Research Article Approximate Multidegree Reduction of λ-bézier Curves

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1 Matheatical Probles in Engineering Volue 6 Article ID 87 pages Research Article Approxiate Multidegree Reduction of λ-bézier Curves Gang Hu Huanxin Cao and Suxia Zhang Departent of Applied Matheatics Xi an University of Technology Xi an 7 China Correspondence should be addressed to Gang Hu; peng gh@6co Received 8 Deceber ; Accepted April 6 Acadeic Editor: George S Dulikravich Copyright 6 Gang Hu et al This is an open access article distributed under the Creative Coons Attribution License which perits unrestricted use distribution and reproduction in any ediu provided the original work is properly cited Besides inheriting the properties of classical Bézier curves of degreen the corresponding λ-bézier curves have a good perforance in adjusting their shapes by changing shape control paraeter In this paper we derive an approxiation algorith for ultidegree reduction of λ-bézier curves in the L -nor By analysing the properties of λ-bézier curves of degree n a ethod which can deal with approxiating λ-bézier curve of degree n+by λ-bézier curve of degree ( n)is presented Then in unrestricted and C C constraint conditions the new control points of approxiating λ-bézier curve can be obtained by solving linear equations which can iniize the least square error between the approxiating curves and the original ones Finally several nuerical exaples of degree reduction are given and the errors are coputed in three conditions The results indicate that the proposed ethod is effective and easy to ipleent Introduction Bézier curves are one of the ain atheatical odels in CAD/CAM syste Degree reduction of Bézier curve is an iportant technique in geoetric coputation and geoetric approxiation and has great significance for shape design Firstly it is ebodied in data transfer and exchange between CAD systes or in CAD syste because the highest allowable degree of Bernstein polynoial for curve is generally different in various CAD systes or odels Next degree reduction of curve is favorable for data copression With the popularization of digitized and network product design data counication between design systes becoes quite frequent and geoetric data in design syste has coe to ass Therefore the operation of degree reduction attracts a good deal of attention The issue of degree reduction of Béziercurvesisconcerned with the solution of the following proble: for a given Bézier curve R n (t) of degree n with Bézier points {r i } n i= findanapproxiatebézier curve R (t) of lower degree where < nwiththesetofbézier points { r i } i= sothat R n and R satisfy boundary conditions at the end points and the error between R n and R is iniu For degree reduction of Bézier curves any scholars have done a lot of research that can be classified into three categories: geoetry of approxiate control point 8 algebraic eans of basis function transforations 9 and B net and constrained optiization 6 Watkins and Worsey 9 presented an algorith for generating (n )st degree approxiation to nth degree Bézier curve Eck investigated a coplete algorith for perforing the degree reduction within a prescribed error tolerance by help of subdivision Chen and Wang investigated the proble of optial ultidegree reduction of Bézier curves with constraints of endpoints continuity Zheng and Wang proved that the proble of finding a best L -approxiation over the interval for constrained degree reduction is equivalent to that of finding a iniu perturbation vector in a certain weighted Euclidean nor Using the transforation atrices Lu and Wang presented a ethod for the best ultidegree reduction with respect to t t -weighted square nor for the unconstrained case Tan and Fang proposed three ethods for degree reduction of interval generalized Ball curves of Wang-Said type Degree reduction of Bézier curves has been conducted according to different nors ostly L - nors for both unconstrained and constrained conditions In general unconstrained degree reduction gives lower error than the constrained one However Bézier curves are often a part of a piecewise curve so constrained degree reduction is preferred

