Curve and surface construction based on the generalized toric-Bernstein basis functions
aa r X i v : . [ c s . G R ] A p r Curve and surface construction based on the generalizedtoric-Bernstein basis functions
Jing-Gai Li, Chun-Gang Zhu ∗ School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China.
Abstract
The construction of parametric curve and surface plays important role in computer aidedgeometric design (CAGD), computer aided design (CAD), and geometric modeling. In thispaper, we define a new kind of blending functions associated with a real points set, calledgeneralized toric-Bernstein (GT-Bernstein) basis functions. Then the generalized toric-Bezier(GT-B´ezier) 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 generalizationsof the classical rational B´ezier curves and surfaces and toric surface patches. Furthermore,we also study the properties of the presented curves and surfaces, including the limitingproperties of weights and knots. Some representative examples verify the properties andresults.
Keywords:
Curve and surface design, B´ezier curve and surface, Basis functions, Bernsteinbasis functions, Toric surface patches
1. Introduction
Representing curves and surfaces of geometric shapes is an essential but challenging taskof Computer Aided Geometric Design (CAGD) and Computed Aided Design (CAD) ([1, 2]).Curves and surfaces are commonly represented by the parametric method and the implicitmethod. To be specifically speaking, the parametric method is characterized by its advan-tages of easy plotting, generalization and splicing. Parametric curves and surfaces have goneexperienced the developments of polynomial parametric curve (eg. Ferguson curves), B´ezier ∗ Corresponding author
Email addresses:
[email protected] (Jing-Gai Li), [email protected] (Chun-GangZhu)
Preprint submitted to Elsevier April 11, 2019 ethod, B-spline method and NURBS method ([1, 2]). In this development process, theconstruction of basis functions(also called blending functions) of a curve and a surface playsa key role. The history of parametric curve and surface is essentially the history of the basisfunctions, which has gone through the polynomial power basis, Bernstein basis, B-spline ba-sis, and their related rational and extended forms. Therefore, the basis functions are the coreof curve and surface construction. A set of basis functions with good properties inevitablymakes the parametric curve and surface possessing powerful vitality and application value.In 1912, S.N. Bernstein [3] constructed a set of polynomial sequences to prove the Weier-strass approximation theorem. The key of Bernstein’s proof is the construction of the basisfunctions, which are called Bernstein basis functions. In 1959, de Casteljau [4] firstly appliedthe Bernstein basis functions to the curve representation and then P. B´ezier used them forgeometric shape representation, namely B´ezier curve. The B´ezier curve uses the control poly-gon to represent the curve, which is a major breakthrough in the curve representation methodin CAGD and CAD. The B´ezier method develops rapidly and becomes the core method ofgeometric shape representation because of its good geometric properties and algorithms. It isworth mentioning that the Bernstein basis functions and their generalizations are widely ap-plied in various fields with its good properties, and thus become an important factor affectingmathematics, applied mathematics and related subjects in the 20th century [5].In recent years, the parametric curves and surfaces construction based on different ba-sis functions have been studied by many scholars. Chen and Wang et al. [6, 7] defined theC-B´ezier curve and the C-B spline curve (NUAT B-spline curve) by extending the spaceof mixed algebra and trigonometric polynomial. The C-B´ezier curve and C-B spline curveintroduce shape parameters and adjust the shape of the curve by controlling the changesof control points and parameters. Oru¸c and Phillips [8] defined a q -B´ezier curve based onthe q -Bernstein operator which was constructed by Phillips [9]. They studied the rational q -B´ezier curve and the tensor product q -B´ezier surface, proved the properties of the curveand surface and gave the corresponding subdivision form further. Han et al. [10] constructeda new generalization of B´ezier curves and the corresponding tensor product surfaces over therectangular domain .These curves and surfaces are based on the Lupa¸s q -analogue of Bern-stein operator. Compared with q -B´ezier curves and surfaces based on Phillips q -Bernsteinpolynomials, these curves and surfaces show more flexibility in choosing the value of q andsuperiority in shape control of curves and surfaces. Schaback [11] gave an introduction tocertain techniques for the construction of surfaces from scattered data, which emphasis isputting on interpolation methods using compactly supported radial basis functions. Gold-man and Simeonov [12] studied the properties of quantum Bernstein bases and quantumB´ezier curves by introducing a new variant of the blossom.In 2002, Krasauskas [13] proposed a new multisided surface modeling method namely toricsurface based on the toric ideals and toric varieties defined by a given set of integer lattice2oints. The basis functions constructing the toric surface are called the toric-Bernstein basisfunctions (also called the toric-B´ezier basis functions). The toric surface is degenerated intorational B´ezier curve if the set of lattice points are constrained to a one-dimensional integerpoints, and the tensor product and the triangular B´ezier surfaces are also special forms of thetoric surface. Garc´ıa-Puente et al. [14] explained the geometric significance of the weights oftoric surfaces, which is called the toric degeneration property.Most of the basis functions constructing curves and surfaces defined above are representedin non-negative integer power forms. At present, some researchers have put many efforts onthe construction of curves and surfaces based on basis functions with rational or irrationalnumber powers . The multiquadric(MQ) function is a radial basis function (RBF) withthe rational number power form which is widely used in numerical analysis and scientificcomputing [11]. In 2015, Zhu et al. [15] extended the Bernstein basis functions and thenconstructed αβ -Bernstein-like basis with two exponential shape parameters α and β withreal number degrees.Garc´ıa-Puente and Sottile [22] showed that tuning a pentagonal toric patch by latticepoints e A (see Fig. 1(b)) instead of A (see Fig. 1(a)) to achieve linear precision, where e A contains three non-integer points. In 2008, Craciun et al. [16] studied the theory of toricvarieties defined by generally real lattice sets, which are applied in algebraic statistics knownas toric model [17] and studied the geometric properties of toric surfaces. In 2015, Postinghelet al. [18] presented the degenerations of real irrational toric varieties defined by generally realnumber set. Li et al. studied the T-B´ezier curve constructed by the real points preliminaryin [19]. (a) Lattice points A (b) Lattice points e A Figure 1: Lattice points A and e A In this paper, inspired by above methods, especially by [18], we present a kind of gener-alized toric-Bernstein basis functions of real number powers for any given real number set,3nd then construct a new kind of parametric curve and multisided surface based on the gen-eralized toric-Bernstein basis functions. The properties of presented curves and surfaces arealso studied.The rest of this paper is organized as follows. In Section 2, the generalized toric-Bernsteinbasis functions are defined and the properties of the basis are discussed. Then, a class ofgeneralized B´ezier curve is constructed in Section 3, which is the generalization of the classicalrational B´ezier curve. In Section 4, we construct a new kind of generalized toric surface bybivariate generalized toric-Bernstein basis functions. At last, we conclude the whole paperand 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 integerlattice points A and boundary functions of the convex hull(lattice polygon) ∆ A of A . Whenthe lattice polygon ∆ 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 afterthe parameter transformation. So they are the generalizations of the classical Bernstein basisfunctions. In this section, we generalize the toric Bernstein basis functions to finite set ofreal points, and give the definitions of generalized toric-Bernstein basis functions in one andtwo dimensions. Consider A = { a , a , · · · , a n } ⊂ R with a ≤ a ≤ · · · ≤ a n − ≤ a n , and ∆ A =[ a , a n ]. Obviously, the endpoints of ∆ A are points a and a n and we assume a < a n . Set h ( t ) = k ( t − a ) and h ( t ) = k ( a n − t ), where k , k are positive real numbers such that h ( t ) ≥ , h ( t ) ≥ , t ∈ ∆ A . Then basis functions indexed by A can be constructed as follow. Definition 1.
