Hndaw Publshng Corporaton Appled Mathematcs Volume 3, Artcle ID 739, 9 pages http://dxdoorg/55/3/739 Research Artcle Quas-Bézer Curves wth Shape Parameters Jun Chen Faculty of Scence, Nngbo Unversty of Technology, Nngbo 35, Chna Correspondence should be addressed to Jun Chen; chenun88455579@63com Receved October ; Revsed 7 February 3; Accepted 4 February 3 Academc Edtor: Juan Manuel Peña Copyrght 3 Jun Chen Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted The unversal form of unvarate Quas-Bézer bass functons wth multple shape parameters and a seres of correspondng Quas- Bézer curves were constructed step-by-step n ths paper, usng the method of undetermned coeffcents The seres of Quas-Bézer curves had geometrc and affne nvarablty, convex hull property, symmetry, nterpolaton at the endponts and tangent edges at the endponts, and shape adustablty whle mantanng the control ponts Varous exstng Quas-Bézer curves became specal cases n the seres The obvous geometrc sgnfcance of shape parameters made the adustment of the geometrcal shape easer for the desgner The numercal examples ndcated that the algorthm was vald and can easly be appled Introducton The Bézer curve γ t) lsted as follows has a drect-vewng structureandcanbecomputedusngasmpleprocess;t s also one of the most mportant tools n computer-aded geometrc desgn CAGD) Consder γ t) n P B n t), t [, ] ) Here, Bernsten bass functons {B n t)}n are defned as: B n t) n ) t)n t,,,,n ) Gventhattheshapeofthecurvescharacterzedby the control polygon, the desgner always adusts the control pont {P } n when necessary However, n the actual process, desgnng the geometrcal shape s usually not completed at one tme The desgner prefers to have more satsfactory geometrcal shapes by mantanng control polygon, whch allows hm or her to make mnute adustments on the shape of the curve wth fxed control ponts The ratonal Bézer curve γ t) lstedasfollowssanatural choce to meet ths requrement [] γ t) n P B n t) n P ω B n t [, ] 3) t), By assgnng a weght ω for each control pont P,the desgner can adust the shape of the curve by changng the value of the weghts {ω } n [, 3] Although the ratonal Bézer curve has good propertes and can express the conc secton, t alsohasdsadvantages,suchasdffcultynchoosngthevalue of the weght, the ncreased order of ratonal fracton caused by the dervaton, and the need for a numercal method of ntegraton In addton, the algebrac trgonometrc/hyperbolc curve γ 3 t) wth the defnton doman α as the shape parameter s afeasblemethod[4 6] Consder γ 3 t) n P u n t), t [, α] 4) The smple form of the algebrac trgonometrc/hyperbolc curve γ 3 t) can express transcendental curves eg, spral and cyclod) that cannot be expressed by the Bézer curve Nevertheless, the bass functons {u n t)}n nclude trgonometrc/hyperbolc functons, such as sn t, cost, snht, and cosh t So, the algebrac trgonometrc/hyperbolc curve s ncompatble wth the exstng NURBS system, thereby restrctng ts applcaton n the actual proect In vew of the fact that the expresson of the parametrc curvesdetermnedbythecontrolpontsandthebassfunctons, the propertes of such functons dentfy the propertes of the curve wth ts fxed control ponts Therefore, several