2 Matheatical Probles in Engineering Although Bézier curves have now becoe a powerful tool for constructing free-for curves in CAD/CAM they have their own disadvantages Specifically the shape of a Bézier curve is well-deterined by its control points after choosing the basis functions 7 In recent years in order to iprove the perforance of Bézier curves any scholars have constructed soe new curves which are siilar to the Bézier ones by introducing paraeters into basis functions; see 8 These new curves share any basic properties with the Bézier ones Furtherore they hold the property of flexible shape adjustability Yan and Liang 8 constructed a new kind of basis function by a recursive approach; thus a kind of paraetric curves with shape paraeter is defined which are called λ-bézier curves These new curves have ost properties of the corresponding classical Bézier curves Moreover the shape paraeter can adjust the shape of the new curves without changing the control points Focusing on degree reduction of λ-bézier curves we study the corresponding proble by the least square ethod and obtain the new control points as well as the shape paraeter of approxiating λ-bézier curves The reainder of the paper is organized as follows The definition and properties of λ-bézier curves are introduced in Section In Section we give the proble description of approxiating degree reduction In Section we present the least square degree reduction of λ-bézier curve Nuerical exaples are given in Section and we present soe applications At last a short conclusion is given in Section 6 The Definition and Properties of λ-bézier Curves Extension of Basis Function The definition of extension Bernstein basis functions is given as follows 8 Definition Let λ ;foranyt thepolynoial functions b (t; λ) =( λt+λt ) ( t) b (t; λ) =t( t) ( + λ λt + λt ) b (t; λ) =( λ+λt )t are called the extension Bernstein basis functions of degree associated with the shape paraeter λ For any integer n (n )thefunctionsb in (t; λ) (i = n)defined recursively by () Theore The extension Bernstein basis functions of degree n can be expressed explicitly as b in (t; λ) =(+ Ci n +Ci n Ci n C i n C i n ti ( t) n i (i=n) where n C i n = n!/i!(n i)! Construction of λ-bézier Curves λ Ci n Cn i λt + λt ) Definition Given control points i (i = n;n ) in R or R then p (t; λ) = n i= () i b in (t; λ) t λ () is called a λ-bézier curve of degree n with shape paraeter λ wherebasisfunctionsb in (t;λ) (i = n;n )are defined by () (see Definition in 8) When the shape paraeter λ is equal to zero λ-bézier curves degenerate to the classical Bézier curves λ-bézier curve inherits ost properties of the classical Bézier curve such as convex hull property geoetric invariance syetry and the following terinal property: p (; λ) = p (; λ) = n dp (t; λ) = (n+λ)( dt P ) t= dp (t; λ) = (n+λ)( n dt P n ) t= Because of introducing paraeter λ λ-bézier curves have ore powerful expressiveness than the classical Bézier curves Figure shows graphs of λ-bézier curves with the sae control polygon but different shape paraeters Figure (a) shows the curves generated by the extension Bernstein basis functions with n = and p (t; ) (solid lines) p (t; ) (dashed lines) and p (t; ) (dot-dashed lines) respectively Figure (b) shows the curves generated by the sae basis functionsasinfigure(a)withn=and p (t; ) (solid lines) p (t; ) (dashed lines) and p (t; ) (dot-dashed lines) respectivelyfrothefigureswecan seethatλ-bézier curves approach the control polygon when the shape paraeter is increasing () b in (t; λ) = ( t) b in (t; λ) +tb i n (t; λ) t () Proble Description Proble Given control points { i }n+ i= Rs (s=) λ- Bézier curve of degree n+is expressed as follows: are called the extension Bernstein basis functions of degree n In the case k= or k>lwesetb kl (t; λ) = p (t; λ) = n+ i= i b in+ (t; λ) t (6)

3 Matheatical Probles in Engineering λ= λ= λ= (a) λ-bézier curve of degree λ= λ= λ= (b) λ-bézier curve of degree Figure : λ-bézier curves with the sae control polygon but different shape paraeters where {b in+ (t; λ)} n+ i= are basis functions of degree n+and λ is global shape paraeter Given control points {P i } i= Rs thecorrespondingλ-bézier curves of degree ( n)are p (t; λ) = i= P i b i (t; λ) (7) such that the distance iniizes between p (t; λ) and p(t; λ) in certain distance function d(p (t; λ) p(t; λ)) Here we are interested in obtaining explicit expression of approxiating λ-bézier curves p(t; λ) sowechoosethe following least square distance function: d (p (t; λ) p (t; λ)) = p (t; λ) p (t; λ) dt (8) Then we can convert Proble into s subprobles and every subproble leads to a iniized coponent function: d (p k (t; λ) p k (t; λ)) = p k (t; λ) p k (t; λ) dt (k=s) Let p (t; λ) = (p (t; λ) p (t;λ)p s (t; λ)) and p(t; λ) = (p (t; λ) p (t;λ)p s (t; λ)); then (8) is deterined by the following forula: d(p (t; λ) p (t; λ)) = s k= d (p k (t; λ) p k (t; λ)) / (9) () For subdistance function d (p k (t; λ) p k(t; λ))itissufficient to iniize d(p (t; λ) p(t; λ)) Therefore we can just study the proble of iniu coponent function in the following Proble Given a series of real nubers { i }n+ i= fro which we will deterine λ-bézier functions of degree n+ f (t; λ) = n+ i= i b in+ (t; λ) () where i denotes a coponent of vector i thenitis necessary to find real nubers {P j } j= with the corresponding λ-bézier functions of degree f (t; λ) = j= P j b j (t; λ) () such that d (f (t; λ) f(t; λ)) f (t; λ) f(t; λ) dt iniizes by least square distance In order to deterine the approxiate function f(t; λ) priarily we ai to obtain the coefficients {P j } j= Least Square Degree Reduction of λ-bézier Curves The Approxiate Degree Reduction of λ-bézier Curves under Unrestricted Condition Theore 6 If coefficients {P j } j= of approxiating functions f(t; λ) are solutions of Proble the vector P = (P