Let A = { a , a , · · · , a n } ⊂ R with a ≤ a ≤ · · · ≤ a n − ≤ a n and a < a n .Then, for any point a i in A , we define the generalized toric-Bernstein (GT-Bernstein) basisfunctions as β a i ( t ) = c a i h ( t ) h ( a i ) h ( t ) h ( a i ) , t ∈ ∆ A , (1) where coefficient c a i > and a i is called knot. The rational form of the GT-Bernstein basis β a i ( t ) is T a i ( t ) = ω a i β a i ( t ) P ni =0 ω a i β a i ( t ) , t ∈ ∆ A , (2)where ω a i > emark 1. The GT-Bernstein basis { β a i ( t ) } defined by equation (1) depends on the selectionof the coefficients k and k . Since any positive real numbers can be selected, if there is nospecial explanation, we set k = k = 1 . We will show that the curve defined by { β a i ( t ) } isindependent of k and k in Section 3. It can be seen that the GT-Bernstein basis can be degenerated into the classical univariateBernstein basis by simple transformation if a i = i, c a i = n n (cid:0) ni (cid:1) ( i = 0 , , · · · , n ). For a i = in ( i = 0 , , · · · , n ), let k = k = n and select coefficients properly, then the GT-Bernsteinbasis degenerates into the univariate Bernstein basis too. For A = { a , a , · · · , a n } ⊂ Z , thebasis functions defined by (1) degenerate into the toric-Bernstein basis functions defined in[13]. Therefore, the GT-Bernstein basis is the generalization of Bernstein basis and toric-Bernstein basis. Example 1.
Let a = 0 , a = √ , a = , a = √ , a = 1 and c a = , c a = 1 , c a = , c a = , c a = . By (1), we have β a ( t ) = 12 (1 − t ) , β a ( t ) = t √ (1 − t ) − √ ,β a ( t ) = 32 t (1 − t ) , β a ( t ) = 710 t √ (1 − t ) − √ , β a ( t ) = 910 t, and the basis functions β a i ( t ) on ∆ A = [0 , are shown in Fig. 2. The changes of basisfunction β a ( t ) while coefficient c a varying as shown in Fig. 3 (the coefficients of curvesfrom bottom to top are . , . , . , . respectively), which shows that the coefficient mainlyaffects the function value of the basis function at each point. However, the changes of thebasis function β a ( t ) when its corresponding knot changes are shown in Fig. 4 (the knotscorresponding to curves from left to right are √ , , √ respectively), which means that theknot mainly affect the positions of the maximum point of the basis function. From the Definition 1 and rational form (2), some properties of the basis functions { T a i ( t ) } can be obtained directly as follows. Theorem 1.
The rational GT-Bernstein basis functions defined in (2) have the followingproperties: (a)
Nonnegativity. T a i ( t ) ≥ , t ∈ ∆ A , i = 0 , , · · · , n . (b) Partition of the unity. P ni =0 T a i ( t ) ≡ . (c) Normalized totally positive (NTP). The rational GT-Bernstein basis { T a i ( t ) } ni =0 is a NTPbasis. This property is proved recently by Yu et al. [20]. igure 2: GT-Bernstein ba-sis Figure 3: Effect of co-efficient changing on GT-Bernstein basis Figure 4: Effect of knot changingon GT-Bernstein basis (d) Endpoints property. At the endpoints of [ a , a n ] , we have T a i ( a ) = (cid:26) , i = 0 , , i = 0 , T a i ( a n ) = (cid:26) , i = n, , i = n, (e) Degeneration property. The GT-Bernstein basis { β a i ( t ) } degenerates to the classicalBernstein basis for A = { , , · · · , n } or A = { , n , · · · , } after proper parametertransformation, and to toric-Bernstein basis for A ⊂ Z . Therefore, the rational GT-Bernstein basis degenerates to rational Bernstein basis for A = { , , · · · , n } or A = { , n , · · · , } after proper parameter transformation. Yu et al. [20] presented the following result for GT-Bernstein basis.
Theorem 2.
