Approximation of Parametric Functions by Bicubic B-spline Functions. Majid Amirfakhrian a, Sahar Didab b.

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1 Joral of Aerca Scece ;9() htt://wwwofaercasceceorg Aroxato of Paraetrc Fctos by Bcbc B-sle Fctos Mad Arfakhra a Sahar Ddab b a Deartet of Matheatcs Islac Azad Uversty Cetral Tehra Brach Tehra Ira arfakhra@actbacr b Deartet of Matheatcs Islac Azad Uversty Cetral Tehra Brach Tehra Ira Abstract: I ths aer we roose a ethod to aroxate a araetrc D fcto by bcbc B-sle fctos [Mad Arfakhra Aroxato of Paraetrc Fctos by Bcbc B-sle Fctos J A Sc ;9():9-96] (ISS: 545-) htt://wwwofaercasceceorg 7 Keywords: artal dfferetal eatos arabolc eatos Radal bass fcto collocato ethod Itrodcto Creatg freefor srfaces s a challegg task eve wth advaced geoetrc odelg systes The roble of covertg the dese ot sets rodced by laser scaers to sefl geoetrc odels s referred to as srface recostrcto Paraetrc crves are wdely sed dfferet felds sch as coter grahcs (CG) coter aded geoetrc desg (CAGD) coted ercal cotrol (CC) systes [ ] Oe basc roble the stdy of araetrc crves s to aroxate a crve wth lower degree crve segets For a gve dgtal crve there exst ethods to fd sch aroxate crves effcetly [ 4 5 6] If the crve s gve by exlct exressos ether araetrc or lct these ethods are stll sable However soe ortat geoetrc featres sch as sglar ots caot be reserved I ths aer we wll focs o cotg aroxate srfaces whch ca aroxate the gve srface to ay recso a slar strategy for araetrc crves The rest of ths aer s orgazed as follows I Secto we trodce the roble soe otatos ad relary of B-sles are gve Secto I Sectos 4 ad 5 we gve rereset the least sares ethod for costrctg crves ad srfaces Secto 6 there are soe exales whch sed to llstrate the ethod I secto 7 the aer s coclded Paraetrc B-sles The x y ad z coordates of a crve s rereseted araetrc for as x x( y y( () z z( where the araeter t rages over a rescrbed set of vales The derlyg core of the B-sle s ts bass or bass fctos The orgal defto of the B- sle bass fctos ses the dea of dvded dffereces ad s atheatcally volved Carl de boor establshed the early 97's a recrsve relatosh for the B-sle bass By alyg the Lebz theore de boor was able to derve the followg forla ( ) () where s the -th B-sle bass fcto of order s a eber of odecreasg set of real bers also called as the kot seece ad s the araeter varable Ths forla shows that the B-sle bass fctos of a arbtrary degree ca be stably evalated as lear cobatos of bass fctos of a degree lower The obvos defg featre of the bass fcto s the kot seece The kot seece s a set of o-decreasg real bers The varable reresets the actve area of the real ber le that defes the B-sle bass It takes kots or tervals to defe a bass fcto Sce the bass fctos are based o kot dffereces the shae of the bass fctos s oly deedet o the kot sacg ad ot secfc kot vales Soe of the roertes of the B-sle bass fctos are: The s of the B-sle bass fctos for ay araeter vale wth a secfed terval s always eal to ; e htt://wwwofaercasceceorg 9 edtor@aercasceceorg

2 Joral of Aerca Scece ;9() htt://wwwofaercasceceorg Each bass fcto s greater or eal to zero for all araeter vales Each bass fcto has oly oe ax vale There are three dfferet ethods cooly sed to araetrze odel crve data; for chord legth ad cetretal These ethods are dscssed below Ufor Ths s the slest tye of araetrzato where the kot sacg s chose to be detcal for each terval Tycally kot vales are chose to be sccessve tegers: For ay cases however ths ethod s too slstc ad gores the geoetry of the odel data ots Chord Legth Ths araetrzato s based o the dstace betwee the data ots The kot sacg s roortoal to the dstace betwee the data ots Eato () reflects ths relatosh Ths araetrzato ore accrately reflects the geoetry of the data ots () whch s the -th doa kot s the -th data ot ad s the ber of kot terval Cetretal Ths araetrzato s derved fro a hyscal aalogy It seeks to sooth ot varato the cetretal force actg o a ot oto alog the crve Ths reres the kot seece to be roortoal to the sare root of the dstace betwee the data ots as show Eato (4) (4) Other araetrzato ethods have bee vestgated All these ethods have certa crcstatal advatage over the others There s a trade-off betwee geoetrcal reresetato ad cotato te Tycally chord legth araetrzato reslts a very good corose I ay evet each araetrzato reslts a dfferet shae of the crve B-sle srfaces B-sle srfaces are a exteso of B-sle crves The ost coo kd of a B-sle srface s the tesor rodct srface The srface bass fctos are rodcts of two varate (crve) bases The srface s a weghted s of srface (two desoal) bass fctos The weghts are a rectaglar array of cotrol ots The followg Eato () shows a atheatcal descrto of the tesor rodct B-sle srface [ v [ v v ] ] ( () where S ( s a B-sle srface as a fcto of two varables 's are cotrol ots s the -th bass fcto of order as a fcto of ( s the -th bass fcto of order as a fcto of v ad v are eleets of the two kot seeces related to the varables ad v resectvely For ost coter aded desg roses as the case of the crve S ( s a vector fcto of two araetrc vales ad v A atheatcal descrto of ths relatosh s show below Eato () x y z ( ( ( () where x y ad z are coordates odel sace The rectaglar array of cotrol ots fors what s called a cotrol et Slar to the B-sle crve the B-sle srface aroxates the shae of the cotrol et Fgre shows a bcbc B-sle fcto htt://wwwofaercasceceorg 9 edtor@aercasceceorg

3 Joral of Aerca Scece ;9() htt://wwwofaercasceceorg Fgre A bcbc B-sle fcto Slar to the B-sle crve the B-sle srface s also a etwork of olyoal eces Each ece of the B-sle srface s a two desoally rereseted art of a srface or atch As wth a B- sle crve each atch of a B-sle srface ay be rereseted by a erodc relatosh rovded the kot sacg s for each drecto Ths s a for B-sle srface If the kot seeces are ot forly saced the the srface s o-for The bass fctos wold the have to be evalated by the recrsve relatosh The ofor atch Eato () ca be rereseted atrx for The ew Least Sares Method for Costrctg Crves ad Srfaces Gve a kot vector U { } r r the assocated B-sle fctos are defed as follows: < ( ) (4) otherwse ad (4) ( ) for ad r A B-sle crve wth cotrol ots s the defed as C( ) [ ] (4) whch 's are cotrol ots Wth data ots oe ca fd a terolato B-sle crve I ay case oe eeds to assg a locato araeter to each of the data ots defe a kot vector U ad fally cote the cotrol ots [8 4] The locato araeters ca be assged based o the chord legth as or by sg a cetretal ethod as The kot vector U ca be defed as r r ad A stadard terolato roble s to solve a lear syste C( ) Whe there are data ots e { } wth > the corresodg locato { } ad the kot vector U ca also be araeters derved fro the data ots { } a slar way Sose that the ew aroxato crve corresodg to U s C () the the least-sares ethod s to solve the ew cotrol ots by zg C( ) Usally the least-sares ethod rodces well-behaved reslts coared to those of the stadard terolato ethod bt t caot esre that the resltg crve exactly terolates the data ots { } [] 4 Costrctg the crve ad srefaces The seed crve s costrcted as a cbc Bezer crve ad ca be wrtte as htt://wwwofaercasceceorg 94 edtor@aercasceceorg

4 Joral of Aerca Scece ;9() htt://wwwofaercasceceorg A( ( ( t ( t t where { } are the cotrol ots Sose that the gve crve has two ed ots ad ad the corresodg taget vectors at the ed ots are t ad t resectvely Fro the taget costrat at the ed ots we have t t Whe the vales of ad are detered the corresodg cbc Bezer crve s the defed The geoetrc Herte ethods sch as the oe [7] ca be sed for deterg the vales of of ad The least-sares ethod ca also be sed whch s to ze A( C( t ( t t ) dt where t ad t are the araeters of ots ad o the gve crve C ( resectvely For the D case we also se the er ot terolato ethod whch s to select a er ot where the gve crve ad the aroxato crve are taget wth each other Sose that the er ot of the gve crve s ( x y ) ad t s the corresodg taget vector of ( t x t y ) the gve crve at Let A ( be ( X ( Y ( ) The we have X ( x Y ( y (5) X ( t y Y ( t x The eato syste (5) has three kow varables; e ad t ad three eatos as well The frst two eatos the eato syste (5) are lear wth resect to ad The ters ad ca the be drectly solved as ( ad ( Sbstttg ( ad ( to the thrd eato of the eato syste (5) we obta a varate eato t whch ca be slfed to a varate cbc olyoal eato H ( A bref overvew of related detals ca be fod Aedx By solvg H ( we fally obta the vales of t ad Ths the resltg aroxato cbc Bezer crve s also obtaed A stadard terolato roble s to solve a lear syste v ) The least-sares ethod s to solve the ew cotrol ots by zg v ) Usally the least-sares ethod rodces well-behaved reslts coared to those of the stadard terolato ethod bt t caot esre that the resltg crve exactly terolates the data ots } { 5 ercal Exales Exale By cosderg U { } V { } ad the data ots () () () () () () P () () () () () (4) we have a srface wth the shae were show Fgre Fgre Paraetrc aroxato htt://wwwofaercasceceorg 95 edtor@aercasceceorg

5 Joral of Aerca Scece ;9() htt://wwwofaercasceceorg The flatted shae s show Fgre : Fgre The flatted araetrc aroxato () Exale By cosderg U { } V { } ad () () () P () () () () () (4) we have a srface wth the shae were show Fgre 4 Fgre 4 Paraetrc aroxato Coclso I ths work we exted a ethod of aroxatg crves by least sares to cote a srface Ackowledgeet Ths work was sorted by Islac Azad Uversty Cetral Tehra Brach (IAUCTB) Refereces C Baa F Berard ad G X Atoatc recostrcto of srfaces ad scalar felds fro D scas Coter Grahcs (SIGGRAPH â 95 Proceedgs) (995) 9-8 X D Che W Maa ad J C Pa Cbc B- sle crve aroxato by crve clag Coter-Aded Desg 4 () 5-54 X Che W Ma ad J Yog J Pal Ier ot terolato ethod wth taget drecto costrat for laar crve aroxato Coter Aded Geoetrc Desg 4 B Crless ad M Levoy A voletrc ethod for bldg colex odels fro rage ages Coter Grahcs (SIGGRAPH â 96 Proceedgs) (996) - 5 C De Boor A Practcal Gde to Sles (ew York Srger-Verlag 978) 6 L Fag ad D Gossard Mltdesoal crve fttg to orgazed data ots by olear zato Coter-Aded Desg 7 () (995) K H o llg ad J Koch Geoetrc Herte terolato Coter Aded Geoetrc Desg (6) (995) J Hoschek ad D Lasser Fdaetals of coter aded geoetrc desg Lodo: AK Peters; 99 9 P Laret-Gegox ad M Mekhlef Otzato of a URBS reresetato Coter-Aded Desg 5 () (99) L Wag Pealzed least sares aroxato Itergrah desg docet T Lyche ad K Morke A data-redcto strategy for sles wth alcatos to the aroxato of fctos ad data IMA Joral of ercal Aalyss 8 (988) 85-8 D Meek B Og ad D Walto Costraed terolato wth ratoal cbcs Coter Aded Geoetrc Desg () () 5-75 H Park A error-boded aroxate ethod for reresetg laar crves B-sles Coter Aded Geoetrc Desg (5) (4) L Pegl Tller W The URBS book Secod ed ewyork Srger-Verlag 997 // htt://wwwofaercasceceorg 96 edtor@aercasceceorg