Sliding-Windows Algorithm for B-spline Multiplication

Transcription

1 Slng-Wnows Algorthm for B-splne Multplcaton Xanmng Chen School of Computng Unversty of Utah Rchar F. Resenfel School of Computng Unversty of Utah Elane Cohen School of Computng Unversty of Utah ABSTRACT In aton to B-splne multplcaton, that s, fnng the coeffcents of the prouct B-splne of two other B-splnes, beng a symbolc operaton tself, t s an mportant prerequste to many other symbolc computaton operatons on B-splnes. Most algorthms for B-splne multplcaton use nrect approaches such as noal nterpolaton or computng the prouct of each set of polynomal peces usng varous bases. The orgnal rect approach s complex, an whle B-splne blossomng proves another rect approach that can be straghtforwarly translate from mathematcal equaton to mplementaton, the algorthm oes not scale well wth egree or menson of the nvolve tensor prouct B-splnes. In ths paper, we present a new blossomng base algorthm to B-splne multplcaton to aress the above efcences, the slng wnows algorthm. Categores an Subect Descrptors H.4 [Informaton Systems Applcatons]: Mscellaneous; D..8 [Software Engneerng]: Metrcs complexty measures, performance measures General Terms Delph theory Keywors NURBS multplcaton, Slng-Wnow Algorthm, Blossomng. INTRODUCTION In aton to B-splne multplcaton, that s, fnng the coeffcents of the prouct B-splne of two other B-splnes, (Prouces the permsson block, copyrght nformaton an page numberng. For use wth ACM PROC ARTICLE- SP.CLS V.6SP. Supporte by ACM. Contact author. beng a symbolc operaton tself, t s an mportant prerequste to many other symbolc computaton operatons on B-splnes. There have been several theoretcally base rect algorthms an several nrect approaches that have been propose to performng ths symbolc computaton. The frst theoretcally proven constructve algorthm for rect B-splne multplcaton was presente by Morken n [9] usng the screte B-splne representaton. Elber an Cohen [0] use the prouct B-splne representaton to symbolcally query an analyze secon orer fferental surface propertes. They use Elber s multplcaton approach [8] base on samplng the prouct by samplng each multpler B-splne an nrectly formng the prouct B-splne usng noal nterpolaton. Uea [7] reporte a rect approach for B-splne multplcaton base on a blossom representaton of B-splnes, an prove ts equvalence to Morken s earler screte B-splne approach. However, E. Lee [8], who was also aware of the straghtforwar blossom-base rect approach at the tme, observe that the neffcency of the blossom-base rect approach renere t mpractcal for CAD systems; nstea, he propose a blossom-base nrect approach that converts B- splne bass representaton to a power bass representaton, performs multplcaton convolng coeffcents, an then converts back to B-splne bass representaton va the e Boor- Fx formula [5]. As the whole process, especally the thr stage of back-converson, s computatonally expensve, the author evelope a new scheme to evaluate the coeffcents of the prouct B-splne a group at a tme by computng a chan of blossoms. Pegl an Tller [0], explotng the well-known result on Bézer multplcaton [4, 4], prove another nrect approach that converts B-splnes va knot nserton to composte Bézers, performs Bézer multplcaton, an then converts back to B-splnes va knot removal. Of varous approaches for B-splne multplcaton, Uea s blossom-base approach oes not perform bass converson, an only convex affne combnaton s use n the process of constructng new prouct B-splne coeffcents. Whle etale comparson to other approaches on numercal accuracy an numercal stablty, to our best knowlege, has not yet been reporte so far, t seems that blossom base rect approach at least lacks the uncertanty on these two mportant numercal ssues when splne bass converson or extrapolaton s performe n some stage of the multplcaton algorthm. Unfortunately, the approach s not very

2 effcent whch s a maor rawback as the egree an menson ncrease. Several authors (for example [7] have observe that brute force mplementaton of many blossom-base B-splne algorthms suffers from severe neffcency f the nvolve recursve blossom evaluaton has a combnatoral characterstcs, whch s exactly the case for B-splne multplcaton. Usng some kn of assocate table to look up an reuse prevous partal result of recursve blossom evaluaton s an obvous speeup strategy, as was also note n [7]. However, partal result reuse only won t help much for B-splne multplcaton nvolvng hgh egrees an especally hgh mensons, whch s becomng more an more common n recent years. Ths current tren of usng hgh mensonal tensor prouct B- splnes s mostly ue to the earler ntroucton of robust B-splne subvson base ratonal constrant solvers [6, 3] that enable one fnally to solve many complex geometry problems. For examples, 4-mensonal B-splne multplcaton s carre out for the computaton of varous geometrc enttes nclung, bsector surfaces [], b-tangent curves an flecnoal curves [9], accessble regons for 5-axs machnng [, 3], offset surface self-ntersecton [4], an perspectve slhouette of a general swept volume [5] etc. 5- mensonal B-splne multplcaton s requre n the trackng of eformng surface/surface ntersecton [], an even 7- mensonal B-splne multplcaton has to be performe to fn the trple-pont sngularty of eformng surface/surface ntersecton []. In ths paper, we present a smple an effcent algorthm, calle slng-wnows algorthm, for blossomng-base B- splne multplcaton. Prelmnary comparson shows a ramatc speeup compare to the brute force mplementaton of blossom-base rect