Appled Mathematcs Table : Propertes of the bass functons and the curves wth shape parameters Property [7] [8] [9] [] [] [] Thspaper Bass functons wth multple shape parameters Nonnegatvty Partton of unty Symmetry Multple shape parameters Lnear ndependence Degeneracy Curve wth multple shape parameters Geometrc and affne nvarablty Convex hull property Symmetry Interpolaton at the endponts Tangent at the end edge The property of symmetry n [8 ]sbasedontheshapeparameters knds of polynomal bass functons wth shape parameters [7 ] andthecorrespondngcurvehavebeenconstructed as follows By lettng {b, t; λ,λ,,λ m )} be + polynomal functons of degree called order and degree )and {P } be +ponts n spaces, the parametrc curve wth multple shape parameters {λ } m s constructed as follows: P t; λ,λ,,λ m ) P b, t; λ,λ,,λ m ) 5) For the sake of concson, the notatons {b, t; λ,λ,,λ m )} and Pt; λ,λ,,λ m ) wll be replaced by {b, t)} and Pt) And{b, t)} and Pt) wll be called Quas-Bernsten bass and Quas-Bézer curve, respectvely Wth the extra degree of freedom provded by the shape parameters {λ } m n {b, t)},thecurvept) can be freely adusted and controlled by changng the value of {λ } m nstead of changng the control ponts {P } The exstng works are compared n detal n Table The constructon of the bass functons wth shape parameters s the key step n [7 ] Although many knds of bass functons wth shape parameters have been obtaned n the exstng research, two problems need to be solved ) In all exstng research, the bass functons wth shape parameters are ntally gven, and whether or not these functons and the correspondng curves have nherted the characterstcs of the Bernsten bass functons and the Bézer curve, respectvely, s examned However, the method of obtanng the complex expressons of the bass functons remans unclear Are these bass functons obtaned through ntuton or through an amless attempt? ) There are numerous known bass functons wth shape parameters n varyng forms Is there a type of Quas-Bernsten bass functon, whch makes exstng bass functons wth shape parameters be ts specal case? To answer the prevous two questons, ths paper uses the method of undetermned coeffcents, whch clarfes the constructon process of the Quas-Bernsten bass functons A seres of Quas-Bernsten bass functons are fnally obtaned, renderng the exstng bass functon wth shape parameters as ther specal case Quas-Bézer Curve Notaton Frst, the followng vectors are ntroduced: b, b, t),b, t),,b, t)), P P, P,,P n ) T Equaton 5)canberewrttenas 6) P t) b, P 7)
Appled Mathematcs 3 Gven that {b, t)} are polynomals wth degree, theycanbeseenasthelnearcombnatonofthebernsten bass functons {B t)} wth degree gven by b, B M,, B B t),b t),,b t)), M, m ),, Thus, as long as the elements n the matrx M, are determned, the Quas-Bernsten bass functons {b, t)} wth order and degree are completely constructed Except for several elements that can be determned n M,, the rest are shape parameters of the Quas-Bernsten bass functons and the Quas-Bézer curve Here, the matrx M, s called the shape parameter matrx Constructon of the Shape Parameter Matrx M, The +) +) elements of m n M, must be determned so that {b, t)} and Pt) become the Quas-Bernsten bass functons and the Quas-Bézer curve, respectvely Determnaton of m accordng to the Characterstcs of the Quas-Bernsten Bass Functons The Quas-Bernsten bass functons {b, t)} wth order and degree must satsfy the characterstcs of nonnegatvty, normalzaton, symmetry, lnear ndependence, and degeneracy Proposto nonnegatvty) A suffcent condton for t),,, ) s b, 8) m,,,,,,, ) 9) Proof It s known that b, t) m B t)) m )B t) m )B t) m )B t) B t) ) Accordng to the lnear ndependence of the Bernsten bass functons {B t)}, the necessary and suffcent condton for b, t) s m,,, ) Note By combnng 9) and), m satsfes m,,,,,,, ) Proposton 3 symmetry) The necessary and suffcent condton for b, t) b, t),,,) s gven by m m n,,,,,,,, ) 3) Proof Accordng to the symmetry of the Bernsten bass functons {B t)} of B t) B t), thefollowng can be derved: Proof