Suppose k = k = k and set a ≤ t < t < · · · < t n ≤ a n to be an anyincreasing sequence. Then the collocation matrix of { β a i ( t ) } ni =0 at t < t < · · · < t n M (cid:18) β a , · · · , β a n t , · · · , t n (cid:19) = ( β a j ( t i )) i =0 , , ··· ,nj =0 , , ··· ,n (3) is a strictly totally positive matrix. Since the basis { β a i ( t ) } ni =0 defined by equation (1) may do not hold the property ofpartition of the unity on ∆ A for arbitrary positive coefficients, we present a method tochoose coefficients by Theorem 2, which makes the basis { β a i ( t ) } has partition of the unityon a given increasing sequence a ≤ t < t < · · · < t n ≤ a n .Given an increasing sequence a ≤ t < t < · · · < t n ≤ a n , we have the following systemof equations: n X j =0 β a j ( t i ) = c a h ( t i ) h ( a ) h ( t i ) h ( a ) + · · · + c a n h ( t i ) h ( a n ) h ( t i ) h ( a n ) = 1 , i = 0 , · · · , n. (4)6f we write C = ( c a , · · · , c a n ) T and = (1 , · · · , T , then we obtain M (cid:18) β a , · · · , β a n t , · · · , t n (cid:19) C = . (5)It’s clear that the basis { β a i ( t ) } ni =0 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. Forthe bivariate generalized toric-Bernstein basis in Section 2.2, the method for selection of thecoefficients is similar to the univariate case. Consider a finite set of real points A = { a , a , · · · , a n } ⊂ R , Let ∆ A be the convex hullof A . The lines defined by edges φ i of ∆ A are h i ( u, v ) = ξ i u + η i v + ρ i , where h ξ i , η i i is thenormal vector of φ i towards inside of ∆ A such that h i ( u, v ) ≥ , ( u, v ) ∈ ∆ A , i = 1 , · · · , r .We construct the generalized toric-Bernstein basis functions as follows. Definition 2.
Let A = { a , a , · · · , a n } ⊂ R be a finite collection of real points, and set ∆ A to be the convex hull of A . Then, for any point a i in A , we call β a i ( u, v ) = c a i h ( u, v ) h ( a i ) · · · h r ( u, v ) h r ( a i ) , ( u, v ) ∈ ∆ A , (6) is the bivariate generalized toric-Bernstein (GT-Bernstein) basis function, where c a i > isthe coefficient and a i is called knot. The rational form of the GT-Bernstein basic function β a i ( u, v ) is T a i ( u, v ) = ω a i β a i ( u, v ) P ni =0 ω a i β a i ( u, v ) , ( u, v ) ∈ ∆ A . (7)where ω a i > Remark 2.
In (6), the basis function depends on the choice of coefficients, and the coefficientscan vary from case to case. If there is no special explanation, we set c a i ≡ .For A = { a , a , · · · , a n } ⊂ Z , the basis { β a i ( u, v ) } defined by (6) degenerates to thetoric-Bernstein basis in [13]. In particular, for A = { ( i, j ) ∈ Z | i + j ≤ k, i ≥ , j ≥ } ,if we choose proper coefficients, then the GT-Bernstein basis degenerates to the bivariatetriangular Bernstein basis. Analogously, for A = { ( i, j ) ∈ Z | ≤ i ≤ m, ≤ j ≥ n } , theGT-Bernstein basis degenerates to the bivariate tensor product Bernstein basis if coefficientschosen properly. xample 2. Let e A = { (0 , , (1 , , (0 , ) , ( , ) , (2 , , (0 , , ( , , (2 , } (see Fig. 1(b)).By (6), the GT-Bernstein basis defined by e A are given as below. β (0 , ( u, v )=(3 − u − v )(2 − u ) v , β (1 , ( u, v )=(2 − u ) v u, β (0 , ) ( u, v )=(2 − v ) (3 − u − v ) (2 − u ) v ,β ( , ) ( u, v )=(2 − v ) (3 − u − v ) (2 − u ) v u , β (2 , ( u, v )=(2 − v ) vu , β (0 , ( u, v )=(2 − v ) (3 − u − v ) (2 − u ) ,β ( , ( u, v ) = (2 − v ) (3 − u − v ) (2 − u ) u , β (2 , ( u, v ) = (2 − v ) (3 − u − v ) u . Three of the basis functions are shown in Fig. 5. We further set each weight ω a i = 1 ,then the rational forms of these three basis functions on ∆ A are shown in Fig. 6. (a) β (0 , ( u, v ) (b) β ( , ) ( u, v ) (c) β ( , ( u, v )Figure 5: GT-Bernstein basis functions(a) T (0 , ( u, v ) (b) T ( , ) ( u, v ) (c) T ( , ( u, v )Figure 6: Rational GT-Bernstein basis functions φ i ( i = 1 , · · · , r ) of the convex hull ∆ A are ordered counterclockwise andlet V i be vertex of ∆ A where two edges φ i and φ i +1 meet, ( i = 1 , · · · , r ). The indiceswill be treated in a cyclic fashion: for instance, φ = φ r , φ r +1 = φ and so on. Denoteby ˆ φ i = φ i ∩ A the intersection of A and φ i . Note that { V i } ri =1 and ˆ φ i are subsets of A respectively, i = 1 , · · · , r .From Definition 2 and rational form (7), we can obtain the following properties of thebasis functions { T a i ( u, v ) } directly. Theorem 3.
The rational forms of the GT-Bernstein basis functions defined in (7) have thefollowing properties: (a)
Nonnegativity. T a i ( u, v ) ≥ , ( u, v ) ∈ ∆ A , i = 0 , , · · · , n . (b) Partition of the unity. P ni =0 T a i ( u, v ) ≡ . (c) Boundary property. When ( u, v ) is constrained on the edge φ j of ∆ A , all basis functions β a i ( u, v ) and T a i ( u, v ) with indices a i ∈ A \ ˆ φ j vanish, that is: β a i ( u, v ) = 0 , a i ∈ A \ ˆ φ j , ( u, v ) ∈ φ j .β a i ( u, v ) = 0 , a i ∈ ˆ φ j , T a i ( u, v ) = 0 , a i ∈ A \ ˆ φ j , ( u, v ) ∈ φ j .T a i ( u, v ) = 0 , a i ∈ ˆ φ j , (8) (d) Corner points property. At the vertices of ∆ A , we have (cid:26) T a i ( V i ) = 1 , a i = V i ,T a i ( V i ) = 0 , a i = V i . (9) (e) Degeneration property. For A = { a , a , · · · , a n } ⊂ Z , the basis defined by (6) degen-erates into toric-Bernstein basis defined in [13]. In particular, the GT-Bernstein basisdegenerates to the bivariate triangular Bernstein basis for A = { ( i, j ) ∈ Z | i + j ≤ k, i ≥ , j ≥ } , and to the tensor product Bernstein basis for A = { ( i, j ) ∈ Z | ≤ i ≤ m, ≤ j ≥ n } , if coefficients selected properly. . Generalized Toric-B´ezier Curves For given control points and weights, we can use the Bernstein basis functions to con-struct the classical rational B´ezier curve. The classical rational B´ezier curve has many goodproperties, such as convex hull property, boundary property, and affine invariance. In thesame way, the basis functions defined by (2) can be used to define a new class of rationalcurves.