approach wth partal result reuse only, whch s not a surprse at all because of the reucton of combnatoral complexty ntrouce n ths paper. The more mportant task, though, s to compare the presente algorthm to other tratonal nrect approaches on both effcency an numercal stablty ssues, whch wll be scusse n a comng report. In ths paper, we focus only on llustratng the etals of the slng-wnows algorthm. The rest of the paper s organze as follow. After a bref revew on the basc prncples of multplyng two B-splnes va blossomng n Secton, Secton 3 presents a much better formulaton of the blossom representaton of prouct B-splne. Secton 4 evelops an ncremental algorthm of knot subsequence enumeraton. After a bref revew on general n-mensonal B-splne multplcaton n Secton 5, Secton 6 constructs, for the two factor B-splnes, a par of n-mensonal array of blossom values,.e. the so-calle blossom meshes. A par of wnows are constructe, n Secton 7, to nex nto a par of sub meshes of the blossom meshes that s use to compute a corresponng control pont of the prouct B-splne, an the entre control ponts are compute by slng ths wnow par over the entre blossom meshes. After a bref scusson on the slng wnow szes an ther rrelevant elements n Secton??, the paper fnally conclues n Secton 9.. B-SPLINE PRODUCT VIA BLOSSOMING Ths secton revews the basc blossomng prncples use for multplcaton of two B-splne functons as note n [8] an reporte n etals n [7]. General ntroucton on blossom can be foun n [,, 6, 3, 6]. Gven an unvarte B-splne functon G of egree, efne by, a knot vector v,, v, v,, v,, v r,, v r, v r,, v r {z } {z } {z } {z } n = 0<n + 0<n r + n r= where v, v,, v r are stnct knot values n ascenng orer ; an a seres of control ponts, whch are exactly the blossom g of G evaluate at a seres of knot sequences, ( g(v,, v, g(v,, v, v,, g(v r,, v r ( {z } {z } {z } Each knot sequence above has a length of an s calle a -knot-sequence, or smply -sequence or even abbrevate as -seq n ths paper. The -sequences appeare n ( actually are qute specal n the sense that they take consecutve knots from the knot vector (, an ther corresponng blossom values are exactly the control ponts. Ths s calle ual functonal property of the B-splne blossomng. These specal -sequence s calle, n ths paper, to be ual to ther corresponng control ponts, or smply ual -sequence. Dual -sequences have a natural lnear orer, wth two neghborng -sequences shft one poston w.r.t. the corresponng knot vector (. Now, let us conser another B-splne functon b G (wth blossom bg of egree b, efne smlarly wth knot vector, bv,, bv, bv,, bv,, bv s,, bv s, bv s,, bv s {z } {z } {z } {z } bn = b 0<bn b + 0<bn s b + bn s= b an control ponts (.e., blossom bg evaluate at ual b - sequences, (3 bg(bv,, bv, bg(bv,, bv, bv,, bg(bv s,, bv s (4 {z } {z } {z } b b b Assumng v = bv = u an v r = bv s = u t, the prouct of G an b G s another B-splne functon F of egree D = + b, wth knot vector u,, u, u,, u,, u t,, u t, u t,, u t, (5 {z } {z } {z } {z } m =D 0<m D+ 0<m t D+ m t =D where, m = n + bn = m t = n r + bn s = + b = D, an for 0 < < t, we have the followng table for computng the multplcty of the prouct knot vector. The prouct knot vector s erve smply base on contnuty conseraton. On the other han, by ual functonal property, the control ponts of prouct B-splne s agan the The nternal + multplcty s allowe for representaton of C B-splnes. The en multplcty s, snce the extra knot, f + were use, has no effect on any algorthms on B-splne [4].

3 m = n + b m = bn k + 3 m = max(n + b, bn k + f u = v for some but s absent from b G s knot vector f u = bv k for some k but s absent from G s knot vector f u = v = bv k for some an k Table : Multplcty of Prouct Knot Vector blossom f of F evaluate at ther ual (D-sequences, that s, all sequences of D consecutve knots from the prouct knot vector (5. For any constructve B-splne algorthms, no close form of f s requre, so long as we can evaluate t at these ual (D-sequences. Conserng the symmetrc an mult-affne property of the blossom f, ths can be acheve by symmetrzng va subset enumeraton; specfcally, for any ual (D-sequence k k k D (Eq.(7 of [7] an Eq.( of [8] f(k, k,, k D = X g(k, k,, k b bg(k, k,, k b {,,, } + b, (6 where {,,, } s any -subset of {,,, D, D = + b }, an {,,, b } s the complmentary b -subset. In the equaton, a sngle evaluaton of the blossom f at a ual (D-sequence of the prouct B-splne, s expane to `D evaluatons of blossom g of the frst factor B-splne at -subsequences, an blossom bg of the secon factor B-splne at b -subsequences. The -subsequence an b -subsequence are such name because they are subsets of the orgnal ual (D-sequence from the prouct knot vector. Notce that, on the other han, they are smply -sequence an b -sequence that conssts of consecutve knots from some refne knot vectors of the two factor B-splnes G an b G, respectvely. In another wors, wth respect to the orgnal knot vector of the corresponng factor B-splne, they are generally not ual sequences, an, n ths paper, are usually explctly calle factor sequences to reflect the fact that they are sub sequences of ual sequences from the prouct knot vector an are use as oman value for the evaluaton of the blossom functon of the corresponng factor B-splne. Furthermore, any blossom value of g at a factor -sequence n the summaton can be expane as some convex affne combnaton of blossom values at another two -sequences, an the process s performe recursvely untl the consere -sequence conssts of consecutve knots from G s orgnal knot vector, that s, untl the consere -sequence s a ual -sequence that s ual to some orgnal control pont of G. Exactly the same proceure s apple on the recursve evaluaton of blossom bg. What s ust escrbe n the last paragraph, plus a strategy of partal result reuse wth table look-up, s bascally Uea s algorthm for B-splne multplcaton. However, ue to ts combnatoral characterstcs, ths smple mplementaton s very neffcent as rghtly observe by Lee who also presente an alternatve nrect B-splne multplcaton algorthm [8] at the same year. Actually, t s a qute common observaton (see for example [7] that, whle varous B-splne formulaton va blossom representaton are elegant, any smple an nave mplementaton woul typcally result n algorthms of hgh complexty. For any practcal CAD system nvolvng hgh egree an especally hgh menson, such a smple mplementaton of blossom-base B-splne multplcaton approach woul most lkely be one of the obstacles to achevng nteractve performance. 3. BETTER FORMULATION OF B-SPLINE MULTIPLICATION IN BLOSSOM REP- RESENTATION In ths secton, we prove a much better formulaton of the blossom representaton of prouct B-splnes; the reformulaton s almost trval mathematcally yet t s very sgnfcant from the perspectve of computer mplementaton, an, to our best knowlege, t has never before been observe an use n the specfc context for B-splne multplcaton. 3. Prouct B-splne Represente n Subsets of Mult-sets Recall that t s not uncommon for a non-escenng knot vector to have nternal multplcty greater than, resultng contnuty less than the maxmal possble (.e. the egree mnus. Ths s especally true for B-splne multplcaton (cf. Table, where each knot n the prouct knot vector has ts multplcty rase by the egree of ether of the two factor B-splnes. Therefore, n Eq. (6, we are actually enumeratng all subsets of a mult-set. Usng a superscrpt m of a knot value u to enote the same knot value u repeate m tmes, prouct knot vector (5 can be re-wrtten as u m, u m,, u ms s, (7 an, Eq. (6 can be reformulate by enumeratng all the factor sequences as subsets of the gven ual knot sequence as a mult-set as; specfcally f(u n, u m + +,, um, un = X w g(u λ, u λ + +,, uλ bg(ub λ λ, u b + +,, ub λ + b, (8 where the summaton takes over all {λ,, λ }, an for l =,,, an w = Y λ l 0, b λ l 0 λ l + λ b l = n l f l = or, or m l otherwse X λ l =, X bλ l = b l l l λ l + b λ l λ l = n λ Y <l< m l λ l n λ (9 3

4 Notce that the ual D-sequence, as a consecutve knot sequence from the prouct knot vector (7, oes not necessarly have maxmal multplcty at ts two en ponts,.e., 0 < n m an 0 < n m 3. Reucton of Combnatoral Complexty The total -subset of a set of carnalty D s + b = D = Db wth lower bouns of [7], «D an D That s, f navely usng Eq. (6 rectly, we woul en up wth evaluatng at least max (D/, (D/ b b factor sequences for computaton of each control pont of the prouct B-splne, not mentonng that more knot sequences have to be ealt wth when each factor -sequence or -sequence b n the summaton s recursvely evaluate. On the other han, by Eq. (8, the total stnct -sequence (an the complmentary b -sequence of a ual D-sequence wll be ramatcally reuce, wth the upper boun to be constant or lnear wth respect to the prouct B-splne egree for 3 most common cases as lste n Table. Appenx. 8 also shows that the average stnct -sequences of a ual D-sequence s of the orer O(D. D-Sequence case u D b «b Pars of Factor Sequences case u n v D n mn(+, b +, +n, +D n case 3 u n v D σ w σ n ( + n( + σ n ( + σ Table : Number of Pars of Factor Sequences Shown are number of stnct pars of factor -sequence an factor b -sequence of a D-sequence n ts 3 typcal forms; where D,, b are egrees of the prouct B-splne, the frst an secon factor B-splnes, respectvely; where σ s the contnuty at the break pont v; an where 0 < n < D for case, an 0 < n < σ for case 3. For example, let us conser a egree 3 factor B-splne an a egree 8 factor B-splne multplyng nto a egree prouct B-splne. A -sequence of u 9 v, has factor 3-sequences of (u 3, u v, u v, an corresponng factor 8-sequences of (u 8, u 7 v, u 6 v, that s, a total of mn( +, b +, + n, + D n = mn(4, 9, 0, 3 = 3 stnct pars of factor sequences. Notce that, whle countng all the factor sequences, wth fxe length an b respectvely, we can gnore any one knot an countng all the possble multplctes of other knots. For case, knot multplcty of factor -sequence cannot excee any of +, + n, + D n, whle that of factor b-sequence cannot excee any of b +, +n, +D n, hence the maxmal number of stnct pars as shown n the table. For case 3, f v exsts n both factor B-splne knot vectors, then σ < an σ < b, an consequently both ( + n an ( + σ n are less than + an b + ; then, the stnct pars of case 3 s exactly ( + n( + σ n whch has a maxmum as shown n the table at n = σ. To unerstan why a D-sequence of a egree D prouct B- splne typcally cannot have 4 or more stnct knot values as shown n the table, we nee a bref revew of the followng characterstcs of NURBS symbolc computaton[5, 8, 7, 0, 3],. Prouct B-splne has a egree that s the sum of the egrees of the factor B-splnes; an ratonal B-splne fferentaton oubles the egree.. The contnuty of a resultng B-splne from any NURBS symbolc operatons, wll never ncrease an wll ecrease by for fferentaton. 3. The stnct knot set of a resultng B-splne from any NURBS symbolc operatons s the sum of the stnct knot sets of all the operans. 