Interpolating Solid Orientations with a C -Continuous B-Spline Quaternion Curve

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1 Iterpoltg Sold Orettos wth C -Cotos B-Sple Qtero Crve Webg Ge *, Zhg Hg, d Gopg Wg Dept. of Compter Scece d Techology, eg Uversty, Beg, Ch h: dgewb@grphcs.p.ed.c {dgewb,hz,wgp}@grphcs.p.ed.c Abstrct. A lgorthm s preseted to costrct C -cotos B-sple qtero crve whch terpoltes gve seqece of t qteros o the rotto grop SO. We preset method to eted B-sple terpolto crve to SO. The problem s essetlly to fd the qtero cotrol pots of the qtero B-sple terpolto crve. Althogh the ssocted costrt eqto s o-ler, we c get the ccrte qtero cotrol pots ccordg to two ddtol rles for qtero compttos S. I ddto, we provde pot serto method to costrct terpolto crves tht hve locl modfcto property. The effectveess of the lgorthm s verfed by pplyg t to some emples. Keywords: C -cotos, terpolto, qtero, B-sple crve, compter mto. Itrodcto Qteros hve bee trodced s sefl represetto for terpoltg D oretto compter mto; especlly the t qteros re most stble for cotrollg the rotto for both obect models d vrtl cmers []. The t qteros hve my dvtges sch s they re free from sglrtes; d they re compttolly more effcet th the mtr represetto of D rotto [8]. Ths, the qesto of how to costrct t qtero crves plys mportt role compter mto [-, 8-]. I compter mto, t s fdmetl problem to geerte smooth moto for rgd body so tht the geerted moto terpoltes gve seqece of eyfrme postos d orettos [8]. The rgd moto c be represeted s cotos trsltol d rottol motos. The rotto grop SO s obted s proectve spce of the t qtero spce S. For the terpolto of ey frme postos R, there re my well-ow techqes vlble sch s B- sple d Bezer. However, t s rther dffclt to eted them to S. Shoeme [] provded scheme for dervg Bezer cotrol pots from seqece of qteros. Ths provdes terpoltg sple for qteros, bt the costrcto s somewht more complcted. Nelso [] presets ew method to costrct * Correspodg thor. K.-c. H et l. Eds.: Edtmet 7, LNCS 9, pp. 5, 7. Sprger-Verlg Berl Hedelberg 7

2 Iterpoltg Sold Orettos wth C -Cotos B-sple Qtero Crve 7 qtero w-sple crve. Lee d Sh [] preset ew scheme tht pples flter ms or covolto flter to oretto dt. Bt my of commo propertes R geometrc modelg bre dow eve the smple o-eclde spce S, for emple, the de Cstel type costrcto of cbc B-sple qtero crves does ot preserve C -cotty [8]. There re lso my other methods to costrct crves S, bt most of the crves re C cotos qtero crves [,,]. Becse of the dscotty secod dervtves, there my occr lrge glr ccelertos t the crve ots, whch hve desrble effects o geertg trlly-loog rottos, ths, hgh degree cotty s lso mportt fctor for the qtero crves compter mto [8]. I ths pper, we preset oe ew pproch to costrct the C cotos B-sple qtero crve by sg of the ddtol qtero opertors S. Km etc. [9] preseted geerl crve costrcto scheme tht eteds sple crve R represeted by bss form to smlr oe S. Bt whe costrctg terpolto qtero crve [8], they do t provde method to compte the ccrte cotrol pots becse of the o-lerty of the eqto. Isted they ppromte the cotrol pots. Wth or method, we c get the ccrte cotrol pots. I ddto, we provde pot serto method to costrct terpolto crves tht hve locl modfcto property. The rest of ths pper s orgzed s follows. I secto, we revew the prevos costrcto method of B-sple qtero crves. Secto descrbes or lgorthm for compte the cotrol pots of t qtero crves sg proposed rles. Secto demostrted some epermetl reslts. Flly, we coclde secto 5. relmres Gve cotrol pots {p },,, the B-sple crve t of order s defed by: t p B t where B t s the bss fcto of B-sple of order, d the correspodg ot vector s T{t,, t - }. The B-sple crve c be reformlted s follows: ~ ~ t p B t p p B t where B t f t < t < t ~ B t B t f t t f t t Km etc. [9] preseted geerl crve costrcto scheme tht eteds sple crve R represeted by bss form to smlr oe S. By replcg the

