ISSN 46-69 Eglad UK Joual of Ifomato ad omputg Scece Vol No 6 pp -8 Fag of Paametc Qutc Sples Yuau Wag Shagha Isttute of Spots Shagha 48 ha School of Mathematcal Scece Fuda Uvesty Shagha 4 ha { P t )} ( Receved July accepted Oct ) Abstact I ths pape we gave a fag algothm fo paametc qutc sple cuves Ths ew algothm gves a clea modfcato to the bad pot's posto taget vecto ad secod taget vecto We also poved that ths algothm s of eegy optmzato Keywods: cuve fag qutc paametc sple eegy optmzato Itoducto I the feld of cuve fag thee ae two ma poblems Oe s how to quatfy the faess of a cuve whch teds to be a dveget poblem Nevetheless accodg to the pcple of eegy optmzato [] a beautful desg s ofte smple shape I ths pape we tae the eegy ctea as a tool to udge the cuve's faess Aothe poblem s the methods of fag Usually fag methods ca be classed as teactve ad automatcal oes I geeal automatcal methods ca obta a faste appomato to a optmum tha teactve oes [] Automatc fag algothm ca also be futhe classfed as local o global fag methods Global oes have bette global fag esult but t s tme-cosumg because of ts lage calculatos Local oes oly have local fag effect but they ae qucly to pefom I paametc sple fag Kellade poposed a local fag method to ufom paametc cubc sples [] Polaoff eteded the mehod to o-ufom paametc cubc sples [4] ad late a automatc fag algothm was peseted [] I ou othe pape [6] we mpoved Polaoff's fag algothm to cubc paametc sples by chagg the bad pots' posto ad ts taget vecto I ths pape we peset a local fag algothm fo qutc paametc sples The bad cotol pots ae modfed usg a local eegy optmzato The pape s ogazed as follows: secto we gave the fag algothm to paametc qutc sple cuves; secto we peseted a smple eample; secto 4 we gave the cocluso The fag algothm to paametc qutc sple cuves We suppose that fo some tege a paametc qutc sple cuve passes though data pots ( '' The t s assumed that the cuve eeds to be faed at oe bad data pot fo th some ( < < ) Suppose a pot o the segmet of the qutc sple ca be ewtte as () ( ) t a tt t t t t t Δ () t t '' ( t ) '' t t ''( t ) '' the If the ed codtos ae ( ) ( ) ( ) ( ) a ae gve by P E-mal addess: 84@fudaeduc wy8@yahoocomc Publshed by Wold Academc Pess Wold Academc Uo
Y Wag et al: Fag of Paametc Qutc Sples '' a a a a b b b ( 4 Δ Δ ) Δ a ( b b Δ b Δ ) a (6b b Δ b Δ ) 4 4 Δ Δ () Whee b Δ Δ b Δ b t Now we use W ( () t ) dt to appopate the cuve's teal sta eegy Ad accodg to t ou fag ctea the smalle the cuve's eegy s the moe fa the cuve s to fa the cuve by modfyg the bad pot P s equal to mmze the cuve's sta eegy We'll gve ou fag algothm by mmzg the cuve's sta eegy the followg ( ) ( ) ( ) ( t t ) th 6 4 t t ( ) 6 ( ) 4 ( ) ( ) t t Sce t a a t t a t t a so the secto cuve's sta eegy W s W () t dt a a t t a t t a t t dt Fom equal () we ca see a a a4 a cludg ot cludg th ( ) I othe wods oly the cotol pots P P ca affect the secto cuve's sta