Definition 3.
Given real points set A = { a , a , · · · , a n } , control points B = { b a i | a i ∈A} ⊂ R , and weights ω = { ω a i > | a i ∈ A} , the rational parametric curve P A ,ω, B ( t ) = n X i =0 b a i T a i ( t ) = n X i =0 b a i ω a i β a i ( t ) P ni =0 ω a i β a i ( t ) , t ∈ ∆ A . (10) is called the generalized toric-B´ezier curve (GT-B´ezier curve for short) of degree n .The n-edgepolyline polygon is obtained by sequentially connecting two adjacent control points of B witha straight line segment, is called control polygon. Remark 3.
Although the GT-Bernstein basis defined by equation (1) depends on the selectionof the coefficients k and k , the GT-B´ezier curve is independent on the choice of thesetwo parameters. It can be known from the results in [18], the GT-B´ezier curve defined bythe equation (10) is obtained by the projection (the projection is related to the weights andthe control points) of the high-dimensional real projective toric variety defined by the A = { a , a , · · · , a n } . Given point set A , for different coefficients k and k , after unitizing thecorresponding toric variety and eliminating the constant in the projective space, the toricvarieties are identical, then the GT-B´ezier curve defined by point set A is also the same. Formore Details refer to [14, 18].The degree of n of curve in Definition 3 is just the number of forms in the curve, one lessthan 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 , a = √ , a = , a = √ , a = 1 } as show in Example 1 ,weights ω a = 1 , ω a = 10 , ω a = 20 , ω a = 6 , ω a = 5 and control points b a = (0 , , b a =(0 . , . , b a = (2 , , b a = (3 . , . , b a = (4 , . Suppose c a i = 1( i = 0 , · · · , , then thequadratic GT-B´ezier curve is P A ,ω, B ( t ) = X i =0 b a i T a i ( t ) , t ∈ [0 , , and the curve is shown in Fig. 7. igure 7: Quadratic GT-B´ezier curve From the properties of the GT-Bernstein basis functions associated with A = { a , a , · · · , a n } ⊂ R , some properties of the GT-B´ezier curve can be obtained as follows: (a) Affine invariance and convex hull property . Since the basis (2) have the propertiesof nonnegativity and partition of the unity, these show that the corresponding GT-B´ezier curve (10) has affine invariance and convex hull property. (b) Endpoints interpolation property . This property follows directly from the cornerpoints property of the basis (2), that is P A ,ω, B ( a ) = b a , P A ,ω, B ( a n ) = b a n . (c) Progressive iteration approximation (PIA) property . The GT-B´ezier curve hasPIA property from the result in [23] because its basis { T a i ( t ) } ni =0 is a NTP basis. (d) Degeneration property . If a i = i (or a i = in ), ( i = 0 , , · · · , n ) and k = k = n ,then the GT-B´ezier curve (10) degenerates into the classical rational B´ezier curve afterreparameterization and coefficients selected properly. For A = { a , a , · · · , a n } ⊂ Z ,the GT-B´ezier curve (10) is the toric B´ezier curve defined in [21], which is exactly theone-dimensional form of the toric surface defined in [13]. (e) Endpoints tangent vectors . For k = 1 a − a , k = 1 a n − a n − , P ′A ,ω, B ( a )= c a k k − k k ( a n − a ) − k k ω a ( b a − b a ) c a ω a , P ′A ,ω, B ( a n )= c a n − k k − k k ( a n − a ) − k k ω a n − ( b a n − b a n − ) c a n ω a n . (11)We can see the tangent vectors at the end points of curve P A ,ω, B ( t ) are parallel to −−−−→ b a b a and −−−−−→ b a n − b a n respectively. And this property can be used to construct G continuouspiecewise GT-B´ezier curve. (f ) Multiple knot property . When a knot in A tends to its adjacent knot, the followingresults describe the limit property of the GT-B´ezier curve, which also show the resultingGT-B´ezier curve defined by A with multiple knots. Theorem 4.
Suppose c a i = 1( i = 0 , · · · , n ) . When knot a k (0 ≤ k < n ) approachesto its adjacent knot a k +1 , the limit of GT-B´ezier curve P A ,ω, B ( t ) of degree n defined in(10) is exactly the GT-B´ezier curve e P e A , e ω, e B ( t ) of degree n − , defined as lim a k → a k +1 P A ,ω, B ( t )= e P e A , e ω, e B ( t )= P k − i =0 ω a i b a i β a i ( t )+ e ω a k +1 e b a k +1 β a k +1 ( t )+ P ni = k +2 ω a i b a i β a i ( t ) P k − i =0 ω a i β a i ( t )+ e ω a k +1 β a k +1 ( t )+ P ni = k +2 ω a i β a i ( t ) , (12) where e ω a k +1 = ω a k + ω a k +1 , e b a k +1 = ω ak ω ak + ω ak +1 b a k + ω ak +1 ω ak + ω ak +1 b a k +1 , e A = { a , · · · , a k − , a k +1 , · · · , a n } , e B = { b a , · · · , b a k − , e b a k +1 , b a k +2 , · · · , b a n } and e ω = { ω a , · · · , ω a k − , e ω a k +1 , ω a k +2 , · · · , ω a n } .Proof. When a k (0 ≤ k < n ) tends to a k +1 , we havelim a k → a k +1 β a k ( t ) = lim a k → a k +1 ( t − a ) a k − a ( a n − t ) a n − a k = ( t − a ) a k +1 − a ( a n − t ) a n − a k +1 = β a k +1 ( t )12hus, lim a k → a k +1 n X i =0 b a i T a i ( t ) = e P e A , e ω, e B ( t )= lim a k → a k +1 n X i =0 b a i ω a i β a i ( t ) P ni =0 ω a i β a i ( t )= P 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 ) P 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 )= P 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 ) P i = k,k +1 ω a i β a i ( t )+( ω a k + ω a k +1 ) β a k +1 ( t ) . Let e ω a k +1 = ω a k + ω a k +1 , e b a k +1 = ω ak ω ak + ω ak +1 b a k + ω ak +1 ω ak + ω ak +1 b a k +1 , we can obtainlim a k → a k +1 P A ,ω, B ( t ) = e P e A , e ω, e B ( t ) = P i = k,k +1 ω a i b a i β a i ( t )+ e ω a k +1 e b a k +1 β a k +1 ( t ) P i = k,k +1 ω a i β a i ( t )+ e ω a k +1 β a k +1 ( t ) , where e A = { a , · · · , a k − , a k +1 , · · · , a n } , e B = { b a , · · · , b a k − , e b a k +1 , b a k +2 , · · · , b a n } and e ω = { ω a , · · · , ω a k − , e ω a k +1 , ω a k +2 , · · · , ω a n } . This leads to prove the result. Example 4.
