HYPERBOLIC PARABOLOID SPLINE FOR SPATIAL DATA FITTING AND APPROXIMATION

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1 Joural of ure ad Appled Mathematcs: Advaces ad Applcatos Volume 8 Number ages 5-9 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA FIING AND AROXIMAION FENGFU ENG ad JIENG NIU School of Mathematcs ad Computatoal Scece Gul Uversty of Electroc echology Gul Guagx 5. R. Cha e-mal: pegfegfu@tom.com.c Abstract A method to costruct the free sple curve ad surface s preseted ths paper whch proposes four ratoal bass fuctos each of them cludes a commo cotrol hadle ad two varables. Oe of the varables s determed by a fucto a set wth a sgle parameter ad the other s defed by a mult-tegral upper lmt fuctos about the prevous varable. For a spatal cotrol polygo of four pots a sple curve ca be produced by the bass fuctos whch ca approxmate the cotrol edges at wll va chagg the cotrol hadle. Addtoally the authors cosder the propertes of shapepreservg ad the sple curves at the edpots. By costrag the cotrol hadle for the sple the edpot curvatures are less tha some threshold at both edpots. For a rage of pots cosequetly oe approxmately G -cotuty curve ca be coected by ths sple curves wthout other codtos requred. Mathematcs Subject Classfcato: 65D7 65D7. Keywords ad phrases: bass fuctos ratoal sple curve approxmato G -cotuty tegral sple. Receved August 9 Scetfc Advaces ublshers

2 6 FENGFU ENG ad JIENG NIU. Itroducto here are maly avalable to costruct the sple curves ad surfaces wth parametrc ad algebrac forms [ 6-8] the area of computer aded geometrc desg (CAGD) ad computato. he great advatages of the latter are ts more freedom trscally smpler represetato ad better geometrc operato etc but ts structure s dffcult to uderstad ad the computg s hard to apply wdely. herefore at preset the former became a rreplaceable method the geometrc desg due to the coveece of costructg sple curves/surfaces specally of whch the sple fucto are a cetral ssue. I the meatme ratoal parametrc sple s ofte used to costruct the approxmate sple for a cotrol polygo subdvso ad the model recostructo whch have excellet propertes as ce effect of approxmato easy to mplemet ad wth shape hadles for humamache teracto [5 ] etc. he curret research of ratoal sple becomes oe of the hot spots CAGD ad a umber of authors have vestgated the ratoal polyomal sple such as the paper [] troduces a ratoal sple that used for offset curve dfferet shapes sple curves are arse a famly of a plae ratoal curves depedg o a parameter. Besdes shape-preservg curves ad surfaces desg are also a ey ssue for the ratoal sple [8]. I addto the complex ratoal sples are recalled the lterature [5] ad the algebractrgoometrc bleded sples s studed [9]. I order to produce a more farg sple other words to reach farg as G -cotuty usually the curvature of a sple curve (surface) s maly cosdered at the coecto pots each segmet ad eve the whole sples []. A defto of the twsted cubc s developed was troduced the lterature [7]. he lterature [] pots out that ratoal cubc allow modfcato of ther fulless eve whe the ed tagets are ept fxed ths s the reaso why they are occasoally preferred to stadard cubc CAGD. hs should serve as a easy ad tutve troducto ad help the potetal user to choose a sutable represetato. hs wor starts from the papers [ ] where the authors preset a approach to ft a rage