Here, b, t) m B t) s known to have been extracted from 8) Based on the non-negatvty of the Bernsten bass functons {B t)}, a suffcent condton for the non-negatvty of the Quas-Bernsten bass functons {b, t)} s the non-negatvty of the elements m n M, Hence,m must satsfy 9) Note Clearly, there s no row wth all elements beng n M, Inotherwords, m,, ) ) Proposto normalzaton) The necessary and suffcent condton for t) s gven by b, b, t) b, t) B B B B B t) m,n B t) m, B t) m n, t) m n, m ),,,, t) m t) m t) m 4) m,,, ) ) Accordng to the lnear ndependence of the Bernsten bass functons {B t)}, the necessary and suffcent condton for b, t) b, t) s 3)
4 Appled Mathematcs Proposton 4 lnear ndependence) The necessary and suffcent condton for the lnear ndependence of {b, t)} s gven by Proof It s known that k b, t) r M, ) + 5) k m B t)) k m )B t) 6) Determnaton of m accordng to the Characterstcs of the Quas-Bézer Curve Proposton 6 nterpolaton at the endponts) The necessary and suffcent condton for P) P and P) P n s gven by m, m,,, ), ) m n,, m n,,,, ) ) Proof Clearly, the necessary and suffcent condton for P) P s the Quas-Bernsten prmary functons {b, t)} satsfyng the followng: Accordng to the lnear ndependence of the Bernsten bass functons {B t)}, the necessary and suffcent condton for k b, t) s gven by k m,,, ) 7) It s known that,, ) {, b, b, ),b, ),,b, )) ) M m,,m,,,m n,) T s defned as the th column vector of M, Equaton7)sequvalentto k M Thus, the necessary and suffcent condton for the lnear ndependence of {b, t)} s also the lnear ndependence of the column vectors {M } of the matrx M, Consequently, the necessary and suffcent condton for the lnear ndependence of {b, t)} s that the rank of the shape parameter matrx M, satsfes rm, ) + Note 3 When 5)strue, Proposton 5 degeneracy) If the elements {m }, n, the matrx M, are represented by 8),theQuas-Bernsten bass functons {b, t)} wth order and degree are degenerated nto the Bernsten bass functons {B t)} wth order m ) { n, max, n ) )) mn,), {, ; {, else 8) ) Proof When the elements {m }, n the matrx M,, are represented by 8), the followngs obtaned: B B M, 9) Comparng 9) wth8), Proposton 5 s proven Note 4 If, M, s an dentty matrx here B ),B ),,B )) m ),,,,,) m ),, m,m,,m,n,m n ) 3) Thus, the necessary and suffcent condton for b, ), b, ),,b, )),,,)s ) Smlarly, the necessary and suffcent condton for P) P n s the Quas-Bernsten prmary functons {b, t)} satsfyng the followng: It s known that b, ) {,,, n b, ),b, ),,b, )) B ),B ),,B )) m ),,,,,) m ),, m n,,m n,,,m n,,m n, ) 4) 5) Thus, the necessary and suffcent condton for b, ), b, ),,b, )),,,)s ) Hence, the necessary and suffcent condton for P) P n s ) Note 5 When {b, t)} have the property of symmetry, )sequvalentto) accordng to 3)
Appled Mathematcs 5 Proposton 7 tangent edges at the endponts) The necessary and suffcent condton for P ) P P, P ) P n P n s gven by m +m, m, m,3,, ), m n, +m n,, m n,, m n, Proof It s known that P ) P t) t,,, ) b, t),b, t),,b, t)) t P, P,,P n ) T B t),b t),,b t)) t m ),, P, P,,P n ) T,,,,) m ),, 6) 7) P, P,,P n ) T m m,m m,,m n m n ) P, P,,P n ) T 8) Clearly, the necessary and suffcent condton for P ) P P s m m )/m m ),m m,3,, ) whch verfes 6) Smlarly, the necessary and suffcent condton for P ) P n P n s 7) Note 6 When {b, t)} have the property of symmetry, 7)sequvalentto6) accordng to 3) 3 Form of Shape Parameter Matrx M, All shape parameter matrxes that satsfy 9), ), 3), 5), ), ), 6), and 7)havethefollowngform: m m m m m,n m n M, ) m n m,n m m m m ) n +) +) 9) Here, m are varable shape parameters that satsfy m,3,,[ +) ]), m <, m,3,,[ +) ],,,,n ) 3) 3 The