4 Matheatical Probles in Engineering P P ) T satisfies linear systes AP = bwhere A =(a ij ) ++ b =(b b b + ) T a i+j+ b i (t; λ) b j (t; λ) dt b j+ n+ i= Proof By Proble we obtain i b in+ (t; λ)b j (t; λ) dt (ij=) () Let e j+ =(a j+ a j+ a +j+ ) T (j = )and suppose c j+ e j+ =c j= a a a + +c + +c a + a + a ++ a a a + + = (8) S=d (f (t; λ) f(t; λ)) That is f (t; λ) f(t; λ) dt n+ i= i b in+ (t; λ) P j b j (t; λ) dt j= () j= c j+ a i+j+ c j+ b j (t; λ) b i (t; λ) dt j= = (i=) (9) Let S/ P j = (j=); then the above equations canbesiplifiedtothefollowingones: i= P i b i (t; λ) b j (t; λ) dt n+ i= i b in+ (t; λ)b j (t; λ) dt (j=) () Furtherore () can be represented in atrix for by calculation which is described as follows: where A =(a ij ) ++ b =(b b b + ) T a i+j+ b i (t; λ) b j (t; λ) dt b j+ n+ i= AP = b (6) i b in+ (t; λ)b j (t; λ) dt (ij=) (7) We then get the following forula: c j+ b j (t; λ) dt j= = i= c j+ b j (t; λ) j= c i+ i= c i+ b i (t; λ)dt c j+ b j (t; λ) b i (t; λ) dt = ; j= () thus j= c j+b j (t; λ) Since{b (t; λ) b (t;λ) b (t; λ)} are linearly independent in interval t we have c j+ (j = ) which eans vectors {e e e + } are linearly independent and then solutions of linear systes (6) are uniquely deterined The Approxiate Degree Reduction of λ-bézier Curves under C Constraint Condition When approxiating degree reduction we expect to satisfy C continuity that is aintaining interpolation of terinal points so two equations P = and P = n+ are deterined The reaining control points are deterined by the following theore Theore 7 If coefficients {P i } i= of approxiating functions f(t; λ) are solutions of Proble and aintain C continuity the vector P = (P P P ) T satisfies linear systes

5 Matheatical Probles in Engineering AP = b except for two equations P = and P = n+ for terinal points where where A =(a ij ) A =(a ij ) b =(b b b ) T a ij b i (t; λ) b j (t; λ) dt b j n+ i= i b in+ (t; λ)b j (t; λ) dt ( b (t; λ) + n+ b (t; λ))b j (t; λ) dt (ij= ) () b =(b b b ) T a ij b i (t; λ) b j (t; λ) dt b j n+ i= i b in+ (t; λ)b j (t; λ) dt ( b (t; λ) + n+ b (t; λ))b j (t; λ) dt (ij= ) () Let e j = (a j a j a j ) T (j = )and suppose Proof According to the condition of C continuity that is f (; λ) = f(; λ) and f (; λ) = f(; λ)itiseasytoobtain two equations P = and P = n+ Then by applying Proble we get S=d (f f)= f (t; λ) f(t; λ) dt n+ i= i b in+ (t; λ) P j b j (t; λ) dt j= () That is c j e j =c j= a a a +c +c a a a a a a + = (6) Let S/ P j = (j= )Equation()canbe siplified to the following equations: j= c j a ij c j b j (t; λ) b i (t; λ) dt = j= (i= ) (7) i= P i b i (t; λ) b j (t; λ) dt n+ i= i b in+ (t; λ) b (t; λ) () Because {b (t; λ) b (t;λ)b (t; λ)} are linearly independent in interval t thevectors{e e e } as in Theore 6 are linearly independent Thus solutions of linear systes () are uniquely deterined and aintain C continuity n+ b (t; λ)b j (t; λ) dt Furtherore these equations can be represented in atrix for as follows: AP = b () The Approxiate Degree Reduction of λ-bézier Curves under C Constraint Condition When approxiating degree reduction if C continuity is aintained (ie four equations P = P = n+ P = +((n++λ)/(+λ))(p P ) and P = n+ ((n + + λ)/( + λ))(p n+ P n ) are specified) the reaining control points are deterined by the following theore Theore 8 If coefficients {P i } i= of approxiate functions f(t; λ) are solutions of Proble and aintain C continuity