Consider the curve P A ,ω, B ( t ) defined as in Example 3. Let knots A = { a =0 , a = √ , a = , a = √ , a = 1 } , weights ω a = 1 , ω a = 10 , ω a = 20 , ω a = 6 , ω a = 5 ,control points b a = (0 , , b a = (0 . , . , b a = (2 , , b a = (3 . , . , b a = (4 , and c a i = 1 ( i = 0 , · · · , . If a approaches a , then the changes of the GT-B´ezier curve areshown in Fig. 8. We can see that the limit curve lim a → a P A ,ω, B ( t ) coincides with thetarget curve e P e A , e ω, e B ( t ) , which verifies the Theorem 4. Theorem 4 indicates that the GT-B´ezier curve of degree n degenerates into the GT-B´ezier curve of degree n − e A = { a , · · · , a k − , a k +1 , · · · , a n } , control points e B = { b a , · · · , b a k − , e b a k +1 , b a k +2 , · · · , b a n } and weights e ω = { ω a , · · · , ω a k − , e ω a k +1 , ω a k +2 , · · · , ω a n } when a k = a k +1 . The following corollary generalizes Theorem 4, and gives the limit ofGT-B´ezier curve with multiple knots. The proof of the corollary is similar to Theorem4 and will be omitted here. Corollary 1.
Suppose c a i = 1 ( i = 0 , · · · , n ) . When knots a q , a q +1 , · · · , a q + k − (0 ≤ q < n, < k ≤ n + 1 − q ) approaches to the knot a q + k − , the limit of GT-B´ezier curve a) Initial curve (b) Limit curveFigure 8: Limits of the quadratic GT-B´ezier curve of single knot P A ,ω, B ( t ) of degree n defined in (10) is exactly the GT-B´ezier curve of degree n − k +1 as lim a q , ··· ,a q + k − → a q + k − P A ,ω, B ( t ) = e P e A , e ω, e B ( t )= P q − i =0 ω a i b a i β a i ( t )+ e ω a q + k − e b a q + k − β a q + k − ( t )+ P ni = q + k ω a i b a i β a i ( t ) P q − i =0 ω a i β a i ( t )+ e ω a q + k − β a q + k − ( t )+ P ni = q + k ω a i β a i ( t ) , where e ω a q + k − = ω a q + ω a q +1 + · · · + ω a q + k − , e b a q + k − = ω aq e ω aq + k − b a q + · · · + ω aq + k − e ω aq + k − b a q + k − , e A = { a , · · · , a q − , a q + k − , · · · , a n } , e B = { b a , · · · , b a q − , e b a q + k − , b a q + k , · · · , b a n } and e ω = { ω a , · · · , ω a q − , e ω a q + k − , ω a q + k , · · · , ω a n } . Example 5.
Consider the curve P A ,ω, B ( t ) defined as in Example 3. If a → a and a → a , then the changes of the GT-B´ezier curve are shown in Fig. 9. The limitcurve is constructed by knots e A = { a = 0 , a = , a = 1 } , control points e B = { b a =(0 , , e b a = ( . , ) , b a = (4 , } and weights e ω = { ω a = 1 , e ω a = 36 , ω a = 5 } . Wecan see that the limit curve coincides with the target curve together, which verifies theresult of Corollary 1. (g) Toric degeneration property . For each t ∈ ∆ A , we have the limiting property ofGT-B´ezier curve while a single weight of curve tends to infinity, that islim ω ai → + ∞ P A ,ω, B ( t ) = b a t = a , b a i t ∈ ( a , a n ) , b a n t = a n . a) Initial curve (b) Limit curveFigure 9: Limits of the quadratic GT-B´ezier curve with multiple knots And this property can be derived from weight property of rational B´ezier directly.Fig. 10 shows the limit curve of GT-B´ezier curve defined in Example 3 with ω a → + ∞ . Figure 10: Limit of GT-B´ezier curve with ω a → + ∞ . Next, we consider the property of GT-B´ezier 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 )) ∈ R . Wedenote P λ = conv { ( a i , λ ( a i )) | a i ∈ A} the convex hull of the lifted points. Each edge ofthe convex hull P λ has a normal vector pointing to the outer side. We call it the upperedges of P λ if the last coordinate of the normal vector is positive. If we project theseupper edges back vertically into R , they can cover ∆ A and form a regular subdivisionΓ λ of ∆ A induced by λ [14]. 15e group together the points of A that are in the same subset of the Γ λ and on thesame upper edge of the P λ . Then we get a decomposition of A , which is called regulardecomposition S λ of A induced by λ . For each subset F of S λ , we can use the weights ω | F = { ω a i | a i ∈ F } and the control points B| F = { b a i | a i ∈ F } to define a newGT-B´ezier curve P F ,ω | F , B| F on ∆ F = conv { a i ∈ F | a i ∈ A} by Definition 3. The unionof these curves P A ,ω, B ( S λ ) = [ F∈S λ P F ,ω | F , B| F 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 a i | a i ∈ A} . These weights are used to define the map P A ,ω λ ( x ) , B ( t ) = P ni =0 x λ ( a i ) ω a i b a i β a i ( t ) P ni =0 x λ ( a i ) ω a i β a i ( t ) , t ∈ ∆ A . (13)The image of ∆ A under this map is a GT-B´ezier curve with a parameter x , denoted as P A ,ω λ ( x ) , B . We have the following result. Theorem 5.