4. Although t can also be use for moelng, for example constructng a surface from a B-splne curve an a B-splne offset vector, NURBS symbolc computaton typcally fns ts play at the later stages of a CAD applcaton for example, of the analyss an verfcaton of the CAD moel where the egrees coul be easly n the 0s n contrast to the typcally cubc when the moelng s ntally starte, an where the menson coul be well beyon n contrast to -mensonal curves an -mensonal surfaces when the moelng s starte. Ths s exactly where effcency of NURBS computaton, especally the most funamental NURBS multplcaton, s of great concern, an ths paper takes ths as the typcal stuaton of NURBS symbolc computaton. Therefore, we have Remark. In any non-trval NURBS symbolc computaton, the multplcty of a knot s D σ, where D s the most lkely ever ncreasng egree, whle σ s the most lkely ever ecreasng contnuty. For example, accorng to [3], the square curvature κ of a ratonal quartc B-splne plane curve s another ratonal B-splne wth egree as hgh as 96, an the egree stll s 6 even after all the egree reucton strateges for NURBS symbolc computaton, as etale n that paper, are apple; on the other han, the contnuty s only. Furthermore, we have Remark. In any CAD applcaton nvolvng a seres of NURBS symbolc operators, the B-splnes typcally all have the same stnct knot vectors, except probably the ones n the very frst stages of NURBS symbolc computaton. 4

5 Now conserng the fact that a typcal CAD applcaton starts moelng wth quarc, cubc, quartc an quntc (wth cubc beng the most common, by Remark, the contnuty of erve B-splnes n the later stages, for example of the analyss an verfcaton of the moels, must be less than 5. Conser the contnuty σ at knot v,. f σ <, then, we only have the frst an secon cases n Table. Incentally, f the contnuty s all zero for all knot breaks, then the multplcaton s reuce to the smple Bézer case [4, 4], an no nee for the algorthm as presente n ths paper.. f σ =, then, we have all 3 cases n Table. 3. f σ >, then, we woul have more cases than the 3 lste n Table. However, any D-sequence s stll qute unlkely to have 4 (or more stnct knot values; otherwse u α v m w m x β woul most lkely be a sequence of sze larger than D, because par of lnearly orere lsts of factor sequences, whch are use later for the computaton of prouct B-splne control ponts. 4. Enumeraton of Subsequences of a Sngle Dual Sequence By Eq. (8, to compute a control pont of the prouct B- splne, or equvalently, the blossom at the corresponng ual D-sequence, the blossoms of the two factor B-splnes nee to be evaluate at a whole set of factor knot sequences that are subsets of the ual D-sequence of the prouct vector. Thus, subset enumeraton s an essental operaton here. We nee the followng subsequence enumeraton algorthm to generate subsequences of a gven sequence wth certan lnear orer, whch we wll scuss n etals later. The algorthm s presente for easy llustraton, wth actually better mplementaton. Algorthm. Enumerate subsequences of a sequence m + m (D σ = D + (D σ D where D σ > 0 for most stuatons by Remark. Ths assumpton of at most 3 stnct knots for any D- sequence s further ustfe by Remark whch says that, most lkely, factor B-splnes have the same stnct knot vectors. Then, a D-sequence wth 4 stnct knots as above woul yel an mmeate contracton as m + m D by case 3 of Table. Input (u λ u λ Output L (p, q Begn. L (p, q (q-seq to enumerate ts (p-subseqs lst of all (p-subseqs of (u λ u λ In concluson, Eq. (8 allows us a ramatc reucton of a typcally combnatoral number of factor sequences to an at most lnearly boune (wth respect to egrees number of factor sequences for most typcal cases. It shoul be note, though, that navely usng the earler equaton (6, t woul stll be possble to avo the unnecessary re-evaluaton of the blossom on the same knot sequence that appears to be fferent; however, specal care an extra computaton has to be taken so that all the fferent guses of the actually same knot sequence wll assocate to the same (alreay evaluate blossom value store n some type of assocate ata structure. Also, t typcally takes far less computaton tme to multply a value smply by w than lookng the value from an assocatve table an ang the value w tmes; ths s partcularly true from an asymptotc complexty pont of vew because w has some sort of combnatoral orer of ncrease wth respect to the nvolve B-splne egrees. An of course, for very hgh egrees, the unnecessary memory consumpton, whch s also combnatoral wth respect to the egree, s a concern as well. 4. INCREMENTAL ALGORITHM FOR KNOT SUBSEQUENCE ENUMERATION In ths secton, we evelop an algorthm for enumeratng subsequences,.e., factor sequences, of ual knot sequences of the prouct vector. The consecutve factor sequence enumeraton wth respect to consecutve ual sequences of the prouct vector s ncremental, an furthermore, results n a En. If (u λ u λ = ( 3. Else If p = 0, a empty strng ( to L (p, q For each k =,, λ (a L (p k, q k enumeraton of (p k- subseqs of (q k-seq (u λ u λ (b For each (p k-seq (X L (p k, q k, appen (X u k to the en of L (p, q 4. Incremental Subsequence Enumeraton for All Dual Sequences It seems that each tme Eq. (8 s use to compute a new control pont of the prouct B-splne, Algorthm ( has to be apple to enumerate all the factor sequences. However, a performance enhancement s realy avalable here by observng that the subsequence enumeraton of two neghborng ual (D-sequences shares most of ther subsequences, an ther corresponng weghts as well. Ths s true smply because the two neghborng ual (D-sequences shft only one poston wth respect to the prouct knot vector. Specfcally, 5