Characterstcs of the Quas-Bézer Curve In summary, the Quas-Bézer curve Pt) based on the Quas- Bernsten bass functons {b, t)} has the characterstcs lsted as follows: a) shape adustablty: the shape of the Quas-Bézer curve can stll be adusted by mantanng the control ponts b) geometrc nvarablty: the Quas-Bézer curve only reles on the control ponts, whereas t has nothng to do wth the poston and drecton of the coordnate system; n other words, the curve shape remans nvarable after translaton and revolvng n the coordnate system; c) affne nvarablty: barycentrc combnatons are nvarant under affne maps; therefore, 9) and) gve the algebrac verfcaton of ths property; d) symmetry: whether the control ponts are labeled P P P n or P n P n P, the curves that correspond to the two dfferent orderngs look the same; they dffer only n the drecton n whch they are traversed, and ths s wrtten as P b, t) P n b, t), 3) whch follows the nspecton of 3); e) convex hull property: ths property exsts snce the Quas-Bernsten bass functons {b, t)} have the propertes of non-negatvty and normalzaton; the Quas-Bézer curve s the convex lnear combnaton of control ponts, and as such, t s located n the convex hull of the control ponts; f) nterpolaton at the endponts and tangent edges at the endpont: the Quas-Bézer curve Pt) nterpolates the frst and the last control ponts P) P and P) P n ; the frst and last edges of the control polygon are the tangent lnes at the endponts, where P ) P P and P ) P n P n 4 GeometrcSgnfcanceoftheShapeParameters Accordng to 9), when m,,,,,,,, ) ncreases, b, t) and b, t) ncrease as well; specfcally, Pt) comes close to the control ponts P and P n The geometrc sgnfcance of the shape parameters s shown n Secton 3
6 Appled Mathematcs P 8 8 6 6 4 4 4 6 8 P P 3 4 5 m m 4 m 8 Bézer curve m m 4 m 8 a) b) Fgure : Quas-Bernsten bass functons and Quas-Bézer curves when and 3 3 Numercal Examples Example The shape parameter matrx M 3, s constructed from 9) The correspondng Quas-Bernsten bass functons and the Quas-Bézer curves wth dfferent shape parameter m are gven as follows: M 3, m m ), m m m < 3) The geometrc sgnfcance of the shape parameters m s shown n Fgure As the value of the shape parameter m ncreases, the elements n the second column of M 3, decrease Accordng to 8), the second Quas-Bernsten bass functon b,3 t) decreases So, the correspondng Quas- Bézer curve moves away from the control pont P see Fgure b)) Example The shape parameter matrx M 4, s constructed from 9) The correspondng Quas-Bernsten bass functons and the Quas-Bézer curves wth dfferent shape parameters m and m are gven as follows: m m M 4, m m m ), m m m <, m 33) The geometrc sgnfcance of the shape parameters m and m s shown n Fgure When we ncrease the value of m and keep m unchanged, the elements n the second column of M 4, decrease Accordng to 8),thesecondQuas- Bernsten bass functon b,4 t) decreases Compare the blue curvewththeredone,andwewllfndthatthequas-bézer curve moves away from the control pont P see Fgure b)) If we ncrease the value of m and keep m unchanged, smlarresultsalsoobtanedcomparetheredcurvewth the green one, and we wll fnd that the Quas-Bézer curve moves away from the control pont P see Fgure b)) Example 3 The shape parameter matrx M 3,3 s constructed from 9) The correspondng Quas-Bernsten bass functons and the Quas-Bézer curves wth dfferent shape parameter m are gven as follows: M 3,3 m m ), m m m < 34) The geometrc sgnfcance of the shape parameters m s shown n Fgure 3 As the value of the shape parameter m ncreases,theelementsnthesecondandthethrdcolumnof M 3,3 decrease Accordng to 8),thesecondQuas-Bernsten bass functon b 3,3 t) and the thrd Quas-Bernsten bass functon b 3,3 t) decrease So, the correspondng Quas-Bézer curve moves away from the control ponts P and P see Fgure 3b)) Example 4 The shape parameter matrx M 4,3 s constructed from 9) The correspondng Quas-Bernsten bass