th th eegy W By cotast f we chage the cotol pot P oly the secto cuve's sta eegy W W wll be affected ad all othe W ( ) wll be uchaged So W( ew) W( ) Let W deote the vaato of the cuve's eegy W ew W The Δ ( ) ( ) ( ( ) ) ( ) ( ) Δ W W ew W W ew W W ew W W ew W Fo cofeece we gve the followg sgal: { P( t '' )} ~ the pmtve data pots { P( t ( ew) ( ew) ( ew ))} () t ~ the pmtve tepolatg sple cuve ( ew)() t ~ the ew tepolatg sple cuve '' ~ the modfed data pots Δ P ( δ ) ξ ~ the chage of P δ ( δ ) δ y δ z ~ the chage of the posto of P ( ) y z ~ the chage of the fst devatve of P ξ ( ξ ξ ξ ) ~ the chage of the secod devatve of P Whee y z Now we gve ou algothm: P( ew) P ΔP that s ( ew) δ ( ew) ( ew ) ξ () JI emal fo cotbuto: submt@cogu
Joual of Ifomato ad omputg Scece (6) pp -8 Ad ad So Δ Δ δ ( ) 6 Δ Δ ( Δ Δ ) ( Δ Δ) 4 ( Δ ) Δ ( ) Δ Δ 8 Δ Δ ( ) ξ 6 ( Δ ) 6 Δ 4 ( a a ) 4( a4δ a 4Δ ) ( aδ a Δ ) 4 96 64 6 4( a a ) a Δ a4δ a 4Δ aδ a Δ 6 4 8 4 ( a4δ a 4Δ ) aδ a Δ Lemma hagg P to P ( ew) all othe pots uchaged the chage of teal sta eegy s ( ) W * W I (4) δ ξ () 4 ( ) ( a a ) ( ) ( ) 4 a 4 a 4 a a I δ ξ Δ Δ Δ Δ δ 4 96 64 6 4( a a ) a Δ a4δ a 4Δ aδ a Δ 6 4 8 4 ( a4δ a 4Δ ) a Δ a Δ ξ 9 δ ( ) Δ Δ ξ Δ Δ Δ Δ 6 δ δ ξ Δ Δ Δ Δ t ( ( )( ) Poof: Fo W( ew) ew t dt otcg As ected befoe ) t ( ew)( t) 6 ( t t ) ( t t ) ( t t ) 4 a a a a ( ) ( ) ( ) t ( ) 6 4 t W ew t t t t t t a a a a dt t y t z ( ew)( t) a ( t t ) ( ) ( ( ) ) ( ) ( ) Δ W W ew W W ew W W ew W W ew W ( ()) ( ()) ( ()) W t y t z t dt W W W Sce JI emal fo subscpto: fo@cogu
4 Y Wag et al: Fag of Paametc Qutc Sples whee W t dt the same to W y Wz Now suppose that P s chaged by ΔP tag the t t ( ()) -compoet of (fo claty we also deote t wth ) We ca see that each s a fucto of a a a but ot of y y y ; z z z So the chage of -compoet W ( ew) W ca be epessed usg Taylo fomula W ( ew) ( ew) ( ew) ( ew) W ( ) ( ) ( δ ξ ) ( ) W ( ew) W ( W ( ew)) δ ( W ( ew)) ( W ( ew)) ξ ( W ( ew)) δ ( W ( ew))! ( W ( ew)) ( W ( ew)) ( ( ( ) δ W ew δξ W ew ξ R δ ξ By calculatg we ca get ( a ) ( a ) ( ) a 6 ( a ) ( a ) ( a ) Δ Δ Δ 8 ( a ) ( a ) ( a ) 4 4 4 4 Δ Δ Δ 6 ( a ) ( a ) ( 4 ) a Δ Δ Δ The 4 ( W ( ew)) 4 a a4δ a Δ 4 64 ( W ( ew)) 4 a a4δ aδ 6 4 ( W ( ew)) a4δ aδ 4 84 6Δ ( W ( ew)) ( W ( ew W ew Δ Δ ( W ( ew)) ( W ew Δ 6 ( W ( ew)) ( W ew Δ ( W ( ew)) ( W ew All othe hghe devatves of W ( ew ) to R δ ξ s zeo Theefoe ae zeo so ( ) ξ JI emal fo cotbuto: submt@cogu
Joual of Ifomato ad omputg Scece (6) pp -8 ( ( ) ( ) ( ) W ( ew) ew ew ew W ad smlaly we ca get { [ 4 4 64 a 4a4 Δ a Δ 4 4 δ a a Δ a Δ 6 4 4 84 6 4 a4 Δ aδ ξ δ ξ δ Δ! Δ Δ Δ 44 δ ξ ξ Δ ( ( ) ( ) ( ) W ( ew) ew ew ew W { δ [ 4 96 a 4a Δ a Δ 4 a a Δ a Δ 6 8 4 4 84 6 a Δ a 4Δ a Δ ξ δ Δ! Δ Δ 4 4 4 44 δ δ ξ ξ Δ Δ add the above two equals ad eplace wth ξ we get W ( ew ) W I ( δ ξ ) whee 4 ( ) ( a a ) ( ) ( ) 4 a 4 a 4 a a I δ ξ Δ Δ Δ Δ δ 4 96 64 6 4( a a ) a Δ a4δ a 4Δ aδ a Δ 6 4 8 4 ( a4δ a 4Δ ) a Δ a Δ ξ 9 δ ( Δ Δ ) ξ Δ Δ Δ Δ 6 δ δ ξ Δ Δ Δ Δ { } Theoem Assume a paametc qutc sple cuve passg though pots P( t ) Δ t t > We also assume that oly oe bad pot P ( < < ) eques to be modfed ad all othe pots ae good hagg to P ew the ew cuve tepolatg { P( ew) ( t ( ew) ( ew) ( ew) )} Let P ( ) s of eegy optmzato Poof: Fom the Lemma the chage of the teal sta eegy s W ( ew ) W I ( ) δ ξ JI emal fo subscpto: fo@cogu