6 . Conser any (-subsequence (u λ of the current (D-sequence u λ (u n u m + + u m un. If λ < n, then t s stll a val (-subsequences of the next ual (D-sequence of the prouct vector, whch s (u n u m + + u m, f n < m (0 (u n u m + + u m um u +, f n = m un + Furthermore, by Eq. (9, the assocate weght s upate smply by a scale factor of n λ = n λ n λ n λ n λ n, f n = m, otherwse n + λ = n λ n λ n n + n λ + ( Note that the next ual (D-sequence of the prouct vector n Eq. (0 actually starts wth u m + + f n =, but Eq. ( hols as well n ths specal case.. New -subsequences of the next ual D-sequence (0, must be of the form (X u n +, f n < m, (X u +, f n = m. ( where X an X are compute recursvely by Algorthm ( for subsets problems wth reuce sze. Specfcally, X s any ( n -subsequences of the followng (D n -sequence (u n u m + + u m, an X s any ( -subsequences of the followng (D -sequence (u n u m + + u m un. On the other han, the assocate weght s ntalze by a rect computaton of Eq. (9 Fnally, we gve the etale algorthm below to ncrementally enumerate all subsequences par (one for each factor B-splne for each ual D-sequence of the prouct knot vector, where each enumerate subsequence s further tagge wth ts assocate weght. Fg. 4. llustrate a snapshot of output of the algorthm. Algorthm. Enumerate factor sequences of all ual sequences of a prouct B-splne Input,,D b egrees of st, n factor an prouct B-splnes u m u ms s prouct knot vector, where m = m s = D Output L SEQ Lseq Lseq LP Begn Lst of ual (D-seq of prouct B-splne Lst of factor (-seq of st factor B-splne Lst of factor ( -seq b of n factor B-splne Lst of pars of weghte ntervals of Lseq an Lseq. L SEQ { (u D }, Lseq { (u }, Lseq { (u b }. Intalze LP to contan only par of ntervals, beng the whole Lseq wth ts only sequence tagge wth `D weght, an the whole Lseq wth ts only sequence tagge wth weght, respectvely. 3. For both L SEQ an LP, copy back element an appen to the en so that to generate ntal values for current ual D-sequence, SEQ, an ts corresponng par of weghte ntervals, (, c. 4. Upate SEQ by Eq. (0 5. For each factor (-sequence n, clear ts assocate weght w to 0 f t s not a val (-subsequences of SEQ; otherwse, scale w accorng to Eq. (. 6. Apply the same proceure to factor ( b -sequences n c, clear weghts bw f necessary but o not scale. 7. By Eq. (, generate new factor (-sequences, appen to Lseq, an exten the rght en of to the en of Lseq. 8. Wth the same proceure, generate new factor ( b -sequencer, an upate Lseq an c accorngly. 9. Sle forwar left ens of ( c untl the frst tagge w ( bw 0 0. If seq u D s, go back to Step (3. En 4.3 Backwar Lexcographc Orer of Subsequence Enumeraton Subset enumeratng s a well-stue classcal topc on combnatoral algorthms; the reason to gve the etale Algorthm ( earler s manly to llustrate the specfc orer of the resultng enumeraton. From the for loop n step (3 of Algorthm (, t s obvous that subsequences wth less multplcty of the back element are enumerate frst, an ths observaton apples recursvely. Therefore, lookng backwar (.e. from rght to left at each enumerate subsequence, the enumeraton s n lexcographc orer w.r.t. the alphabet of stnct ascenng knots. We call ths backwar lexcographc orer. Furthermore, as all the ual D-sequences of the prouct vector are also terate, n Algorthm (, n ths backwar lexcographc orer, specfcally by shftng one knot 6

7 to the rght n the prouct knot vector, then, appenng the newly generate factor sequences to the current lsts factor sequences, Lseq an L cseq, wll not change ther backwar lexcographc orer. ŵ = w = k + k +D L S EQ Lseq Lŝeq Fgure : Illustratng Subsequences Enumeraton of Algorthm Shown for a n-mensonal prouct B-splne are some nformaton specfc to the horzontal recton nclung:. the current lst L SEQ of ual D-sequences of the prouct vector n the consere recton. the corresponng slces (columns, for the surface case of control ponts ue to L SEQ ; ths s to be compute later by Algorthm the two current lsts, Lseq an Lseq, of relevant sequences (w.r.t. the two factor B-splnes n the consere recton that are subsequences of ual D-sequences of the prouct vector generate so far; the fnal lsts are gong to be use n Algorthm (4 to compute the control meshes of the two factor B-splnes. Hghlghte are. the current ual D-sequence of the prouct knot vector to be consere for subsequence enumeraton,. ts subsequence enumeraton represente as two ntervals nto Lseq an Lseq respectvely, wth each knot sequence n the ntervals further assocate wth normalze weghts. Elements n the nterval par are pare together as shown by ashe lnes. All such nterval pars are use n Algorthm (5 to construct a slng wnows to compute, n orer, all the control ponts of the prouct B-splne. Notce that all other nterval pars corresponng to earler ual D-sequences are not shown n the mage. 4.4 Reverse Parng of Knot Sequences of Two Factor B-splnes We have been workng on a par of subsequence enumeraton so far. Notce that, n Algorthm (, subsequence enumeraton of the secon factor B-splne goes nepenently to that of the frst factor B-splne, an wth exactly the same proceure, except that the assocate weghts ether 0 or serve to only ncate whether or not the sequence s a genune subsequence of the D-sequence of the prouct knot vector. Ths s unerstanable that we have alreay assocate the real weght to the other subsequence of the consere par. Now a natural queston s: how are -sequences pare wth b -sequences? It turns out that the parng s really smple. Specfcally, by Algorthm (, each ual D-sequence of the prouct knot vector s parttone nto a factor -sequence an a complmentary factor b -sequence n varous ways, wth all the possble factor -sequences forms an nterval of Lseq, an all the possble factor b -sequences forms another nterval c of Lseq. Because of the backwar lexcographc orer of both Lseq an L SEQ, the frst -sequence n, beng the smallest one, has to be pare wth the last b -sequence n c, beng the largest one, an the process goes recursvely whle skppng all the nval factor sequences wth 0 assocate weghts. Fg. 4. llustrates, among others, the so-calle reverse parng. 