3 8 W. Ge, Z. Hg, d G. Wg crve t wth the qtero crve Qt, d the cotrol pot p wth the qtero cotrol pot q, d the smmtos wth the qtero mltplctos, the correspodg B-sple qtero crve wth cmltve bss form s formlted s follows: t B t B q q q Q t ~ ~ The B-sple qtero crve s C - cotos d loclly cotrollble by movg the cotrol pots {q } [9]. Bsed o the bove eqto, Km etc. [8] preseted method tht c eted B- sple terpolto crve to smlr oe S. Gve seqece of dt pots,,, the pot terpolto c be doe by costrctg form cbc B- sple crve t whch terpoltes ech pot t t. The cbc B-sple t wth cotrol pots p -,,,, s defed by: t B p t 5 The reltoshp forms system of ler eqtos: p p p, for,,. Wth proper bodry codtos for p - d p, there re eqtos for ows. It s esy to compte the ect solto R. Bt whe eteded to B- sple qtero crve, the system of eqtos s o-ler. Km [8] does t offer method to compte the ect solto. Isted they se tertve method to ppromte the solto. I the et secto, we trodce ew method to solve these eqtos d get the ect solto. Qtero Sple Iterpolto. Locl B-Sple Iterpolto As metoed eqtos secto, there re eqtos for ows wth proper bodry codtos for - d : 7

4 Iterpoltg Sold Orettos wth C -Cotos B-sple Qtero Crve 9 Where,,, re the gve pots to be terpolted d,,, re the cotrol pots we wt to compte. Sce the system s strctly dgolly domt, there ests qe solto. Bt we c see tht the terpolto B- sple crve does t hve the locl modfcto property: f we chge y gve pot, ll the cotrol pots wll be chged. So we sggest tht ddtol cotrol pots shold be terpolted t every three or for pots. These ddtol cotrol pots wll gve the qtero terpolto crve locl modfcto property. Also t wll smplfy the comptto of the bove eqtos. We trodce oe method to prodce sch ddtol pots. We se -pot method [] whch s cosdered s clssc method the crve geerto. The - pot scheme wors by repeted refemet of S{, Z}, ccordg to the rle, We st select the secod rle to get ddtol pots, Tht s, ddtol cotrol pots t every three cotrol pots s defed by: 9 Now, the comptto of ech cotrol pot st relted to the three gve pots d two ddtol pots: We c get the solto: , 7, From the bove solto, we c see tht f we chge,,, t wll ffect the cotrol pot,, d or -. The or - wll ffect the cotrol pots or -,,. So oe gve pot wll totlly ffect pots rod t. Net, we wll pply or bove method to costrct the B-sple qtero terpolto crves. We trodce two ddtol comptto rles S ccordg to R.

5 W. Ge, Z. Hg, d G. Wg. The Rles S Accordg to R The cotrol polygo rle: the edge of cotrol polygo S s the gret feror crcle rc betwee them. Tht rc s the crve where the sphere tersects ple throgh the two pots d the org. The ddto rle: s θ s θ q q q q, sθ sθ where cosθ q q 5 The sbtrcto rle: s θ q Θq q cosθ q q cosθ sθ The eqto s solto rle: the solto to the eqto q - q s: s θ q cos θ q q cosθ 7 sθ The rle d rle hve bee descrbed by Shoeme[]. The ddto rle s st forml for sphercl ler terpolto from q to q. The sphercl ler terpolto s the gle terpolto betwee q d q : q X q q q q - Fg.. The comptto of q - q Yo c get sch eqto: q q cos α s α 8 q cos α q s α It c wor ot q s follows: s α s α q q q 9 sα sα where cos q gq. Also we c descrbe the sbtrcto rle d get the eqto s solto the sme wy s follows

6 Iterpoltg Sold Orettos wth C -Cotos B-sple Qtero Crve Sbtrcto rle: q X q q q q Fg.. The comptto of q X q q q cos α s α q cosα q sα It c wor ot q s follows: s α q q cosα q q cosα sα where cos q gq. Now we defe the eqto s solto rle: q - q q q /- Fg.. The solto of eqto: q - q q q cosα sα q cos α s α It c wor ot s follows: s α q cos α q q cosα sα where cos q gq. By the bove for rles, we c compte the solto of the followg eqto, thogh t s o-ler:

7 W. Ge, Z. Hg, d G. Wg 5 s compted sg -pot scheme: 8 Θ. 7 For operto hs Commttve lw d Assoctve lw: : 5:. We c tret s oe qtero, d wor ot : Θ Θ 8 The, we c compte from d from. From the cotrol pots { }d { }, we c get the B-sple qtero terpolto crve se forml. The crve s C cotos s show Fgre, whch s proved by Km etc. [9]. Fg.. Qtero terpolto: the yellow crve s polyle S, whle the red crve s B- sple qtero terpolto crve Epermetl Reslts From the bove reslts, we c pply the B-sple qtero terpolto crve to prodce the sweep moto d sweep sold. By dptg the costrcto method

