Bézier curves with shape parameter *

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1 Wag e al. / J Zhejiag Uiv SCI 5 6A(6): Joural of Zhejiag Uiversiy SCIENCE ISSN hp:// jzus@zju.edu.c Bézier curves wih shape parameer * WANG We-ao ( 王文涛 ) WANG Guo-zhao ( 汪国昭 ) (Deparme of Mahemaics Zhejiag Uiversiy Hagzhou 37 Chia) ww@zju.edu.c Received Ja. 4; revisio acceped Sep. 4 4 Absrac: I his paper Bézier basis wih shape parameer is cosruced by a iegral approach. Based o his basis we defie he Bézier curves wih shape parameer. The Bézier basis curves wih shape parameer have mos properies of Bersei basis ad he Bézier curves. Moreover he shape parameer ca adjus he curves shape wih he same corol polygo. As he icrease of he shape parameer he Bézier curves wih shape parameer approimae o he corol polygo. I he las he Bézier surface wih shape parameer is also cosruced ad i has mos properies of Bézier surface. Keywords: Bézier curve Bézier basis wih shape parameer Bersei basis doi:.63/jzus.5.a497 Docume code: A CLC umber: TP39 INTRODUCTION The Bézier curves ad surfaces form a basic ool for cosrucig free form curves ad surfaces. May basis-lie Bézier basis are preseed. Said (989) ad Goodma ad Said (99) cosruced he Ball basis. Maiar e al.() foud some bases for he spaces { cos si cos si} { cos si} ad { cos si cos si}. Che ad Wag (3) gave he C-Bézier basis i he space { si cos}. Wag ad Wag (4) pu forward Uiform B Splie wih shape parameer which is eeded from Uiform B Splie ad possesses may ecelle properies. I his paper we prese Bézier basis wih shape parameer by a iegral approach. The shape parameer ca adjus he curve s posiio ad he Bézier curves wih shape parameer have mos properies of he Bézier curves. The res of his paper is orgaized as follows. Secio gives a algorihm for cosrucig he basis. Some properies of he Bézier basis wih shape parameer are discussed i Secio 3. Usig his basis * Projec suppored by he Naioal Naural Sciece Foudaio of Chia (No. 37) ad he Naioal Basic Research Program (973) of Chia (No. GCB) we give he defiiio of a Bézier curve wih shape parameer which has mos properies of he Bézier curve i Secio 4. Some figure eamples are show i Secio 4. CONSTRUCTION OF THE BÉZIER BASIS WITH SHAPE PARAMETER We firs give wo iiial fucios (Fig.) 3 N( ) = λ( ) + ( λ)( ) 3 N() = λ + ( λ) () where [ ]. For we defie he Bézier basis wih shape parameer {N () N () N ()} recursively by N () N ( )d δ i = δ i i δi i = N () ( N ( ) N ( ))d N () = N ( )d. () δ

2 498 Wag e al. / J Zhejiag Uiv SCI 5 6A(6):497-5 I hese formulae i i δ = N ( )d i =.... λ ( ) ( ] for 3; whe = λ ( ]. y If λ= he N ()=N () N ()= ad he quadraic Bézier curve wih shape parameer is lie so we mae λ. We choose he wo symmeric iiial fucios i Eq.() i order ha ( N ( ) + N ( ))d = ad N () N () are quadraic fucio of wih shape parameer λ. Defiig he basis fucios i Eq.() recursively esures possessio of may properies ha will be discussed i Secio 3. The parameer λ ca chage he shape of he curve cosruced by his basis ad i paricular whe λ= we ge he Bézier basis for he polyomial space from Eqs.() ad () so we ame i Bézier basis wih shape parameer. Fig. shows he quadraic Bézier basis wih shape parameer PROPERTIES OF THE BASIS N () N () Fig. The wo iiial fucios (λ= ) y N () N () N () Fig. The quadraic Bézier basis wih shape parameer (λ=.5) Properies a he edpois Lemma A he edpois he Bézier basis wih shape parameer has he same properies as he Bézier basis. Tha is for (a) N N ( ) ( ) (b) N N () = () = (3) () = () = ( j) ( ) Ni () = Ni () = j= i ; = i ;. (4) () i (c) Ni () = δ iδ + i δi δi. By Eqs.() ad () i is easy o prove Lemma by iducio o. Liear idepedece I order o chec he idepedece of {N N N } we cosider a rivial liear combiaio α ini () = []. By aig = we ge from Eq.(4) ha α =. Differeiaig he liear combiaio imes we deduce agai from Eq.(4) ha α = for =. Tha is N i () ( ) are liearly idepede. Therefore {N N N } is liearly idepede. Posiiviy Lemma N i () ( ) has o zero o ( ). Proof Usig Eq.() recursively we ge ( ) ( ) N () = ( ) a N () N () = b N () ( ) a b > Ni () = ain() bn i () ab i i> ( i = ) ai b i are cosas ha are idepede of ad λ. Obviously N N has ( ) ( ) oly oe zero o []. Because N () N () = ( ) ( ) i i + λ ( ) ab i i< ad N i is a quadraic fucio ( ) of N i ( i = ) has also oly oe zero o []. By Rolle s Theorem we have ha N i ( ) has a mos zeros o [ ]. We see i Eq.(4) ha N i has zeros o [ ] icludig he i-fold zero a ad he ( i)-fold zero a. So N i () (