6 6 Matheatical Probles in Engineering 7 P 7 P 6 6 P P P P P P P = P = (a) Under unrestricted condition (b) Under C constraint condition P P P P = P = 8 6 (c) Under C constraint condition 6 Figure : Degree reduction with various constraint conditions (fro degree to degree ) Blue solid: the given curve of degree ; red dot-dashed line: the degree-reduced curve of degree the vector P = (P P P ) T satisfies linear systes AP = b exceptforfourequationsp = P = n+ P = + ((n + + λ)/( + λ))(p P )andp = n+ ((n + + λ)/( + λ))(p n+ P n ) for terinal points where A =(a i j ) (P b (t; λ) +P b (t; λ)) b j (t; λ) dt (ij= ) (8) b =(b b b ) T Proof According to the condition of C continuity we get a i j b i (t; λ) b j (t; λ) dt b j n+ i= i b in+ (t; λ) P b (t; λ) P b (t; λ)b j (t; λ) dt f (; λ) =f(; λ) f (; λ) =f(; λ) df (t; λ) = dt t= df (t; λ) = dt t= df(t; λ) dt t= df(t; λ) dt t= (9)

7 Matheatical Probles in Engineering Shape paraeter (a) Under unrestricted condition Shape paraeter (b) Under C constraint condition Shape paraeter (c) Under C constraint condition Figure : graph of degree reduction of λ-bézier of degree It is easy to obtain the following four equations: P = P = n+ P = + n++λ +λ (P P ) P = n+ n++λ +λ (P n+ P n ) Then by Proble we obtain S=d (f f)= f (t; λ) f(t; λ) dt n+ i= i b in+ (t; λ) P j b j (t; λ) dt j= () () Let S/ P j = (j= )Equation()canbe siplified to the following for: i= P i b i (t; λ) b j (t; λ) dt n+ i= i b in+ (t; λ)b j (t; λ) dt (P b (t; λ) +P b (t; λ) +P b (t; λ) +P b (t; λ))b j (t; λ) dt ()

8 8 Matheatical Probles in Engineering P P P P 6 6 P P P 7 P = P P = 7 P (a) Under unrestricted condition P (b) Under C constraint condition P P 6 P P = P = 7 P (c) Under C constraint condition Figure : Degree reduction with various constraint conditions (fro degree 7 to degree ) Green solid: the given curve of degree 7; red dot-dashed line: the degree-reduced curve of degree Furtherore this equation can be represented in atrix for as follows: where A =(a i j ) AP = b () (P b (t; λ) +P b (t; λ)) b j (t; λ) dt (ij= ) () Let e j = (a j a j a j ) T (j = )and suppose b =(b b b ) T a i j b i (t; λ) b j (t; λ) dt b j n+ i= i b in+ (t; λ) P b (t; λ) P b (t; λ)b j (t; λ) dt c j e j =c j= a a a +c +c a a a a a a + = ()