8 Iterpoltg Sold Orettos wth C -Cotos B-sple Qtero Crve desged by [9] for B-sple qtero crves, the C -cotty s grteed whch s sefl the costrcto of the sweep srfce/sold. Emple : gve seqece of obect orettos show s the Fgre. We get the sweep sold s show Fgre b: sweepg wth cbc B-sple qtero crve: the oretto of obects b the sweep sold by the sweep moto Fg. 5. Emple Emple : gve seqece of obect orettos s Fgre5, we get the reslt show s b: sweepg wth cbc B-sple qtero crve: the oretto of obects Fg.. Emple

9 W. Ge, Z. Hg, d G. Wg 5 Coclso b the sweep sold by the sweep moto Fg.. Coted I ths pper, we focs o the B-sple terpolto problem: how to fd the B- sple cotrol pots so tht the B-sple qtero crve terpoltes gve seqece of t qteros. We trodce two ddtol rles for the comptto of qtero S : the sbtrcto rle d the eqto s solto rle. By the for rles, we c compte the cotrol pots ccrtely S. Frthermore, we preset method to prodce the locl modfcto property for the B-sple qtero terpolto crve. Here, there re some sses we shold cosder the ftre, we hd sed the - pot scheme to prodce ew cotrol pot, bt we c lso se the -pot scheme or other method to prodce the ew cotrol pot. I geerl, these terpolted qtero B-sple crves re dfferet by sg of dfferet scheme. Therefore, we wll cote to reserch ther reltoshp d some codtos. Acowledgemets Ths reserch ws spported by Grt No CB79 from The Ntol Bsc Reserch rogrm of Ch97 rogrm, Grt No. 7 d Grt No. 575 from Ntol Ntrl Scece Fodto of Ch, Grt No from Oce 98 Specl rogrm. Grt 959 from Ch ostdoctorl Scece Fodto, Also ws spported by Grt No. AAZ from The Ntol Hgh Techology Reserch d Developmet rogrm of Ch8 rogrm Refereces [] A.Brr, B.Crr, S.Gbrl, d J.Hghes. Smooth terpolto of orettos wth glr velocty costrts sg qteros. I Compter Grphcs roc. of SIGGRAH'9, pges-, 99.

10 Iterpoltg Sold Orettos wth C -Cotos B-sple Qtero Crve 5 [] Dy,N., Lev,D., Gregory,J., -pot terpoltory sbdvso scheme for crve desg. Compter Aded Geometrc Desg, pges: 57-8, 988 [] Gregory M. Nelso. v-qtero Sples for the Smooth Iterpolto of Orettos. IEEE Trsctos o Vslzto d Compter Grphcs Volme, Isse,. [] J.Lee d S.Y.Sh. Geerl Costrcto of Tme-Dom Flters for Oretto Dt. IEEE Trsctos o Vslzto d Compter Grphcs Volme 8, Isse,. [5] J.Schlg. Usg geometrc costrctos to terpolte oretto wth qteros. Grphcs GEMS II, pges 77-8, Acdemc ress, 99. [] K.Shoeme. Amtg rotto wth qtero crves. Compter Grphcs roc. of SIGGRAH'85, pges 5-5,985. [7] Kero T.Mr. Ut qtero tegrl crve: A ew type of fr free-form crves. Competer AIded Geometrc Desg 7, pges:9-58,. [8] M.J.Km d M.S.Km. C-cotos B-sple Qtero Crve Iterpoltg Gve Seqece of Sold Orettos. Compter Amto '95., roceedgs, 995. [9] M.J.Km, M.S.Km, d S.Y.Sh. A Geerl Costrcto Scheme for Ut Qtero Crves wth Smple Hgh Order Dervtves. Compter Grphcs roceedgs of SIGGRAH 95,9: 9-7, Agst 995. [] M.S.Km d K.W.Nm. Iterpolto sold orettos wth crclr bledg qtero crves. Compter-Aded Desg,75:85-98,995. [] Rv Rmmoorth d Al H. Brr. Fst costrcto of ccrte qtero sples. roceedgs of the th l coferece o Compter grphcs d terctve techqes, ges: 87 9, Yer of blcto: 997. [] Smel R. Bss d Jy. Fllmore. Sphercl verges d pplctos to sphercl sples d terpolto. ACM Trsctos o Grphcs Volme, Isse, [] W.Wg d B.Joe. Oretto terpolto qtero spce sg sphercl brcs. I roc. Of Grphcs Iterfce 9, pges -, 99.