The limit of the GT-B´ezier curve P A ,ω λ ( x ) , B as x → ∞ is the regularcontrol curve induced by regular decomposition S λ , 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 [18], the GT-B´eziercurve P A ,ω, B is obtained by the projection of the high-dimensional real projective toricvariety formed by A . Then P A ,ω, B is projection after the composition of a sequence ofmappings A { β ai | a i ∈A} −→ X A ω −→ X A ,ω B −→ P A ,ω, B . For A and weights ω λ ( x ) with parameter x , we can get a family of translated toricvarieties X A ,ω λ ( x ) A { β ai | a i ∈A} −→ X A ω λ ( x ) −→ X A ,ω λ ( x ) . When x → ∞ , X A ,ω λ ( x ) limits to a union of irrational toric varieties in the Hausdorffdistance, which are defined by the all of subset of S λ . That islim x →∞ X A ,ω λ ( x ) = [ F∈S λ X F ,ω | F . 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-B´eziercurve when all the weights tend to infinity. Obviously the control polygon is the regularcontrol curve of GT-B´ezier curve. This property is also called toric degeneration of GT-B´ezier curves. Example 6.
Let A = { , √ , , √ , } , and the lifted values of A by a lifting function λ be (2 , , , − √ , . This induces a regular decomposition of A as ( { , } , { , √ , } ) . The lifted point √ doesn’t lie on any upper edge of the lifting polygon P λ ,then it doesn’tlie on any subset of the decomposition.Fig. 11(a) shows A , the lifted values of A by λ , and the corresponding regular decompo-sition. Fig. 11(b),11(c),11(d) show the toric degeneration of this GT-B´ezier curve for x = 1 . , x = 2 , and x = 3 . The GT-B´ezier curve approaches its regular control curveas the parameter x becomes larger.If λ ′ takes the values of A as { , . , , . , } , then this induces a regular decompositionof A as ( { , √ } , { √ , } , { , √ } , { √ , } ) . The corresponding regular decomposition is shown in Fig. 12(a) and the regular controlcurve is exactly the control polygon of the curve.Moreover λ ′′ takes the values of A as { , , , , } , then the regular decomposition of A is n { , √ } , { √ , } o (see Fig. 12(b)) and the regular control curve is as shown in Fig. 10. (h) Variation diminishing (VD) property . Let d i = a i − a ( i = 1 , · · · , n ) for A = { a , a , · · · , a n } ⊂ R with a ≤ a ≤ · · · ≤ a n − ≤ a n . If d i ( i = 1 , · · · , n ) are rational17 a) A and λ (b) x = 1 . x = 2(d) x = 3 (e) Regular control curveFigure 11: Toric degeneration of GT-B´ezier curve(a) A and λ ′ (b) A and λ ′′ Figure 12: Regular decompositions of A numbers, then d i can be expressed as d i = p i q i ( p i , q i ∈ N ). Let q be the least commonmultiple of q , q , · · · , q n , namely, q = [ q , q , · · · , q n ], then qd i ∈ N \ { } . At this point,we have the following theorem. Theorem 6. If d i = a i − a ∈ Q ( i = 1 , , · · · , n ) , then the planar GT-B´ezier curve P A ,ω, B ( t ) is variation diminishing, which means that the number of intersections of ny straight line with the GT-B´ezier curve P A ,ω, B ( t ) is no more than the number ofintersections of the line with its control polygon.Proof. In order to prove this theorem, we need to use the Cartesian notation rule, whichpresents the upper bound of the number of the positive roots of the polynomial. Forany polynomial f ( t ) = m + m t + · · · + m n t n , if we write Z t> [ f ( t )] to denote thenumber of positive roots of f ( t ) and denote V [ m , m , · · · , m n ] as the number of strictsign changes of polynomial coefficients, then Z t> [ m + m t + · · · + m n t n ] ≤ V [ m , m , · · · , m n ] . Let L denote any straight line, C denote the planar GT-B´ezier curve defined by A ,and write I ( C, L ) to denote the number of times L crosses C . Establish the Cartesiancoordinate system with L as the abscissa axis. Because GT-B´ezier curve is geometricinvariant, we can let ( x a i , y a i )( i = 0 , , · · · , n ) represent the new coordinates of thecontrol points. Let P denote the control polygon and I ( P, L ) denote the number oftimes L crosses P . We only need to prove that I ( C, L ) ≤ I ( P, L ).We set a parameter transformation as u = t − a a n − t , t ∈ ( a , a n ), so that u ∈ (0 , + ∞ ). Thenby the Cartesian notation rule I ( C, L ) = Z a ≤ t ≤ a n " n X i =0 y a i T a i ( t ) = Z a ≤ t ≤ a n " n X i =0 y a i ω a i c a i ( t − a ) t − a i ( a n − t ) a n − a i P ni =0 ω a i c a i ( t − a ) t − a i ( a n − t ) a n − a i = Z
Let A = { (0 , , (1 , , (0 , , (1 , , (2 , , (0 , , (1 , , (2 , } be the integer pointsin the pentagon as shown in Fig. 1(a), and set control points B = { (0 , , , (1 , , , (0 , , , ( , , , (2 , , , (0 , , , ( , , , (2 , , } and weights ω = { , , , , , , , } . Suppose c a i = 1 ( i =0 , · · · , , then we can define a toric surface as shown in Fig. 13(a). This toric surfacedoes not have linear precision, but we can tune it to achieve linear precision. We set e A = { (0 , , (1 , , (0 , ) , ( , ) , (2 , , (0 , , ( , , (2 , } by moving the non-extreme points of A within the pentagon (Fig. 1(b)). The GT-B´ezier surface constructed by e A , ω and B haslinear precision, as shown in Fig. 13(b). The theoretical proof can be found in [22]. (a) Toric surface (b) GT-B´ezier SurfaceFigure 13: Toric Surface and GT-B´ezier Surface From the properties of the GT-Bernstein basis functions, we have the following propertiesof the GT-B´ezier surface. (a) Affine invariance and convex hull property . Since the basis functions (7) possessof nonnegativity and partition of unity, the corresponding GT-B´ezier surface (14) hasaffine invariance and convex hull property.20 b) Degeneration property . When A = { a , a , · · · , a n }⊂ Z , the GT-B´ezier surface associatedof A degenerates to the toric surface defined in [13] by the property of basis(7). Inparticular, the rational B´ezier triangle defined by A = { ( i, j ) ∈ Z | i + j ≤ k, i ≥ , j ≥ } , and the rational tensor product B´ezier surface defined by A = { ( i, j ) ∈ Z | ≤ i ≤ m, ≤ j ≥ n } are special cases of the GT-B´ezier surface. (c) Corner points interpolation property . This property follows directly from the prop-erty at the corner points property of the basis (7), that is P A ,ω, B ( V i ) = b V i , i = 1 , · · · , r ,where V i ∈ A are the vertices of ∆ A . (d) Isoparametric curves property . The isoparametric curves P A ,ω, B ( u ∗ , v ) and P A ,ω, B ( u, v ∗ )of a GT-B´ezier surface are respectively the GT-B´ezier curves. Theorem 7.