3 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA 7 of four spatal pots. I those two papers ther ma dea s to defe a sple o a hyperbolc parabolod the tetrahedro va a factor ad a fucto ad the method starts wth algebrac form but what to be evetually appled s ts parametrc form. he proceedg of mplemetato s trasformg the ufed sple curve to a arbtrarly quadrlateral by the barycetrc coordates [9]. hus ths approach to costruct a sple combes the advatage of algebrac ad parametrc form oetheless the sple has ts lmtato that s ot able to arbtrarly approxmate the cotrol polygo wth four pots. hs deserves further study o the questo about the ratoal form o the hyperbolc parabolod. here s a tutve argumet for us: hs d of sple ca be exteded as a free sple to arbtrarly approxmate a quadrlateral ad the method of mplemetato does as customary Bézer sples. However t should be ratoal form wth a shape hadle ad a smlar fucto ths s what we are cosderg ths paper. he rest of the paper s orgazed as follows. I Secto we aalyze a ruled hyperbolc parabolod wth a par of parameters ad defe a ufed sple o ths ruled surfaced by a real umber ad a fucto a set. I Secto the curvature s propertes of ths sple are studed ad these ut sple are employed to attach a roughly G -cotuty curve to approxmate a cotrol polygo meawhle there s a shape hadle each sple for curve cotrol. I what s to follow the surface employg ths sple s cosdered to approxmate a spatal subdvso Secto. Fally the ma coclusos wll be drow Secto 5.. Bass Fuctos Frst the four b-parameter fuctos are defed by follows: ( u) ( v) α( u v) + ( ) v ( u) v α( u v) + ( ) v uv α( u v) + ( ) v u( v) α ( u v) + ( ) v ()

4 8 FENGFU ENG ad JIENG NIU + where R s a oegatve real umber that wors as a shape hadle for a sple the method hereafter ad u v [ ] are two varables. he four fuctos have the followg characterstcs: () No-egatve: α ( u v) ( ). () Normalzato: α ( u v). () O a hyperbolc parabolod: For arbtrarly > the four fuctos wthout the use of α ( fact α s determed by the others because of α ( u v) ) have relatoshp of α ( α + α ) ( α + ). () α Notce that the surface determed by the Equato () s a sheet of hyperbolc parabolod the coordate system of ( α α α ) whch s a tetrahedro wth the four vertexes ( ) ( ) ( ) ad ( ) (see Fgure ) ad those four vertexes all le the surface determed by the Equato (). herefore we regard the formula () as a ratoal parametrzato of the hyperbolc parabolod. Fgure. Hyperbolc parabolod tetrahedro.

5 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA 9 If the two fuctos v ( t) ad u ( t) are defed o the parameter t [ a b] cosequece oe curve s produced the hyperbolc parabolod. Next we gve the followg parametrc defto to v () t ad determe the other parametrc u ( t) wth v ( t) to loo for the sples o the ruled surface (). Defto. [ a b] v () t wth t [ a b] V s a set of oegatve real fucto a fucto whch satsfes for these codtos: () v() t C [ a b]; () v ( a) v( b) ad v ( t) s ot costat zero for t [ a b]; () () t [ ] v for ay t [ a b]. hs s a tegral seres as followg: u δm b ( b t) u ( ) m t dt a t u m() t δm ( b τ) u ( ) m τ dτ a () t v() t t [ a b] m L m L. Gve a tegral umber m let u( t) um ( t) the fucto u () t has u ( a) u( b) ad u ( t) δm( b t) um ( t) whch s thus at least twce dfferetable fucto ad u ( a) u ( b). At the same tme ts secod dervatves at the edpots are case m u ( a) u ( b) δ > mδm case u ( a) δ ( b a) v u ( b). m a v a s dcatve of ( a) v ad so s v b of v ( b) below. dv v ( t) Furthermore there are ad du δ ( b t) u ( t) m m () v ( t) lm f t a v() t v a ths scheme whch ca refer to the lteratures [ ].