6 Y Wag et al: Fag of Paametc Qutc Sples that s so ( I ( δ )) ξ δ ( I ( δ )) ξ ( I ( δ )) ξ ξ 4 6 84 δ Δ Δ Δ Δ 6 6 δ ( Δ ) Δ ξ Δ Δ δ ξ Δ Δ Δ Δ Δ Δ δ Δ Δ 6 ( Δ Δ ) ( Δ Δ ) ( Δ Δ ) Δ Δ ( Δ Δ ) 4 ( ) 8 Δ Δ ( Δ Δ Δ Δ ) ξ 6 ( Δ ) 6 Δ Also because the Hesso mat of I 4 6 " " " ( I ) ( ) ( ) I I Δ Δ Δ Δ Δ Δ δ δ δ δ ξ " " " 8 H( I) ( I) ( I) ( I) 4 δ ξ Δ Δ Δ Δ " " " ( I) ( I) ( I) ξ δ ξ ξ ξ 6 6 ( Δ Δ) Δ Δ s postve So ( δ ξ ) mmum value of s the oly mmum value of ( ) W( ew) W ew W The ( ) W Theefo we say the ew cuve s of eegy optmzato δ ξ s the oly oollay Especally to ufom qutc sple cuves wth all Δ ( ) the fag algothm s P ( ew) P P Δ δ ξ Whee δ ξ ae epessed as Δ P ( ) JI emal fo cotbuto: submt@cogu
Joual of Ifomato ad omputg Scece (6) pp -8 Eamples 9 δ ( a a ) a 4 a 4 a 4 a ( 9 ) a a a 64 a 4 a 8 4 a a 48 ξ ( a a ) a 4 a 4 a Numeous of fag eamples showed that ths ew method s vald I ths secto we gave a smple eample to llustate ths valdessfg s the pmtve tepolatg cuve Fg s the cuvatue fgue coespodg to Fg It s easy to udge fom Fg that Fg s ot fa Fg s the fgue tepolatg data pots faed by ou method Fg4 s the cuvatue fgue coespodg to Fg lealy ou ew fag method s vey vald - - -6-4 - 4 6 Fg - - - - - -6-4 - 4 6 Fg - - - -6-4 - 4 6 Fg JI emal fo subscpto: fo@cogu
8 Y Wag et al: Fag of Paametc Qutc Sples 6 4 - -4-6 -6-4 - 4 6 4 ocluso Fg 4 I the fag of paametc sple Kellade ad Polaoff composed the fag methods to cubc sple cuves I ths pape based o eegy cteo we gve a ew fag algothm to qutc sple cuves The ew fag algothm ot oly chages the bad pot's posto but also chages t's taget vecto secod taget vecto We also poved the ew algothm s of eegy optmzato Numeous eamples show that ths algothm s vald I futue we'll eted ths algothm to othe sple cuves What's moe as the applcato of suface s moe boadly tha cuve we'll devote ouselves to the fag algothm of the suface Acowledgemets Thas vey much to my advso MYua ao Assocate Pofesso of Fuda Uvesty 6 Refeeces [] Xuefu Wag Fuhua heg ad Ba A Basy Eegy ad B-Sple Itepomato AD 9(996) 48-496 [] N Sapds ad G Fa Automatc Fag Algothm fo B-Sple uves AD (99) -9 [] JAP Kellade Smoothg of ubc Paametc Sple AD (98) -9 [4] J F Polaoff A Impoved Algothm fo Automatc Fag of No-ufom Paametc ubc Sples AD 6(996) 9-66 [] JF Polaoff et al A Automated uve Fag Algothm fo ubc B-sple uve Joual of omputatoal ad Appled Mathematcs (999) ~8 [6] Yuau Wag Yua ao Eegy Optmzato Fag Algothm Of No-ufom Paametc ubc Sples AD&G 9() JI emal fo cotbuto: submt@cogu