5. MULTIPLICATION OF TWO TENSOR PRODUCT B-SPLINES OF GENERAL DI- MENSION N The algorthms we ust gave n the last secton s best unerstoo for unvarate B-splne functons. However, t shoul be note mmeately that they work perfectly well for any n-varate or n-mensonal B-splne functons, when the focus s on any sngle recton or menson. In ths more general settng, a ual sequence of the knot vector n the consere menson, correspons to a whole slce of control ponts (cf. Fg. 4.. Bascally, Algorthm. has to be apple n tmes, once for each menson, an then the resultng n lst pars have to be combne n some way for the purpose of computng the blossoms at varous n-tuples of ual knot sequences of the prouct knot vector. But frst, n ths secton, we nee to brefly revew multplcaton of two B-splnes of general menson n >, takng nto conseraton of all mensons. Conser that the two factor B-splnes G an b G be -mensonal (.e., bvarate tensor prouct B-splnes. Assume that the prouct knot vector n the n menson s v p, v p,, v p t t, (3 an the egrees of the st factor B-splne, the n factor B- splne an the prouct B-splne are, b an, respectvely 7

8 for =,. The mult-mensonal analogy to Eq. (8 s f(u n u m + +,, um, un ; P w w g bg = where + b + b, g bg = g(u λ,, u λ ; w = w = X r=,, bg(u b λ,, u b λ ; n Y bη q k Y η k <r< k<r<l λ r =, m r λ r p r η r vq k k v p k+ k+,, vp l l, vq l l vη k k,, vη l l v bη k k,, v bη l l n bη q l η l X r=k,,l (λ, λ +,, λ, λ + ( b λ, b λ +,, b λ, b λ = (n, m +,, m, n, (η k, η k+,, η l, η l + (bη k, bη k+,, bη l, bη l = (q k, p k+,, p l, q l, an the summaton takes over all η r = {λ,, λ }, {η k,, η l } Smlar equatons exst for the multplcaton of tensor prouct B-splnes of any general mensons n. At ths pont, a few comments about B-splne multplcaton, n comparson to other B-splne operatons, are n orer. A favorable property of tensor prouct B-splnes s that most algorthms for n-mensonal tensor prouct B-splnes are actually of -mensonal nature. Ths s so because a B- splne surface can be regare as a curve of curves, a B- splne volume can be regare as a curve of surfaces ( each of whch, n turn, can be regare as a curve of curves, an so forth. Puttng n another wors, the same algorthmc proceure can be apple to each menson separately. For example, to evaluate a bvarate Bézer surface (the ea s exactly the same for the B-splne case at oman pont (u, v, one apples e Caustelau algorthm to each curve efne by each row of the control ponts at the same D oman pont u, then apples e Caustelau algorthm agan at the same D oman pont v to the curve whose control ponts are the ponts ust evaluate for each row curve. Unfortunately, ths smple menson separaton strategy oes not exten to multplcaton of tensor prouct B-splnes. For example, to multply two B-splne surfaces, there s no way to multply pars of curves efne by approprate rows of control ponts, an then to multply pars of curves efne by approprate columns of the resultng new control ponts. 6. COMPUTING BLOSSOM MESHES OF TWO FACTOR B-SPLINES For the multplcaton of two n-mensonal B-splnes, Algorthm ( has to be apple n tmes, one for each menson. For each {,, n}, the algorthm generates Lseq, a lst of -sequences w.r.t. the -th menson knot vector of the the frst factor B-splne G where s G s egree at -th recton. There are totally n such lsts, an ther set prouct Q Lseq forms an n-mensonal array, each element of whch s a tuple of n factor sequences ( -seq,, n-seq. In ths secton we evaluate the blossom g of G at all such sequence tuples, therefore constructng an n-mensonal array of blossom values, calle blossom mesh of the frst factor B-splne. Of course, another blossom mesh of the secon factor B-splne s constructe as well n exactly the same way. 6. Knot Sequence as Convex Affne Combnaton of Another Sequences Frst, we nee a metho to recursvely evaluate blossom value g at certan -sequence, seq, of the consere factor B-splne G to an expresson nvolvng orgnal control ponts of G; or ually, to recursvely expan seq nto an affne combnaton of another -sequences untl all the fnal -sequences are ual -sequences that correspon control ponts of G. Let seq = XbY where knot b has a larger multplcty than t has n the orgnal knot vector of G (possbly 0, an further let a an c be the left an rght neghbor knots to seq (see etals n Algorthm 3; then b can be expresse an an convex affne combnaton of a an c, b = c b c a a + b a b a b = ( ρa + ρb, where ρ = c a c a, Consequently, the left an rght nterpolatng -sequences L an R are ax Y an X Yc. The etale algorthm, base on multplcty knot vector representaton, s shown below. Algorthm 3. Input Compute Interpolatng Knot Sequences (u λ u λ (-seq to be recursvely nterpolate u m u ms s Ol knot vector (m l = 0 for any new knot u l Output L, R, ρ -seq nterpolatng to (u λ Begn. k frst nex that λ k > m k. If λ <m L (u λ + Else u λ k k u λ u λ by rato ρ L (u su 0 s+ u 0 u λ u λ k k u λ where s < s the frst such nex that m s 0. 8