9 Matheatical Probles in Engineering Shape paraeter Shape paraeter (a) Under unrestricted condition (b) Under C constraint condition Shape paraeter (c) Under C constraint condition Figure : graph of degree reduction of λ-bézier of degree 7 That is j= c j a ij c j b j+ (t; λ) b i+ (t; λ) dt = j= (i= ) (6) Because {b (t;λ)b (t; λ)} are linearly independent in interval t thevectors{e e e }asintheore 6 are linearly independent Therefore solutions of linear systes () exist uniquely and aintain C continuity Nuerical Exaples Exaple Given the shape paraeter λ = and the coordinates of control points { = ( ) = ( 7) =( ) P = ( 6) P = ( ) P = ( )} wecan construct λ-bézier curve of degree Then this curve can be separately reduced to λ-bézier curve of degree respectively under unrestricted and C C constraint condition Control points and errors for approxiating λ-bézier curve of degree toaλ-bézier curve of degree are shown in Table Degree reduction with various constraint conditions fro degree todegreeisshowninfigure We give the approxiation error graphs of degree reduction of degree in three conditions with different shape paraeter as shown in Figure Fro Figure the approxiation error value of degree reduction decreases at first andthenincreaseswhenincreasingtheshapeparaeterthe range of error value is in (a) and those in (b) and (c) are 697 and 89 6 respectively

10 Matheatical Probles in Engineering Table : Control points and approxiation errors with different constraint conditions in Exaple (fro degree to degree ) Constraint condition Control points s Under unrestricted condition Under C constraint condition Under C constraint condition P = ( 868) P = ( 7 66) P = ( ) P = (89 67) P = (86 7) P =( ) P = ( 79 7) P = ( 8 67) P = (7 69) P = ( ) P =( ) P = ( 7 ) P = ( ) P = ( ) P = ( ) d (p (t; ) p (t; )) = 866 d (p (t; ) p (t; )) = d (p (t; ) p (t; )) = 8 Table : Control points and approxiation errors with different constraint conditions in Exaple (fro degree 7 to degree ) Constraint condition Control points Under unrestricted condition Under C constraint condition Under C constraint condition P = ( 77 88) P = (6 89) P = (77 9) P = (86 8) P = (769 6) P = (99 99) P = ( ) P = (68 ) P = (77 8) P = (79 99) P = (7 ) P = (9 ) P = ( ) P = (6667 ) P = (76 7) P = (6 66) P = (7 ) P = (9 ) d (p 7 (t; ) p (t; )) = 67 d (p 7 (t; ) p (t; )) = 96 d (p 7 (t; ) p (t; )) = Exaple Given the shape paraeter λ = and the coordinates of control points { = () = ( ) = ( ) = ( ) = ( 8) = (6 ) 6 = (8 7) 7 = (9 )} wecanconstructaλ-bézier curveofdegree7thenthiscurvewillbeseparatelyreducedto λ-bézier curves of degree under three conditions Control points and errors for approxiating λ-bézier curve of degree toaλ-bézier curve of degree 7 are shown in Table Degree reductions with various constraint conditions fro degree 7 todegreeareshowninfigure We give the approxiation error graphs of degree reduction of λ-bézier of degree 7 in three conditions with different shape paraeter as shown in Figure Fro Figure the approxiation error value of degree reduction increases and slope decreases by increasing the shape paraeter The range of error value is 67 8 in (a) and those in (b) and (c) are 96 8 and 99 respectively Exaple Given shape paraeter λ= and two segents of λ-bézier curves of degree 7 expressing patterned vase then they will be separately reduced to two segents of λ-bézier curve of degree under unrestricted condition Control points and error for approxiating λ-bézier curve of degree to a λ-bézier curve of degree 7 are shown in TableDegreereductionsofthesetwosegentsareshown in Figure 6 In addition approxiation errors with C and C constraint conditions in Exaple are 89 and 7 respectively With the change of shape paraeter we present error graph of degree reduction of λ-bézier of degree 7 which expresses patterned vase in unrestricted condition as is showninfigure7frofigure7theerrorvalueincreases P P P P P P P P P P 6 6 P P Figure 6: Degree reduction of λ-bézier curve of degree 7 which expresses patterned vase in unrestricted condition Green solid and blue solid: the given curve of degree 7; red dot-dashed line: the degree-reduced curve of degree with that of shape paraeter The range of error value is Concluding Rearks λ-bézier curves of degree n have the sae properties as Bézier curves In addition they have better perforance when adjusting their shapes by changing the shape paraeter which includes shape adjustability and better approxiation to control polygon as shown in Figure Furtherore the proble of degree reduction for λ- Bézier curves is studied by least squared approxiation An