Appled Mathematcs 7 P 8 8 6 6 4 4 4 6 8 P P 3 4 5 m, m m 6, m m 6, m 3 Bézer curve m, m m 6, m m 6, m 3 a) b) Fgure : Quas-Bernsten bass functons and Quas-Bézer curves when and 4 8 3 5 P 6 4 5 P 4 6 8 5 P 3 P 3 4 5 m m 5 m 8 Bézer curve m m 5 m 8 a) b) Fgure 3: Quas-Bernsten bass functons and Quas-Bézer curves when 3and 3 functons and the Quas-Bézer curves wth dfferent shape parameter m and m are gven as follows: M 4,3 m m m m m m ), m m ) m <, m 35) The geometrc sgnfcance of the shape parameters m and m s shown n Fgure 4 When we ncrease the value of m and keep m unchanged, the elements n the second and the thrd column of M 4,3 decrease Accordng to 8), the second Quas-Bernsten bass functon b 3,4 t) and the thrd Quas-Bernsten bass functon b 3,4 t) decrease Compare the bluecurvewththeredone,andwewllfndthatthequas- Bézer curve moves away from the control ponts P and P see Fgure 4b)) If we ncrease the value of m and keep m unchanged, smlarresultsalsoobtanedcomparetheredcurvewththe green one, and we wll fnd that the Quas-Bézer curve moves away from the control ponts P and P see Fgure 4b))
8 Appled Mathematcs 8 3 5 P 6 4 5 5 P 4 6 8 P P 3 3 4 5 m, m m 6, m m 6, m 3 Bézer curve m, m m 6, m m 6, m 3 a) b) Fgure 4: Quas-Bernsten bass functons and Quas-Bézer curves when 3and 4 Fgure 5: Three knds of flowers wth sx petals Note 7 Several Quas-Bernsten bass functons for low degree and low order are presented aforementoned The correspondng bass functons for hgher degree and hgher order are defned recursvely as follows [7, ]: b +, + t) t) b, t) +tb, t), 36),,, Here, we set b, t) b, + t) Example 5 Fgure 5 presents three knds of flowers wth sx petals, defned by sx symmetrc control polygons Smlar flowers are obtaned from the same control polygons wth dfferent shape parameters Example 6 Fgure 6 presents three knds of outlnes of the vase, all of whch are smlar to the control polygons So, the Fgure 6: Three knds of outlnes of the vase desgner can make mnute adustments wth the same control polygons by changng the value of the shape parameters 4 Dscusson 4 Specal Cases Several exstng bass functons contanng ust one shape parameter n [7, ] are consdered as the specal cases n ths paper Meanwhle, for the polynomal bass functons wth multple shape parameters n [8 ], the symmetry was not dscussed by authors In fact, when these shape parameters satsfy certan relatons, the correspondng bassfunctonsandcurvesbecomesymmetrcalthen,the curves have geometrc and affne nvarablty, convex hull property, symmetry, nterpolaton at the endponts, and
Appled Mathematcs 9 tangent edges at the endponts, and the correspondng shape parameter matrces are the specal cases of 9) We take [9] as example When the shape parameters satsfy certan relatonsn [9], the shape parameter matrx s M n+,n λ n+ λ n+ λ n+ λ n+ λ n+ λ n+ It s the specal case of 9) ) ) n+) n+) 37) 4 Degree and Order of the Curve In the prevous work, the dfference between the degree and order of the curve s fxed e, n [, ], n [7 9], and n []), and the scope of the curve s also fxed wth the same control ponts However, comparng Fgures a) and b),t s found that the greater the dfference between degree and order, the larger the scope of the Quas-Bézer curve acqured In order to obtan the Quas-Bézer curves wth broader scope wth the same control ponts, the desgner can