Each boundary of the GT-B´ezier surface is a GT-B´ezier curve P ˆ φ i ,ω | ˆ φi , B| ˆ φi ,which defined by control points b a i and weights ω a i by a i ∈ ˆ φ i of corresponding edges φ i ⊂ ∆ A , where i = 1 , · · · , r .Proof. Consider the restriction P ˆ φ,ω | ˆ φ , B| ˆ φ of the GT-B´ezier surface at the fixed edge φ = φ i of ∆ A . Denote V = ( u , v ) = V i − , V = ( u , v ) = V i , and h i ( u, v ) = h ( u, v ) = ξu + ηv + ρ is the equation of φ for simplicity. Let the angle between the edge φ andthe u axis be α . Then tan α = − ξη , andcos α = η p ξ + η . Let σ = | V V | = p ( u − u ) + ( v − v ) .All basis functions β a i ( u, v ) with indices a i ∈ A\ ˆ φ vanishes if ( u, v ) ∈ φ , hence P ˆ φ,ω | ˆ φ , B| ˆ φ depends only on weights and control points indexed by a i ∈ ˆ φ . If a j = ( u j , v j ) ∈ ˆ φ , then h ( a j ) = ξu j + ηv j + ρ = 0, v j = − ρ + ξu j η . Let l j = | a j V | = p ( u j − u ) + ( v j − v ) .By geometric relationship, we have u j = u + l j cos α For the edge equation h k ( u, v ) = 0 for the edge φ k of ∆ A , we evaluate h k ( u, v ) at point a j , h k ( a j ) = ξ k u j + η k v j + ρ k = ηξ k − ξη k η u j + ρ k − η k η ρ = ρ k − η k η ρ + ηξ k − ξη k η u + ( ηξ k − ξη k ) cos αη l j . φ can be expressed as β a i ( u, v ) = c a i r Y k =1 h k ( u, v ) ( ρ k − ηkη ρ + ηξk − ξηkη u ) ( h ( u, v ) ( ηξ − ξη
1) cos αη · · · h r ( u, v ) ( ηξr − ξηr ) cos αη ) l j . Here the first r factors h k ( u, v ) ( ρ k − ηkη ρ + ηξk − ξηkη u ) do not depend on j and can be canceledin the definition of GT-B´ezier surface.When ( u, v ) ∈ φ , then h ( u, v ) = 0, v = − ρ + ξuη . So when ( u, v ) ∈ φ , h k ( u, v ) is univariatefunction of u , written h k ( u ). If we set new variables s = h ( u ) ( ηξ − ξη
1) cos αη · · · h r ( u ) ( ηξr − ξηr ) cos αη , t = σs s , we obtain P ˆ φ,ω | ˆ φ , B| ˆ φ ( u ) = P a j ∈ ˆ φ ω a j b a j c a j s l j P a j ∈ ˆ φ ω a j c a j s l j = P a j ∈ ˆ φ ω a j b a j c a j t l j ( σ − t ) σ − l j P a j ∈ ˆ 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 = u + τ σ cos α (0 < τ < s d τ = dd τ ( h ( u ) ( ηξ − ξη
1) cos αη ) · · · h r ( u ) ( ηξr − ξηr ) cos αη + · · · + h ( u ) ( ηξ − ξη
1) cos αη · · · dd τ ( h r ( u ) ( ηξr − ξηr ) cos αη )= σ r Y k =1 h k ( u ) ( ηξk − ξηk ) cos αη r X j =1 ( ηξ j − ξη j η cos α ) h j ( u ) > , < τ < , and d t d τ = dd τ ( σs s ) = σ (1 + s ) d s d τ > . Hence the reparametrization τ t is monotonic. Also it is easy to check that itpreserves 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 followingtheorem describes the limit property of the GT-B´ezier surface, and demonstrates theconstruction of GT-B´ezier surface by A with multiple knots.22 heorem 8. Suppose c a i = 1( i = 0 , · · · , n ) . When the knot a k (0 ≤ k < n ) approachesto a q (0 ≤ q < n, and q = k ) along line a k a q with the convex hull ∆ A unchanging, thelimit of GT-B´ezier surface P A ,ω, B ( u, v ) defined in (14) is exactly the GT-B´ezier surface e P e A , e ω, e B ( u, v ) , defined as lim a k → a q P A ,ω, B ( u, v )= e P e A , e ω, e B ( u, v )= P i = k,q ω a i b a i β a i ( u, v ) + e ω a q e b a q β a q ( u, v ) P i = k,q ω a i β a i ( u, v ) + e ω a q β a q ( u, v ) , (15) where e ω a q = ω a k + ω a q , e 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 , e A = { a , · · · , a k − , a k +1 , · · · , a n } , e B = { b a , · · · , b a k − , b a k +1 , · · · , e b a q , · · · , b a n } and e ω = { ω a , · · · , ω a k − , ω a k +1 , · · · , e ω a q , · · · , ω a n } .Proof. When a k tends to a q , we havelim a k → a q β a k ( u, v ) = lim a k → a q h ( u, v ) h ( a k ) · · · h r ( u, v ) h r ( a k ) = h ( u, v ) h ( a q ) · · · h r ( u, v ) h r ( a q ) = β a q ( u, v ) . Thus, lim a k → a q n X i =0 b a i T a i ( u, v ) = e P e A , e ω, e B ( u, v )= lim a k → a q n X i =0 b a i ω a i β a i ( u, v ) P ni =0 ω a i β a i ( u, v )= P 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 ) P 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 )= P 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 ) P i = k,q ω a i β a i ( u, v )+( ω a k + ω a q ) β a q ( u, v ) . Let e ω a q = ω a k + ω a q , e 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 obtainlim a k → a q P A ,ω, B ( u, v )= e P e A , e ω, e B ( u, v )= P i = k,q ω a i b a i β a i ( u, v ) + e ω a q e b a q β a q ( u, v ) P i = k,q ω a i β a i ( u, v ) + e ω a q β a q ( u, v ) , where e A = { a , · · · , a k − , a k +1 , · · · , a n } , e B = { b a , · · · , b a k − , b a k +1 , · · · , e b a q , · · · , b a n } and e ω = { ω a , · · · , ω a k − , ω a k +1 , · · · , e ω a q , · · · , ω a n } .23 xample 8. Consider the GT-B´ezier surface defined in Example 7. Let A = { (0 , , (1 , , (0 , ) , ( , ) , (2 , , (0 , , ( , , (2 , } , control points B = { (0 , , , (1 , , , (0 , , , ( , , , (2 , , , (0 , , , ( , , , (2 , , } , weights ω = { , , , , , , , } and c a i = 1 ( i =0 , · · · , . If a = ( , ) approaches a = (1 , , then the changes of the GT-B´eziersurface are shown in Fig. 14.Since the shape of the convex hull ∆ A and control points B are unchanging duringthe process of a tending to a , the original curved surface is stretched like