3 Wag e al. / J Zhejiag Uiv SCI 5 6A(6): ) has o zero o ( ). Lemma 3 The Bézier basis wih shape parameer are posiive o [ ]. Proof Cosider a arbirary Bézier basis fucio wih shape parameer N i () i. From Lemma we coclude ha N i () has o zero o ( ). I oher words N i () is eiher posiive or egaive o he ierval. We coclude from Eq.(4) ha N i () is posiive o ( ). Sice N i () is arbirary we ow ha he Bézier basis wih shape parameer is posiive o [ ]. Lemma 4 The Bézier basis wih shape parameer is ormalized ha is similar. So he proposiio holds by iducio o. GEOMETRIC PROPERTIES OF THE BÉZIER CURVE WITH SHAPE PARAMETER A Bézier curve wih shape parameer p() wih corol pois p i is defied by i i (5) p () = pn () [] where {N i ()} is Bézier basis wih shape parameer. Ni () =. We summarize Lemmas 3 ad 4 i Proposiio. Proposiio The Bézier basis wih shape parameer is a bledig sysem. Symmery Proposiio N i ()=N i ( ) for [ ] ( ). Proof We prove his proposiio by iducio. Whe = he proposiio obviously holds by he defiiio of he Bézier basis wih shape parameer. Assume ha he propery holds for = ha is N i ()=N i ( ). Hece we have Geomeric properies a he edpois The geomeric properies a he edpois of he Bézier curves wih shape parameer ca be easily deduced from hose of he Bézier basis wih shape parameer. (a) p()=p p()=p (6) (b) p ( ) ( ) pini () = (). (7) Cove hull propery The eire Bézier curve wih shape parameer Eq.(5) mus lie iside is corol polygo spaed by p p p. This propery is a cosequece of Proposiio. Fig.3 shows cove hull propery. N ( )d = N ( )d i i i δ i i = N ( )d = N ( )d p p p 3 By leig = we obai δ i =δ i. Therefore for <i<+ we have N ( ) i + + i i δ+ i + i = δ N ( )d N ( )d ( ( )d ) ( ( )d ) δi Ni δi Ni = = N. () i + The proof for he case whe ad + is p Fig.3 Cove hull propery Differeiaio The derivaive p () of degree-(+) Bézier p 5 p 4

4 5 Wag e al. / J Zhejiag Uiv SCI 5 6A(6):497-5 curves wih shape parameer p() is clearly a degree- curve. Such a curve ca be wrie i Bézier curves wih shape parameer-lie form as i i (8) p () = p N () [] where p i ( ) are he corol pois of p (). Differeiaig he fucios i Eq.() ad afer some algebraic maipulaios we fid ha he corol pois of p () i he above form are give by δ (P i+ P i ). Some eamples Figs.4a ad 4b respecively show he heago cosisig of si symmeric corol polygos ad he flowers cosisig of si symmeric quaric ad cubic Bézier curves wih shape parameer for λ= 3 (solid doed ad dashed lies). Fig.4c ad 4d respecively show he square cosisig of four symmeric corol polygos ad he flowers cosisig of four symmeric quaric ad cubic Bézier curves wih shape parameer for λ= 3 (solid doed ad dashed lies). The symbol is he corol poi i all figures. Fig.5 shows degree-6 Bézier curves wih shape parameer for λ=. 5 (solid doed dashed ad dashed-do lies). The figures show ha he Bézier curves wih shape parameer approimae o he corol polygo as he icrease of he shape parameer. (a) (b) (c) (d) Fig.4 The flowers cosisig of symmeric Bézier curves wih shape parameer (a) I si symmeric closed corol polygos; (b) I si symmeric ope corol polygos; (c) I four symmeric closed corol polygos; (d) I four symmeric ope corol polygos

5 Wag e al. / J Zhejiag Uiv SCI 5 6A(6): BÉZIER SURFACE WITH SHAPE PARAMETER Usig he esor produc we ca cosruc Bézier surface wih shape parameer m puv ( ) = b N ( un ) ( v) u v i j i m j j= i which N im (u) N j (v) are he Bézier basis fucios wih shape parameer ad b ij is he corol poi. The esor produc of Bézier surface wih shape parameer has properies similar o hose of he esor produc of Bézier surface. CONCLUSION Differe curves lyig o he Bézier curve of degree- earby ca be creaed by his way i he paper. The Bézier curves wih shape parameer approimae o he corol polygo as he icrease of he shape parameer λ. We ca desig Bézier curves by choosig differe shape parameer i λ ( ) ( ]. Sice Bézier curves wih shape parameer have may of he same of heir properies ad srucure as hose of ordiary Bézier curves ad preserve some pracical geomery properies hey ca more coveiely be used as such. However here are some deficiecies i Bézier basis wih shape parameer such as how o corol he shape parameer ad wha is he geomeric meaig of he shape parameer ec. I he fuure we will research hose problems. Fig.5 Degree-6 Bézier curves wih shape parameer Refereces Che Q.Y. Wag G.Z. 3. A class of Bézier-lie curves. Compuer Aided Geomeric Desig :9-39. Goodma T.N.T. Said H.B. 99. Properies of geeralized Ball curves ad surfaces. Compuer Aided Desig 3(8): Maiar E. Peńa J.N. Sâchez-Reys J.. Shape preservig aleraives o he raioal Bézier model. Compuer Aided Geomeric Desig 5: Said H.B Geeralized Ball curve ad is recursive algorihm. ACM Trasacio o Graphics 8(4): Wag W.T. Wag G.Z. 4. Uiform B Splie wih Shape Parameer. Joural of Compuer-Aided Desig & Compuer Graphics 6(6): Welcome visiig our joural websie: hp:// Welcome coribuios & subscripio from all over he world The edior would welcome your view or commes o ay iem i he joural or relaed maers Please wrie o: Hele Zhag Maagig Edior of JZUS jzus@zju.edu.c Tel/Fa: