loft man s spline is a flexible strip of material, which can be clamped or weighted so it

Transcription

1 . Parametrc B-ple. Sple Cre Sple cre were frt e a a raftg tool for arcraft a hp blg tre. A loft ma ple a flexble trp of materal, whch ca be clampe or weghte o t wll pa throgh ay mber of pot wth mooth eformato. Lobachey etgate b-ple a early a the eteeth cetry; they were cotrcte a coolto of certa probablty trbto. I 96, Schoebrg e B-ple for tattcal ata moothg, a h paper tarte the moer theory of ple approxmato. Goro a Reefel formally troce B-ple to compter ae eg [Far97].. The B-ple ba The erlyg core of the B-ple t ba or ba fcto. The orgal efto of the B-ple ba fcto e the ea of e fferece a mathematcally Parametrc B-ple

2 ole. Carl e boor etablhe the early 97 a recre relatohp for the B- ple ba. By applyg the Lebz theorem, e boor wa able to ere the followg formla for B-ple ba fcto:.,, otherwe = th B-ple ba fcto of orer. = o-ecreag et of real mber alo calle a the ot eqece. = parameter arable. Th formla how that the B-ple ba fcto of a arbtrary egree ca be tably ealate a lear combato of ba fcto of a egree lower. The obo efg featre of the ba fcto the ot eqece. The ot eqece a et of o-ecreag real mber. The arable repreet the acte area of the real mber le that efe the B-ple ba. It tae + ot or teral to efe a ba fcto. Sce the ba fcto are bae o ot fferece, the hape of the ba fcto oly epeet o the ot pacg a ot pecfc ot ale. Parametrc B-ple

3 Aother tghg featre of the B-ple t ablty to hale cae where the ot ector cota cocet ot. Hag cocet ot, or forcg ot to be cocet, a mportat tep the cre ero proce orer to ere that the cre meet certa cotty crtera. The cre ero procere wll be ce more etal later. Fgre. how the relatohp betwee a cbc ba fcto a t ot eqece. Some of the properte of the B-ple ba fcto are: Fgre.: Cbc B-ple Ba Fcto [Far97]. Parametrc B-ple

4 The m of the B-ple ba fcto for ay parameter ale wth a pecfe teral alway eqal to ;.e.,. Each ba fcto greater or eqal to zero for all parameter ale. Each ba fcto ha oly oe maxmm ale.. The B-ple cre B-ple are pecewe polyomal of egree wth C - cotty at the commo pot betwee aacet egmet. B-ple relt by mappg the elemet of a ot eqece parametrc pace to Cartea pace. A ple ealate at a ot relt a cto pot whch the commo pot hare by two aacet egmet. B-ple are completely pecfe by the cre cotrol pot, the cre orer a the B-ple ba fcto a ee the Eqato.: -. = Pot alog the cre a a fcto of parameter = cotrol pot alo ow a the weght or the pot coeffcet. = th B-ple ba fcto of orer. Parametrc B-ple

5 Fgre.: A C Cbc B-ple cre wth t cotrol polygo [Far97]. Each pot o a B-ple a weghte combato of the local cotrol pot, whch form a cotrol polygo eclog the cre. The mber of B-ple ba fcto oboly eqal to the mber of cotrol pot a th mber the meo of the fcto pace. The mber of ot eee to efe th fcto pace eqal to the meo pl t orer. Parametrc B-ple 5

6 Parametrc B-ple 6 A metoe earler, a aatage of parametrc repreetato that t ge the cre coorate-ytem epeece. The B-ple ca th be repreete a a ector of Cartea ale whch are a fcto of the parametrc ba pace a how Eqato.. z z y y x x. B-ple ca be repreete wth repect to ther ot eqece a form or oform. A cre form f the ot pacg betwee all the ot the ame. If the cre form, the acte porto of all the ba fcto form the ame hape oer each teral. If each teral traforme to a teral betwee a, a peroc ba ca be e to ealate each cre egmet. Eqato.5 how a matrx relatohp that e to ealate each teral of a peroc cbc cre. = U M D.5 U = [ ] 6 6 / M D

7 Parametrc B-ple 7 U = moomal ba M = cotat eral traformato matrx D = cotrol pot ole the th teral. If the cre o-form, the ot pacg are oer the ot eqece. The preoly ce recre algorthm Eqato. th eceary to ealate the ba fcto. Eqato.6 how the matrx relatohp for each teral of a cbc o-form B-ple cre the flly expae form. = D.6 = parameter ale; = ot teral T S S R Q R Q P P ; P ; Q ; R ; S D

8 I orer to eterme the pacg betwee the aacet ot a ot ector, fferet parametrzato techqe are e. Parametrzato metho are crcal for the moellg of B-ple ce the pacg of the ot eqece flece the ba fcto a ce before. It amot to efg the legth of each parametrc teral, whch whe mappe to moelg pace, wll efe each cre egmet. There are three fferet metho commoly e to parametrze moel cre ata; form, chor legth a cetrpetal. Thee metho are ce below. Uform Th the mplet type of parametrzato where the ot pacg choe to be etcal for each teral. Typcally, ot ale are choe to be ccee teger a how Eqato.7..7 For may cae, howeer, th metho too mpltc a gore the geometry of the moel ata pot. Chor Legth Th parametrzato bae o the tace betwee the ata pot. The ot pacg proportoal to the tace betwee the ata pot. Eqato.8 reflect th relatohp. Th parametrzato more accrately reflect the geometry of the ata pot. Parametrc B-ple 8

9 Parametrc B-ple 9.8 = th oma ot = th ata pot = ot teral Cetrpetal Th parametrzato ere from a phycal aalogy. It ee to mooth ot arato the cetrpetal force actg o a pot moto alog the cre. Th reqre the ot eqece to be proportoal to the qare root of the tace betwee the ata pot a how Eqato.9. /.9 Other parametrzato metho hae bee etgate [Far97]. All thee metho hae certa crcmtatal aatage oer the other. There a trae-off betwee geometrcal repreetato a comptato tme. Typcally, chor legth parametrzato relt a ery goo comprome. I ay eet, each parametrzato relt a fferet hape of the cre. Parametrc cbc B-ple cre are e th reearch. Lower egree polyomal o ot proe ffcet cotrol of a cre hape, a hgher egree polyomal are comptatoally more cmberome a proe to mercal error.

10 Parametrc B-ple. B-ple Srface B-ple rface are a exteo of B-ple cre. The mot commo of a B- ple rface the teor proct rface. The rface ba fcto are proct of two arate cre bae. The rface a weghte m of rface two meoal ba fcto. The weght are a rectaglar array of cotrol pot. The followg Eqato. how a mathematcal ecrpto of the teor proct B-ple rface. m l,. where, otherwe,, l l l l l otherwe,,

11 , = B-ple rface a a fcto of two arable = cotrol pot = th ba fcto of orer a a fcto of l, = th ba fcto of orer l a a fcto of = Elemet of the ot eqece atfyg the relato, For mot compter ae eg prpoe, a the cae of the cre,, a ector fcto of two parametrc ale a. A mathematcal ecrpto of th relatohp how below Eqato.. m l x, x m l, y, y. m l z, z where x, y a z are coorate moel pace. The rectaglar array of cotrol pot form what calle a cotrol et. Smlar to the B-ple cre, the B-ple rface approxmate the hape of the cotrol et. Fgre. how a bcbc B-ple rface a the correpog cotrol et. Parametrc B-ple

12 Fgre.: A b-cbc B-ple rface a t cotrol et Smlar to the B-ple cre, the B-ple rface alo a etwor of polyomal pece. Each pece of the B-ple rface a two meoally repreete part of a rface or patch. A wth a B-ple cre, each patch of a B-ple rface may be repreete by a peroc relatohp proe the ot pacg form each recto. Th a Parametrc B-ple

13 Parametrc B-ple form B-ple rface. The bcbc cae ecrbe matrx form by Eqato.., = U M D M T V T. where, U = [ ] V = [ ] 6 6 / M,,,,,,,,,,,,,,,, D If the ot eqece are ot formly pace, the the rface o-form. The ba fcto wol the hae to be ealate by the recre relatohp. The oform patch Eqato ca be repreete matrx form. Eqato. how th relatohp for the bcbc cae the compacte form., o,,,,,,,,,,,,,,,,. A how the preo two eqato, the bcbc B-ple rface affecte locally by xtee cotrol pot. Geerally, a pot o ay B-ple rface eterme oly by a local bet of cotrol pot. Alo to be ote the fact that al

14 oparametrc cre o a B-ple rface are B-ple cre themele. For tace, for a rface efe by Eqato., le of cotat,.e., = f, are B-ple cre efe by the Eqato.: f,. f Here f are the cre cotrol pot. Sce th a parameter cre whch le o the rface, t ca be ee from Eqato. that A therefore m,.5 f m f.6 f f Th property e the the to efe oparametrc cre lyg aacet to a ormal to the trm boary for rface terrogato. Smlar to cre, the two ot ector reqre to ecrbe a rface hae to be eterme g oe of the parametrzato techqe ecrbe earler. Howeer, e to the fact that a B-ple rface a teor proct a cotrcte of a array of cotrol pot, there are a mber of pot tace for each al teral, of each ot eqece. The olto typcally to calclate a gle teral tace bae o the aerage of all of the pot tace. Parametrc B-ple

15 Parametrc B-ple 5.5 Dfferetal Geometry The motato for th topc ty to ecrbe the local cre a rface properte le cratre. The ma tool e for the eelopmet of the relt the local coorate ytem, term of whch geometrc properte are ealy ecrbe a te. A ce earler for the cae of a B-ple, a cre E ca be ecrbe parametrcally a z y x, [a,b] R.7 where t Cartea coorate x, y, z, are fferetable fcto of ee Fgre.. It ame that z y x, [a,b] R.8 where the ot eote erate wth repect to. Fgre.: Parametrc cre pace[far97].

16 A trocto of a pecal local co-orate ytem calle the Freet frame, le to a pot o the cre wll gfcatly facltate the ecrpto of the local cre properte at the pot. The frame or trhero ecrbe by three mtally perpeclar ector, t, a b whoe oretato are a trace ot the cre. The ector t calle a the taget ector, calle ma ormal ector, a b calle bormal ector. Fgre.5 epct the Freet frame. Fgre.5: The Freet Frame [Far97]. The mathematcal relatohp of the three ector wth repect to the pot ge by Eqato.9 t, b t, b,.9 where eote the cro proct. Parametrc B-ple 6

17 The plae pae by the pot a the two ector t, calle the oclatg plae O. It eqato ge by r et = r -,, et =, where r eote ay pot o O. It parametrc form O, =. Fgre.6 expla the relato betwee the oclatg plae a the three mtally perpeclar ector a two other plae calle the ormal plae a the rectfyg plae. The ormal a the rectfyg plae are perpeclar to each other a each of them perpeclar to the oclatg plae tr. The ormal a the bormal ector eterme the ormal plae. It that plae whch pae throgh a pot o the cre a perpeclar to the taget le to the cre at that pot. The taget a the bormal ector eterme the rectfyg plae. Lettg the Freet frame ary wth proe a goo ea of the cre behaor pace. The rate at whch the Freet frame moe wth repect to the parameter ge a meare of the Cratre a the Toro of the cre. Cratre the rate of trg of the taget ector a ge by the relato = =. Parametrc B-ple 7

18 Fgre.6: Relato betwee the ormal plae, oclatg plae a rectfyg plae [Far97]. Toro J a meare of the amot of rotato of the oclatg plae. I other wor toro a qatty, whch cate whether the cre twtg raply or lowly. It expree mathematcally a et[,, ] J = J =. Parametrc B-ple 8

19 The cratre, toro a the three Freet frame ector t,, b are relate by the followg et of formla calle the Freet-Serret formla Eqato.. Fgre.7 lltrate thee formla t = +, = - t +J b,. b = -J A pot o a cre where = calle a pot of flecto. Sce th pot ffclt to locate mot practcal cae, aother meare calle the cratre mma employe fg a pot at whch the cratre cloe to zero a th pot ca be coere to be the flecto pot, for all practcal prpoe. Fgre.7: The geometrc meag of the Freet-Serret formla [Far97] Parametrc B-ple 9

20 A cre a to hae cratre mma at a pot where the followg coto hol tre: - - < + -. Smlarly, a cre a to hae cratre maxma at a pot where the followg hol tre: - - > Cotrat-bae B-ple Iero B-ple theory wa orgally eelope to efe a cre or a rface that approxmate a et of ata pot. Howeer, a eger ofte more terete g B-ple for terpolatg a et of ata pot rather tha approxmatg them. The ero metho aree th e by fg the cotrol pot for a B-ple cre or a rface ge a et of ata pot to be terpolate. Th proce typcally cot of ettg p a olg a ytem of lear eqato that are bae o terpolato coto that are reqre by the moel ata. Yamagch ge a olto for the ero of form cbc ple [Yama88] whle Gloama [Glo89] exte th to a o-form cae. Flemg eelope a approach to corporate e cotrat to the et of ata pot to ge the cre or rface a ere leel of cotty [Flem9a] [Flem9b]. Parametrc B-ple

21 Parametrc B-ple For a cbc cre, the ero proce typcally cot of ettg p a olg for + cotrol pot o terpolatg ata pot. eqato ca be geerate bae o thee ata pot cotrat. Th accomplhe throgh a owlege of the behaor of B-ple ba fcto. A preoly ecrbe, the ba fcto are oly locally o zero a ay cbc B-ple teral effecte by, at mot, for ba fcto. At each ot ale, howeer, oly three ba fcto are o-zero. Therefore, the ale of each ata pot ca be repreete by the followg eqato. - -, =...6 Th expreo, howeer, lacg two coto eee to ole for the + cotrol pot. Thee coto, ow a the cre e coto, typcally are expree term of a zero cratre or a tagecy coto. Hece, the cotrol pot ca be fo by olg the followg ytem of eqato a how Eqato.7: S = D.7 coto e coto e e e

22 where, S = Data pot = ba fcto ale at the ot D = cotrol pot Srface ero lghtly more complex tha cre ero a ole olg for the rectaglar array of cotrol pot that terpolate to moel ata pot wth repect to two fferet bae. Th proce ca be ecrbe by the followg Eqato.8: S = D T.8 where S repreet the array of ata pot a e coto both recto. repreet the ba fcto matrx of the form Eqato.7 a a fcto of. repreet the ba fcto matrx wth repect to. D the array of cotrol pot to be ole for. I Flemg approach to b-ple ero, the cocept of ot mltplcty employe. Kot mltplcty amot to ag extra ot at pot where the cotty ha to be chage. For tace, orer to ere taget cotty, oe extra ot reqre at the pecfe pot. If tea a harp corer ee to be moele, the reqre ot mltplcty hol be three two extra ot hol be ae to proe wth pot or C cotty at that corer. Parametrc B-ple

23 Each tme a ot ae, the cotty at that pot rece. Howeer, the meo of the cre chage a a ew ba fcto create. Se a ew ba fcto troce, the mber of cotrol pot to be ole for alo creae whch tr creae the ee for a extra cotrat orer to ole for the ew cotrol pot the ero proce. Thee extra cotrat ca be proe term of tagecy coto or taget ector. For each mltple ot ae to the ot eqece, a taget ector cotrat ca be ae to the lear ytem orer to ole for the extra cotrol pot. Th, for C cotty, oe taget ector reqre, wherea for C cotty, two taget ector are reqre; oe o ether e of the ata pot repreetg the abrpt chage tagecy. Fgre.8 how the effect ag mltple ot ha o the B-ple ba. Whe g cotty cotrat, a ere of tep hae to tae place. Frt, the ata pot are parametrze wth oe of the aalable techqe form, chor legth, etc.. ext, ot mltplcty ae whereer cotty cotrat are reqre. Oce a ew et of ot efe, all ba fcto are calclate o that lear ytem clg cotty cotrat ca be aemble a ole. Parametrc B-ple

24 Fgre.8: Effect of ag mltple ot o the B-ple ba [Flem9a]. Parametrc B-ple