an elas-tic film by the boundary property of the GT-B´ezier surface. Until a = a , the re-sulting surface is defined by e A = { (0 , , (1 , , (0 , ) , (2 , , (0 , , ( , , (2 , } , controlpoints e B = { (0 , , , ( , , ) , (0 , , , (2 , , , (0 , , , ( , , , (2 , , } , weights e ω = { , , , , , , } . (a) a = ( , ) (b) a = ( , ) (c) a = ( , ) (d) Limit surfaceFigure 14: Limit of GT-B´ezier surface with a → a Theorem 8 shows that the limit surface of the GT-B´ezier surface is defined by the e A = { a , · · · , a k − , a k +1 , · · · , a n } , e B = { b a , · · · , b a k − , b a k +1 , · · · , e b a q , · · · , b a n } and e ω = { ω a , · · · , ω a k − , ω a k +1 , · · · , e ω a q , · · · , ω a n } when knot a k (0 ≤ k < n ) approaches to a q (0 ≤ q < n, and q = k ) without the convex hull ∆ A changing. When multiple knotsappear without changing the convex hull ∆ A , we only need to treat it by Theorem 8repeatedly. (f ) Toric degeneration property . Similarly, let λ : A → R be a lifting function to liftthe points a i of A to ( a i , λ ( a i )) ∈ R . We denote P λ = conv { ( a i , λ ( a i )) | a i ∈ A} theconvex hull of the lifted points. Each face of the convex hull P λ has a normal vectorpointing to the outer side. We call it the upper face of P λ if the last coordinate of thenormal vector is positive. If we project these upper faces back vertically into R , theycan cover ∆ A and form a regular subdivision Γ λ of ∆ A induced by λ (see [14]).We group together the points of A that are in the same subset of the Γ λ and on thesame upper face of the P λ . Then we get a decomposition of A , which is called regular24ecomposition S λ of A induced by λ . For each subset F of S λ , we can use the weights ω | F = { ω a i | a i ∈ F } and the control points B| F = { b a i | a i ∈ F } to define a newGT-B´ezier surface P F ,ω | F , B| F on ∆ F = conv { a i ∈ F } by Definition 4. The union ofthese patches P A ,ω, B ( S λ ) = [ F∈S λ P F ,ω | F , B| F is called the regular control surface 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 a i | a i ∈ A} . These weights are used to define the map P A ,ω λ ( x ) , B ( u, v )= P ni =0 x λ ( a i ) ω a i b a i β a i ( u, v ) P ni =0 x λ ( a i ) ω a i β a i ( u, v ) , ( u, v ) ∈ ∆ A . (16)The image of ∆ A under this map is a GT-B´ezier surface with a parameter x , denotedas P A ,ω λ ( x ) , B . We have the following result. Theorem 9.
The limit of the GT-B´ezier surface P A ,ω λ ( x ) , B as x → ∞ is the regularcontrol surface induced by the regular decomposition S λ , 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-B´ezier surface isits regular control surface, and explains the geometric meaning of the limit surface ofthe GT-B´ezier surface when all the weights tend to infinity. And this property is calledtoric degeneration of GT-B´ezier surfaces.
Example 9.
Given point set e A is shown in Fig. 1(b), and the lifted values of e A by λ are shown in Figure Fig. 15(a). The upper hull and the subdivision of ∆ e A by λ areshown in Fig. 15(b), and the regular decomposition S λ is shown in Fig. 15(c).Let control points B = { (0 , , , (1 , , , (0 , , , ( , , , (2 , , , (0 , , , ( , , , (2 , , } and weights ω = { , , , , , , , } corresponding to e A . The toric degeneration processof this GT-B´ezier surface is shown in Fig. 16. This figure also shows the GT-B´eziersurfaces for the parameters x = 5 , x = 100 and x = 600 respectively. As the param-eter x becomes larger, the GT-B´ezier surface approaches its regular control surface inFig. 16(d) (consists of surface patches defined by three triangles and two quadrilaterals). a) λ (b) Upper hull and projec-tion (c) DecompositionFigure 15: Regular decomposition(a) x = 5 (b) x = 100 (c) x = 600 (d) Regular control sur-faceFigure 16: Toric degeneration of GT-B´ezier surface
5. Conclusion and future work
In this paper, we define a new kind of blending functions, called GT-Bernstein BasisFunctions associated with a real number set. And then, we define a new kind of parametriccurve and multisided surface based on the GT-Bernstein basis functions, which are the gen-eralizations of the classical rational B´ezier curves and surfaces, and toric surface patches. Weindicate that the GT-B´ezier curve and surface we presented partially preserve the propertiesof rational B´ezier curves and surfaces. Finally, we also present the limiting properties ofweights and knots.Our further work will be devoted to elevation algorithm and de Casteljau algorithm ofGT-B´ezier curves and surfaces. In addition, the basis defined by the general real numberknots limits the application range of the curve and surface. At present, only the MQ radialbasis functions are investigated deeply in theories and applications, and their degrees are onlyin rational form. In this paper, we present the definition and study the properties of curvesand surfaces theoretically only. How to apply the curves and surfaces for related subjects is26ur work in future too.
Acknowledgements
This work is partly supported by the National Natural Science Foundation of China (Nos.11671068, 11801053).
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