6 FENGFU ENG ad JIENG NIU Meawhle those sple curve whch was cosder a ufed tetrahedro that could be trasformed to dfferet ut wth four cotrol pots by oe ufed tetrahedro. Chose a v () t V [ a b] fucto u( t) um ( t) ( m L) s determed by formula (). We substtute v ( t) u( t) to the relatve varables v u the formula () the the correspodg four bass fuctos are arse ad they are α ( t) α ( t) α ( ) ad α () t t [ a b] respectvely each of whch cludes the commo cotrol hadle. If v () t s tae as a polyomal the u ( t) s also a polyomal ad the four bass fuctos are all ratoal polyomals. For example usg t v () t ( t) t t [ ] () ad worg out u() t u () t t t to the four bass fuctos ad ther shape are llumated Fgure. As a curve o the hyperbolc parabolod those four bass fuctos cota the propertes as the four fuctos defed by formula (). Fgure. Four bass fuctos wth.

7 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA Furthermore they have propertes whe t s a ad b as followg: ad ( a ) α ( b) α ( a ) α ( b) α ( a ) α ( b) α ( a ) α ( b) α α ( ) v α ( b) a a α ( ) v α ( b) a a α ( a) α ( b) v b α ( a) α ( b) v. b hese results lead us to costruct a sple curve or a surface f both v a ad v b are ot zeros.. Bass Fuctos for Sple Curve Let R be a rage of pots whch forms a quadrlateral wth three cotrol edges ad the authors also call the three cotrol edges as a cotrol quadrlateral. Wth the four bass fuctos above we defe the followg sple for the quadrlateral: C ( t ) α ( t) t [ a b] (5) where t s a uque parametrc varable s a cotrol hadle for a sple terms of formula () ad ( t) V a b defed by Defto. v s a fucto [ ]

8 FENGFU ENG ad JIENG NIU.. ropertes of sple curve Wth the defto of sple beg of > ad fucto v () t V [ a b] the followg coclusos about sple curve C ( t ) ca be derved drectoally: () he sple s the covex hull that forms by the four pots. () he sple arse by expresso (5) wth arbtrarly > terpolates the edpots ad (see Fgure whch apples 5 6 v () t ( t) t t [ ] ad u() t u () t t 7.5t + t.5 ). t At the same tme the sples wth dfferet m are of the same propertes at the edpots (see Fgure ). Fgure. Sple curves wth dfferet for quadrlateral.

9 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA Fgure. Sple curves wth dfferet tmes tegral for quadrlateral ad 9. () It mples that the sple s taget to at ad at. Whe s suffcetly small the results of a expermet ca be observed at the small eghbourhood of the two edpots. (v) Wth lm α ( t) ad lm α ( t) t mples that: + t a + t b the greater the more approxmate to the cotrol polygo the sple becames ad that ca approxmate to the cotrol polygo through fte creasg the value of. Fgure shows the varato of the sple curves wth two dfferet ad the same v ( t) wth fucto ()... Curvatures of sples By the deftos of () ad () we ca obta the secod dervatves at the edpots wth m

10 FENGFU ENG ad JIENG NIU a a α ( a) ( b a) δ v v ( a) + ( ) v a α ( a) v ( a) ( ) v α ( a ) α ( a) ( b a) δ v a ad α ( b ) α ( b ) b α ( b) v ( b) ( ) v b α ( b) v ( b) + ( ) v. As a result the absolute curvatures at the edpots of the sple curve are ( ) ( a) ( b a) δ ( ) ( ) ( ) v a ( ) ( b). he smlar cocluso ca be obtaed wth m > ad α ( a) v ( a) + ( ) v a α ( a) v ( a) ( ) v a α ( a ) α ( a ) α ( b ) δ mδm α ( b )

11 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA 5 α ( b) v ( b) ( ) v b α ( b) δ δ v ( b) + ( ) v. m m b he both curvatures at the edpots of the sple curve are ( ) ( a) ( ) ( b) δmδm vb ( ) ( ) ( ). I lteratures [ ] there s ( ) ( ) f we cosder a ufed form as ( ) ( ) ( ) ( ) ad v. For ths reaso the curve coected by these sples wth a v b G -cotuty s also approxmately trasformg to dfferet uts wth four cotrol pots. Employg v () t as () ad u ( t) u ( t) the plot of Fgure 5(a) descrbes the curvature of a sple wth the parameter t whe ad so s Fgure 5(b) whe. (ts data are ( ) ( 5) ( ) ad ( 6 ) ).