9 3. If λ <m R (u λ u λ k k u λ + Else R (u λ u λ k k u λ u 0 + u 0 s u t where t > s the frst such nex that m t 0. Lseq 0 ÓÒØÖÓÐ Å C Lseq 0 Lseq ÐÓ ÓÑ Å B B 4. ρ u k u s u t u s B 0 B 0 En Wthout gong nto any etals, one fnal comment on the comparson of the above presente algorthm wth the approach n [7], where, n our notaton, the nterpolatng rght knot c s the rght neghbor of the X nstea of X Y. Although there wll not be any sgnfcant performance fference f assocate table s use to store an retreve ntermeate result, our metho typcally oes ncur much less levels of recurson. 6. Construct Blossom Meshes In the prevous sub-secton, a factor -sequence s expane as convex affne combnaton of another two -sequences, whch are recursvely expane to an ultmate expresson of some convex affne combnaton of ual -sequences from the orgnal knot vector. On the other han, there s an ual statement of evaluatng blossoms to an expresson of some affne combnaton of orgnal control ponts. Specfcally, by affne property of blossom g, we have, g(,, XbY,, = ( ρ g(,, ax Y,, + ρ g(,, X Yc,, (4 where enotes any knot sequences n all mensons other than the consere one. Notce that, for -mensonal case, the equaton represents an nterpolaton of two blossom values nto the one to be evaluate, an the two nterpolatng blossom values have to be recursvely evaluate, ultmately from the orgnal control ponts that efne the B-splne; on the han, for a general n-mensonal case, Eq. (4 represents an nterpolaton of two slces that s, (n mensonal arrays of blossom values nto the one slce corresponng to the knot sequence XbY (cf. Fg 4., whch means that the same nterpolaton s apple to each corresponng trple of blossom values of the 3 nvolve slces. Such an nterpolaton of slces are carre teratvely for all mensons, ultmately resultng an n-mensonal array of blossom values,.e., the blossom mesh as requre. The etale algorthm s shown below. Algorthm 4. Input Computng Blossom Mesh of a Factor B-splne C Control polygon of st factor B-splne G Lseq 0 Lst of ual seq at recton {,,n} Lseq Lst of factor seq at recton from Algo. Output B n-array of g evaluate at tuples of factor sequence g(lseq 0 [], Lseq 0 [] = g(a 0, B 0 Legen: A 0 A A Lseq 0 ual knot sequence of the orgnal knot vector of the factor B splne n recton ual knot sequence of the orgnal knot vector of the factor B splne n recton factor knot sequence of the factor B splne n recton factor knot sequence of the factor B splne n recton Lseq g(lseq [3], Lseq 0 [] = g(a, B0 g(lseq [3], Lseq [6] = g(a, B Lseq Fgure : Illustratng Blossom Mesh Comp. (of st factor B-splne of Algo. 4 The vertcal slces n the orgnal control mesh are nterpolate, n some way as resulte from the recursve evaluaton of step (b n the algorthm for the horzontal menson. The resultng ntermeate mesh correspons to factor knot sequences n the horzontal menson, whle stll to the orgnal ual knot sequences n the vertcal recton. After the algorthm oes the recursve evaluaton one more tme, on the vertcal menson, the horzontal slces of the ntermeate mesh are nterpolate nto a fnal blossom mesh that correspons to factor knot sequences n both mensons. Begn. SrcMesh C. For each recton =,, n (a DstMesh n-mensonal empty mesh (b For k =,, m, where m s the total elements n Lseq. Usng Eq. (4, recursvely evaluate g(lseq [ ],, Lseq [ ], Lseq [k], Lseq 0 +[ ],, Lseq 0 n[ ] to some affne combnaton of slces (crossng recton from SrcMesh. Appen the evaluate slce to DstMesh along recton. (c SrcMesh DstMesh 9

10 3. B DstMesh Lseq ÐÓ ÓÑ Å B ÐÓ ÓÑ Å B Lŝeq En 7. SLIDING-WINDOWS ALGORITHM The varous algorthms presente so far fnally construct a par of n-mensonal arrays the so-calle blossom meshes of blossom values corresponng to factor -sequences an b-sequences of the frst an the secon factor B-splnes, respectvely. As these factor sequences are exactly those that appear n the rght han se of Eq. (8 for computaton of control ponts of the prouct B-splne, we are now able to compute each control pont of the prouct B-splne rectly by Eq. (8. Furthermore, ue to the consstent backwar lexcographc lnear orer of ual D-sequences of the prouct knot vector, of the lst of factor -sequences of the frst factor B-splne, an of the lst of factor b -sequences of the secon factor B-splnes, an further ue to the assocate weghte factor sequence ntervals corresponng to each D-sequence, we are able to compute the prouct B-splne control ponts one by one n a natural lnear orer whle corresponngly teratng lnearly on the blossom meshes. Frst we orer elements n varous n-mensonal arrays consere n ths paper, n a natural way, by numbers n that correspon to ther mult-nces (,,, n. Then, the control ponts can be compute from a par of wnows, that s constructe from n copes of -mensonal nterval pars as compute by Algorthm ( an that s use to nex nto sub-arrays of the blossom meshes, respectvely. The wnow par never sle backwar n any menson, an ue to the reverse parng property as scusse n Secton?? for -mensonal case, blossom values n the frst wnow are pare wth those n the secon wnow n a reverse lnear orer where the lnearty n the n-mensonal case s specfe as above. Detals are show n Algorthm (5 below. Algorthm 5. Slng-Wnows Algorthm Input B bb LP Output C Notaton sz J Begn For each J Blossom Mesh from Algo. (4 on st factor B-splne Blossom Mesh from Algo. (4 on n factor B-splne Lst of nterval pars from Algo. ( on -th menson n-mensonal control mesh of prouct B-splne Total pars n LP, =,, n n-mensonal mult-nex, where J sz. C[J] 0. Use n nterval pars LP [J ], one per menson {,, n}, to construct B an b B, two sub-arrays of B an b B, resp LP L SEQ Lseq 4 = LP L SEQ Lŝeq Fgure 3: Illustratng Slng-Wnows Algorthm En 3. Use the assocate weghts of LP [J ] to construct a par of n-mensonal weght arrays W an b W, each element of whch s smply the prouct of the corresponng n copes of tagge weghts, one per menson. 4. Lnear terate B an W. Lnear reverse terate b B an b W. Let the terate to be b an w, an b b an bw, respectvely (a Go to next b an w untl w 0 (b Go to next b b an bw untl bw = (c C[J] C[J] + b b b w 8. AVERAGE WINDOW SIZE AND IRREL- EVANT ELEMENTS IN SLIDING WIN- DOWS OF A C σ PRODUCT B-SPLINES In ths sub-secton, we wll erve upper bouns for both the sze of slng wnow an the rrelevant elements t contans. For a egree D C σ prouct B-splne, assumng D > σ, a ual D-sequence of the prouct knot vector has to be of the 0