11 Matheatical Probles in Engineering Before degree reduction After degree reduction Table : Control points and approxiation error under unrestricted condition in Exaple (fro degree 7 to degree ) First segent Second segent First segent Second segent Control points = ( 8) P = (8 7) P = (7 ) P = (6 ) P = ( ) P = ( ) P 6 = ( ) = ( 7) = ( 8) P = ( 7) P = ( ) P = ( ) P = (7 ) P = (7 ) P 6 = (6 ) = (6 7) d(p 7 (t; ) p (t; )) = 8 P = (6 867) P = ( ) P = (68 6) P = (689 66) P = (7 ) P = ( 778) P = (8 867) P = ( 678) P = (7 6) P = (88 66) P = (6696 ) P = ( ) 7 7

12 Matheatical Probles in Engineering Shape paraeter Figure 7: graph of degree reduction of λ-bézier of degree 7 which expresses patterned vase in unrestricted condition algorith for approxiating degree reduction of λ-bézier curves of degree n is provided by adjusting control points under three conditions which can iniize the least square error between the approxiating curves and the original ones Three practical exaples show that the ethod is applicable for CAD/CAM odeling systes We will focus on studying the degree reduction for λ-bézier surfaces in future work Copeting Interests The authors declare that there is no conflict of interests regarding the publication of this paper Acknowledgents This work is supported by the National Natural Science Foundation of China (no and no 67) This work is also supported by the Research Fund of Departent of Science and Departent of Education of Shaanxi China (no JK9) and Research Fund of Shaanxi China (K-) References W Böh G Farin and J Kahann A survey of curve and surface ethods in CAGD Coputer Aided Geoetric Design volnopp 698 G Farin CurvesandSurfacesforCAGD:APracticalGuide Acadeic Press San Diego Calif USA th edition L Dannenberg and H Nowacki Approxiate conversion of surface representations with polynoial bases Coputer Aided Geoetric Designvolno pp 98 Y Ohtake A Belyaev M Alexa G Turk and H P Seidel Multi-level partition of unity iplicits ACMTransactionson Graphicsvolnopp6 7 A R Forrest Interactive interpolation and approxiation by Bézier polynoials Coputer-Aided Designvolno9pp G Farin Algoriths for rational Bézier curves Coputer- Aided Designvolnopp G Chen and G Wang Degree reduction approxiation of Bézier curves by generalized inverse atrices Software (China)volnopp 9 8 L Lu Q Hu and G Wang An iterative algorith for degree reduction of Bézier curves JournalofCoputer-AidedDesign and Coputer Graphicsvolnopp MAWatkinsandAJWorsey DegreereductionofBézier curves Coputer-Aided Design volno7pp M Eck Least squares degree reduction of Bézier curves Coputer-Aided Designvol7nopp8 899 G-D Chen and G-J Wang Optial ulti-degree reduction of Bézier curves with constraints of endpoints continuity Coputer Aided Geoetric Design vol9no6pp6 77 J Zheng and G Wang Perturbing Bézier coefficients for best constrained degree reduction in the L-nor Graphical Modelsvol6no6pp 68 L Lu and G Wang Application of Chebyshev II Bernstein basis transforations to degree reduction of Bézier curves Coputational and Applied Matheaticsvolno pp 68 QTanandZFang Degreereductionofintervalgeneralized ball curves of Wang-Said type Coputer-Aided Design & Coputer Graphicsvolnopp8 98 S Hu J Sun T Jin and G Wang Approxiate degree reduction of Bézier curves Tsinghua Science & Technology vol nopp S Hu Soe geoetric probles of data counications in CAD syste PhD thesis Zhejiang University Hangzhou China X-AHanYCMaandXLHuang Anovelgeneralizationof Bézier curve and surface Coputational and Applied Matheaticsvol7nopp L L Yan and J F Liang An extension of the Bézier odel Applied Matheatics and Coputationvol8no6pp XQQinGHuNJZhangXLShenandYYang A novel extension to the polynoial basis functions describing Bézier curves and surfaces of degree n with ultiple shape paraeters Applied Matheatics and Coputation vol nopp 6 XQinGHuYYangandGWei ConstructionofPH splines based on H-Bézier curves Applied Matheatics and Coputationvol8pp6 67 GHuXMJiandLGuo Quarticgeneralized Bézier surfaces with ultiple shape paraeters and its continuity conditions Transactions of the Chinese Society for Agricultural Machinery vol no 6 pp GHuXQQinXMJiGWeiandSXZhang Theconstruction of λμ-b-spline curves and its application to rotational surfaces Applied Matheatics and Coputation vol 66 no pp 9

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