12 6 FENGFU ENG ad JIENG NIU (a) (b) Fgure 5. Curvatures o parameter t wth dfferet whe u ( t) u ( t).

13 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA 7 Fgures 6(a) ad 6(b) descrbe the curves beg of ( t) u ( t) wth. ad respectvely. u (a) (b) Fgure 6. Curvatures o parameter t wth dfferet whe u ( t) u ( t).

14 8 FENGFU ENG ad JIENG NIU O the bass of the expresso of curvature at the edpots ad what the fgures descrbe above wth a comparso value we ca coclude that () he curvature of oe edpot s zero that eeps fxed ad the other edpot s approxmate to zero wth a great whch eables us to costruct a approxmately G -cotuty curve va attachg some oe as method of Bézer amely oly cosder that the pot to be coected betwee two segmets s three collear wth the two adjacet pots. () Wth a small the sple the curvatures at edpots are a lager alterato ad the curvature s almost equal at the others pots. b Accordg to the smple eergy tegral formula () t dt whch oly a cludes the curvature the eergy of ths sple s thus relatvely small so we ca also use them to produce a farly curve wth may sple uts... Degree reducto ad elevato he method what we put up s ot oly sutable for four cotrol pots whch ca also reduce (elevate) the degree to less (more) tha four pots space. where If ad are cosdered as oe pot the C ( t ) α ( t) t [ a b] α( u v) α( u v) α ( ) u v ( u) ( v) + ( ) v v + ( ) v u( v). + ( ) v After a short calculato we obta the taget vector at the two edpots

15 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA 9 C ( a ) v ( ) C ( b ) v ( ) (6) a b whch possesses a aalogy wth edpots propertes of the four cotrol pots. Usg the same v ( t) ad u( t) u( t) by formula () Fgure 7 descrbes the degree reducto for three pots whch also ca do wth two cotrol pots oly. Fgure 7. Degree reducto to cotrol polygo wth three pots. he scheme of degree elevato s aalogous to the ordary Bézer sple. Assume there are fve pots space L oe sple 5 ( C ) ( ) ca be produced by the prevous four pots ad aother t ( C ) ( ) s doe by the bac four pots. Oe curve defed by t ( ) ( ) ( ) ( ) ( C t u C t + uc ) ( t ) (7) s the form of degree elevato for fve pots ad u u( t) has bee defed above. he formula (7) possesses the edpot propertes as C ( a ) v ( ) C ( b ) v ( ) a b 5 whch s aalogous to the sple tself. he edpot curvatures are

16 FENGFU ENG ad JIENG NIU ( ) ( a) ( b a) δ ( ) ( ) ( ) v a whle m ad ( ) ( b) 5 ( ) ( a) ( ) ( b) 5 δmδm vb ( 5 ) ( 5 ) ( ) 5 whle m >. Fgure 8 shows the degree elevato for fve pots wth v () t t( t) ( t [ ] ) ad u ( t) u ( t). Fgure 8. Degree elevato to cotrol polygo wth fve pots. Utlzg the recursve method we ca geerate a sple to a cotrol polygo wth more tha fve pots. If the degree elevato formula s obtaed for ( > 5) pots the we ca also elevate degree to

17 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA pots as. L A sple ( ) ( ) t C ca be produced by the prevous pots ad aother ( ) ( ) t C s obtaed by the bac pots. Smlarly oe degree elevato curve s defed by ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t u t u t Q C C C α + (8) where [ ( ) ] ( ) ( ).!!! C u u C + Q he formula (8) possesses the edpot propertes of ( ) ( ) ( ) ( ) a v b b C v a C ad the curvatures are ( ) ( ) ( ) ( ) ( ) ( ) δ v a a b a ( ) ( ) b the case of m whle ( ) ( ) a ( ) ( ) ( ) ( ) ( ) b m m v b δ δ the case of. > m I expresso (8) we rewrte the formula through chagg the order of summato ( ) ( ) ( ) t t t Q C β α where β represets ( ) β β β L ad there s a relatoshp betwee β ad α by

