω ~ a q b ~ a q β a q ( u , v ) i k , q ω a i β a i ( u , v ) + ω ~ a q β a q ( u , v ) , (15)

where

ω ~ a q = ω a k + ω a q , b ~ a q = ω a k ω a k + ω a q b a k + ω a q ω a k + ω a q b a q , ω ~ = { ω a 0 , , ω a k 1 , ω a k + 1 , , ω ~ a q , , ω a n } , A ~ = { a 0 , , a k 1 , a k + 1 , , a n } , B ~ = { b a 0 , , b a k 1 , b a k + 1 , , b ~ a q , , b a n } . (16)

Proof

When ak tends to aq, we have

lim a k a q β a k ( u , v ) = lim a k a q h 1 ( u , v ) h 1 ( a k ) h r ( u , v ) h r ( a k ) = h 1 ( u , v ) h 1 ( a q ) h r ( u , v ) h r ( a q ) = β a q ( u , v ) .

Thus,

lim a k a q i = 0 n b a i T a i ( u , v ) = P ~ A ~ , ω ~ , B ~ ( u , v ) = lim a k a q i = 0 n b a i ω a i β a i ( u , v ) i = 0 n ω a i β a i ( u , v ) = i k , q ω a i b a i β a i ( u , v ) + ω a k b a k β a q ( u , v ) + ω a q b a q β a q ( u , v ) i k , q ω a i β a i ( u , v ) + ω a k β a q ( u , v ) + ω a q β a q ( u , v ) = i k , q ω a i b a i β a i ( u , v ) + ( ω a k b a k + ω a q b a q ) β a q ( u , v ) i k , q ω a i β a i ( u , v ) + ( ω a k + ω a q ) β a q ( u , v ) .

Let ω ~ a q = ω a k + ω a q , b ~ a q = ω a k ω a k + ω a q b a k + ω a q ω a k + ω a q b a q , we can obtain

lim a k a q P A , ω , B ( u , v ) = P ~ A ~ , ω ~ , B ~ ( u , v ) = i k , q ω a i b a i β a i ( u , v ) + ω ~ a q b ~ a q β a q ( u , v ) i k , q ω a i β a i ( u , v ) + ω ~ a q β a q ( 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 , 6 5 ) , ( 8 7 , 8 7 ) , (2, 1), (0, 0), ( 6 5 , 0), (2, 0)}, control points 𝓑 = {(0, 2, 0), (1, 2, 4), (0, 6 5 , 2), ( 8 7 , 8 7 , 5), (2, 1, 2), (0, 0, 0), ( 6 5 , 0, 2), (2, 0, 0)}, weights ω = {2, 2, 5, 7, 2, 3, 5, 2} and cai = 1 (i = 0, ⋯, 7). If a3 = ( 8 7 , 8 7 ) approaches a1 = (1, 2), then the changes of the GT-Bézier surface are shown in Figure 14.

Since the shape of the convex hull Δ𝓐 and control points 𝓑 are unchanging during the process of a3 tending to a1, the original curved surface is stretched like an elastic film by the boundary property of the GT-Bézier surface. Until a3 = a1, the resulting surface is defined by 𝓐͠ = {(0, 2), (1, 2), (0, 6 5 ), (2, 1), (0, 0), ( 6 5 , 0), (2, 0)}, control points 𝓑͠ = {(0, 2, 0), ( 10 9 , 12 9 , 43 9 ) , ( 0 , 6 2 , 2 ) , (2, 1, 2), (0, 0, 0), ( 6 5 , 0, 2), (2, 0, 0)}, weights ω͠ = {2, 9, 5, 2, 3, 5, 2}.

Figure 14 
Limit of GT-Bézier surface with a3 → a1.

Figure 14

Limit of GT-Bézier surface with a3a1.

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

P A , ω , B ( S λ ) = F S λ P F , 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

P A , ω λ ( x ) , B ( u , v ) = i = 0 n x λ ( a i ) ω a i b a i β a i ( u , v ) i = 0 n x λ ( a i ) ω a i β a i ( u , v ) , ( u , v ) Δ A . (17)

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 surface P𝓐,ωλ(x),𝓑 as x → ∞ is the regular control surface induced by the regular decomposition 𝓢λ, that is

lim x P A , ω λ ( x ) , B = P A , ω , B ( S λ ) .

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 Δ𝓐͠ by λ are shown in Figure 15(b), and the regular decomposition 𝓢λ is shown in Figure 15(c). Let control points 𝓑 = {(0, 2, 0), (1, 2, 4), (0, 6 5 , 2), ( 8 7 , 8 7 , 18), (2, 1, 2), (0, 0, 0), ( 6 5 , 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).

Figure 15 
Regular decomposition.

Figure 15

Regular decomposition.

Figure 16 
Toric degeneration of GT-Bézier surface.

Figure 16

Toric degeneration of GT-Bézier surface.

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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Received: 2019-06-28
Accepted: 2020-01-18
Published Online: 2020-03-02

© 2020 Jing-Gai Li and Chun-Gang Zhu, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.