ncrease the dfference between the degree and the order 5 Concluson and Further Work In ths paper, a seres of unvarate Quas-Bernsten bass functons are constructed, thereby creatng a seres of Quas- Bézer curves The shape of the seres of curves can be adusted even wth the control ponts fxed The Quas-Bézer curves also possess geometrc and affne nvarablty, convex hull property, symmetry, nterpolaton at the endponts, and tangent edges at the endponts Quas-Bernsten bass functons wth shape parameters have been drectly studed n the prevous research However, n ths paper, each functon has been gradually nferred and constructed usng a clear method of undetermned coeffcents, where each shape parameter s proposed accordng to the propertes of the Quas-Bernsten bass functons and the Quas-Bézer curve Under the premse of satsfyng symmetry, the former bass functons are all consdered as the specal cases n ths paper In the exstng CAD/CAM systems, the trangular Bézer surfaceandthesplnecurvearewdelyusedtheshape parameters also have been brought nto the trangular surface n [ 4] and the splne curve[5 7] The method n ths paper also can be extended to construct the bass functons of the trangular surface and the splne curve wth shape parameters drectly, and more detals can be seen n our other papers submtted Acknowledgments The author s very grateful to the anonymous referees for the nsprng comments and the valuable suggestons whch mproved the paper consderably Ths work has been supported by the Natonal Natural Scence Foundaton of Chna no Y59377) and the Natural Scence Foundaton of Nngbo nos A674 and A69) References [] G Farn, Algorthms for ratonal Bézer curves, Computer- Aded Desgn,vol5,no,pp73 77,983 [] I Juhász, Weght-based shape modfcaton of NURBS curves, Computer Aded Geometrc Desgn, vol6,no5,pp377 383, 999 [3] L Pegl, Modfyng the shape of ratonal B-splnes Part : curves, Computer-Aded Desgn,vol,no8,pp59 58,989 [4] Q Chen and G Wang, A class of Bézer-lke curves, Computer Aded Geometrc Desgn,vol,no,pp9 39,3 [5]JZhang, C-curves:anextensonofcubccurves, Computer Aded Geometrc Desgn,vol3,no3,pp99 7,996 [6] JZhang,FLKrause,andHZhang, UnfyngC-curvesand H-curves by extendng the calculaton to complex numbers, Computer Aded Geometrc Desgn, vol, no 9, pp 865 883, 5 [7] X Wu, Bézer curve wth shape parameter, Image and Graphcs,vol,pp369 374,6 [8]XAHan,YMa,andXHuang, Anovelgeneralzatonof Bézer curve and surface, Computatonal and Appled Mathematcs,vol7,no,pp8 93,8 [9] L Yang and X M Zeng, Bézer curves and surfaces wth shape parameters, Internatonal Computer Mathematcs, vol86,no7,pp53 63,9 [] T Xang, Z Lu, W Wang, and P Jang, A novel extenson of Bézer curves and surfaces of the same degree, Informaton & Computatonal Scence, vol7,pp8 89, [] J Chen and G- Wang, A new type of the generalzed Bézer curves, Appled Mathematcs,vol6,no,pp47 56, [] LYanandJLang, AnextensonoftheBézer model, Appled Mathematcs and Computaton, vol8,no6,pp863 879, [3] J Cao and G Wang, An extenson of Bernsten-Bézer surface over the trangular doman, Progress n Natural Scence,vol7, no3,pp35 357,7 [4] Z Lu, J Tan, and X Chen, Cubc Bézer trangular patch wth shape parameters, Computer Research and Development,vol49,pp5 57, [5] X Han, Quadratc trgonometrc polynomal curves wth a shape parameter, Computer Aded Geometrc Desgn, vol 9, no 7,pp53 5, [6] X Han, Cubc trgonometrc polynomal curves wth a shape parameter, Computer Aded Geometrc Desgn,vol,no6,pp 535 548, 4 [7] X Han, A class of general quartc splne curves wth shape parameters, Computer Aded Geometrc Desgn, vol8,no3, pp5 63,
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