18 FENGFU ENG ad JIENG NIU β γ γ γ α γ γ where γ ( γ γ L ) ad γ C ( u) u ( L ). I fact γ ( L ) are the Berste bass fuctos... Shape-preservg for a quadrlateral a plae I CAGD what we employ to sple curve should be shape-preservg for a covex quadrlateral a plae. Sce a quadrlateral space ca be mapped to a ormal square for example Q ( ) Q ( ) Q( ) ad Q ( ) space the the sple curve terms of formula (5) s C ( t ) ( α ( t) + α ( t) α ( t) + α ( t)) t [ a b] C( t ) (9) R v here α () t + α () t u() t ad we deote α () t + α() t by + ( )v () t the the curve L a plae determed by expresso (9) ca be cosdered as a locus wth pots ( u( t) ( t) ) t [ a b] ad the cha rule gves the secod-order dervatve expresso of L he parametrc curve d du u v tt u t u. () u u( t) v () t t [ a b] mples a fucto v v( u) betwee u ad v. heorem. If v v( u) s a covex fucto whe t [ a b] ad the L s a covex curve the ormal square Q Q Q. Q

19 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA roof. Note that v v( u) C ( ) s a covex fucto the v ( u). By formula () we ca derve that d du d du [ + ( ) v ] vuu [ + ( ) v] ( ) vu [ + ( ) v]. Cosequetly the cocluso holds. Smlarly so s the sple for a quadrlateral degeerated to a tragle. hus we should employ the covex fuctos v v( u) whe u [ ] cosequetly the sple s produced by the bass fuctos wth that s covex-preservg for a plaar covex quadrlateral. I the meatme the curve R ca brg about mootoy f the gve pots are mootoe. heorem. Let { x y }( ) R wth x+ x y + y ( ) the for ay > the curve C ( t ) defed by formula (5) s mootoy. roof. Assume the curve C ( t) be y y( x) R ad let x C ()[ t x] xα () t ad ()[ ] y t y y α () dy C ()[ t y] C t the dx C ()[ t x]. he questo s solved f we ca prove that C ( t) [ y] C ( t) [ x]. Deote x+ x Λ y + y ( ) the C ()[ t x] ( ) α ( t) ( ) α ( t) ω () t

20 FENGFU ENG ad JIENG NIU C ()[ t y] ( Λ Λ Λ ) α ( t) Λ ω () t where α () t ( α () t α () t α ()) t ( ) ad Λ ( Λ Λ Λ ). Cosequetly C ()[ t x] C ()[ t y] ω () t Aω() t () where A ( Λ + Λ ) s a sem-defte matrx of ra less tha. Here we assume ts ozero egevalues be µ ad at least oe of µ them s ozero. Hece there exsts a orthogoal matrx B of to satsfy A µ B µ B. For the sple s a regular curve ad ω ( t ) for arbtrarly t [ a b]. hus the value of () s o-egatve for ay o-egatve Λ ( ) the the above cocluso holds.. Bass Fuctos for Surface here are pots j R ( j L ) whch form a subdvso space. Usg the four bass fuctos we ca employ followg formula to costruct a fttg sple surface: S [ v() t ς] α [ v() t ] α[ vˆ () t ˆ ] t [ a b] ()

21 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA 5 where v () t [ u() t uˆ () t ] s a vector wth two parametrc fuctos V ad ς ( ˆ ) s two shape hadles for a sple surface all of whch are defed by Defto. I addto s a matrx wth j ad α ( α L α ). As show Fgure 9 the sple surfaces are produced wth () t vˆ () t t( t) ( t [ ] ) u( t) u ( t) ad ˆ ( t) u ( t) v u 5 for the mesh X Y [ ] [ ] ad Z. cos( X. ) + cos( Y. ) (the rght of the equato descrbe to use a pot the mesh ad the left s the relatve 6 values of Z) Fgure wth 7 ˆ s produced to a tragle doma. If the orm of ς s great eough the sple surface ca approxmate to the surface of meshes. Fgure 9. G -cotuty surface for rectagle mesh. Fgure. G -cotuty surface for tragle mesh. Wth customary Bézer surfaces method to sttch a surface as descrbed the lterature [6] the osculatg plae of ths sple surface at boudary s the same plae for the two cosecutve sttched surfaces

22 6 FENGFU ENG ad JIENG NIU ad the Gaussa curvature of each agular pot s roughly zero for every uts. Furthermore the Gaussa curvature of a pot the boudary s cosstet for two cosecutve surfaces. herefore the surface sttched by these sple surfaces s a approxmately G - cotuty surface. A approxmately G -cotuty surface sttched by four ut sple surfaces wth 7 ˆ ad use the fuctos as Fgure 9 ad Fgure s llustrated Fgure. Fgure. G -cotuty surface s sttched by four sple surfaces. Employg the scheme to costruct surface a equal-chael tube smulato wth a small orm of ς s descrbed Fgure (a) as well as Fgure (b) wth a greater ad a smaller ˆ whch are all sttched by three pece depedet surface uts.

23 HYERBOLIC ARABOLOID SLINE FOR SAIAL DAA 7 (a) (b) Fgure. wo model surfaces are sttched wth dfferet. 5. Cocluso I ths paper we have preseted a seres of parametrc bass fuctos wth a shape hadle ad a fucto a famly. Usg the bass fuctos a approxmate curve for a quadrlateral ca be produced ad so ca do a surface for a subdvso of. We aalyze the propertes of ts edpots ad ther curvatures. By these ut sples appled to a rage of pots or a meshes space oe approxmately G -cotuty curve or surface s arse wth some free cotrol hadles ad fuctos whch do ot eed other costrats. At the same tme the curve ad surface bear a farg geometrc meag. Compared wth the schemes [ ] ths method avod to loo for the fucto v v( u) wth relatvely complcated defto ad the approach of costructg curves ad surfaces s detcal as that of the customary parametrc Bézer meas.

24 8 FENGFU ENG ad JIENG NIU he ey dea to edow ths represetato wth a shape hadle s that the use of curve ad surface arbtrarly approxmate to a polygo or a subdvso ad the fuctos have the multple of the hadle ad teror fucto v (). t Addtoally otce that some of the four cotrol pots for a ut may collapse a segmet fact the sple became a straght le whe for ths reaso we also ca utlze t for a tragle or a straght le. O the other had f v ( t) s ot a polyomal the drawbac of the provded sple s that t requres more calculato amout tha ratoal polyomal sple. I vew of ths f oe employ a o-polyomal fucto v ( t) for a curve or a surface the the fucto + ( )v bass fuctos () ca be expaded to polyomal at a + b t wth the aylor formula ad the lmted terms are chose terms of the requred precso to tae the place of v ( t). hs wor puts ts emphass o troducg the sple for four pots space however t ca be exteded to arbtrary approxmate a polygo wth pots. We th that the quatfcato approaches s dffcult to cosder for a ratoal form. It s a terestg problem ad curretly uder studyg of the authors ad s left for the future study. Acowledgemet he research wor the paper s supported by project Key Research roject Hgher Educato Isttutos of Guagx (Grat No. ZD5). Refereces [] J. G. Alcazar O the dfferet shapes arsg a famly of plae ratoal curves depedg o a parameter Comput. Aded Geom. Des. 7() () [] C. Bajaj ad G. Xu A-sples: Local terpolato ad approxmato usg C -cotuous pecewse real algebrac curves Computer Scece echcal Report urdue Uversty (99) 9-.

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