11 form u D, or u α v D α, or u α v D σ w β, where α + β = σ, where, u, v, w, s the stnct knot vector. Assumng C σ at every knot break, the subsequence enumeraton of some consecutve ual D-sequences have the pattern as shown n Table 8. u σ v D σ u σ v D σ w u σ v D σ w v D σ u σ v σ u σ v σ+ u σ v σ w u σ v σ+ u σ v σ+ w u σ v σ w u σ 3 v σ+3 u σ 3 v σ+ w u σ 3 v σ+ w... u v uv w uv 3 w v v w v w v σ w σ 0 0 P (σ P = = (σ + (σ 3(σ (σ + Table 3: Factor -Sequences of Some Consecutve Dual D-Sequences The frst column shows the subset -sequences of ual D- sequence u σ v D σ, an all the rest shows the corresponng subsets -sequences of ual D-sequences followng u σ v D σ. Notce that, wthn each column, the multplcty of the frst knot u s always ecreasng, whle there s a ump between the last sequence n the prevous column an the frst sequence n the current column. Due to ths pattern, we can see that the ual sequence u σ v D σ w, shown at the thr column, has a -sequence n ts backwar lexcographcally orere subset nterval that s actually not one of ts subset. Ths rrelevant element, u σ v D σ w, s on the top of the prevous column, an t s n the nterval because all the -sequences, except the top two, come before u σ v D σ w yet are stll subsets of the consere ual sequence u σ v D σ w. Smlarly, the subset nterval of the next ual D-sequence wll have rrelevant element from ts prevous column, an rrelevant elements from the next prevous column. Ths pattern contnues untl ual D-sequence u 0 v D σ w σ, whch has rrelevant elements of,,, an σ from all ts prevous columns, from rght to left, respectvely. Therefore, the total rrelevant elements of ual sequences from s, = w σ [u σ v D σ w 0, u 0 v D σ w σ ] (5 σ X σ X( + ( + + = =. (6 Now conser the ual sequences n (v D σ w σ, v w D (7 because all ual sequences n the range have v multplcty no less than, the last factor -sequence of the frst column,.e., v s the same left en of all the corresponng subset ntervals. Hence, the total rrelevant elements of ual sequences from (v D σ w σ+ to v w D s, (++ +σ (D σ = The next range of ual sequences s. (σ σ (D σ (8 [v w D, v σ w D +σ (9 In ths seres, the ecreasng multplcty of v effectvely moves the left en of subset ntervals from v, to v w, untl v σ+ w D +σ,.e., move along the secon to last row n the table. Therefore, we have very smlar stuaton as that of the frst range we consere earler; specfcally the total rrelevant elements are, X σ X( + ( + + = =σ For the last range, we nee conser = (0 [v σ w D +σ, v σ w D σ ( each of whose ual D-sequence has no rrelevant element. Startng at v σ w D σ, the whole whole process wll repeat tself. Therefore, the average number of rrelevant elements s σ P (+ + (σ σ (D σ = D σ ( where D σ s the total ual sequences n each cycle wth the above repeatng pattern. Wth some ervaton ths s whch s approxmately σ(σ (3D σ 6(D σ σ(σ f egrees are much larger than the contnuty. The last row n Table?? shows the sze of subsets of each corresponng ual D-sequences, usng the result of case 3 from Table??. (σ +, (σ, 3(σ, 4(σ,, (σ, σ + Further usng the result of case from Table??, all the rest ual D-sequences have ther subset szes n the followng pattern, σ+, σ+3,, mn(, b +,, mn(, b +,, σ+3, σ+ Assumng the contnuty σ s far less than the total egree D, by summng the prevous expressons an vng by D σ (the total ual sequences n the cycle, the average sze of subsets can be shown to be approxmately

12 + mn(, b f mn(, /D b f mn(, /D b 0.5 an s always boune by O(D. Although the above ervaton on average subset sze an average rrelevant element number s base on the assumpton of C σ contnuty for all break ponts, the constant orer, wth respect to the prouct egree, of rrelevant element, an the lnear orer of subset sze reman true even ths assumpton s relaxe, because the repeatng cycle,.e., D σ of approxmately D. Notce that the repeatng cycle, as scusse so far, s D σ that s approxmately D. Therefore, for a C σ an egree D prouct B-splne, even f not very knot break has the maxmal an thus the same contnuty C σ, t can stll be state that average wnow sze n the slng wnows algorthm s of the orer O(D, whle the average number of rrelevant elements (.e., wth 0 assocate weghts n a slng wnow s constant wth respect to D. 9. CONCLUSION We have presente n ths paper a blossom-base rect B-splne multplcaton algorthm. The so calle slngwnows algorthm s conceptually very smple. By constructng two n-mensonal mesh of blossom values of the two factor B-splnes, the control ponts of the prouct B- splne can be compute smply by slng a par of wnows (as sub-arrays on the two blossom meshes, an the blossom values n the wnows are pare, multple, an fnally affne combne nto the control ponts. The algorthm s motvate by the effcency ssue of NURBS symbolc computaton nvolvng B-splnes of hgh egrees an especally hgh mensons, whch we beleve s a current tren n the CAD communty ue to the ncreasng eman on tasks beyon smple moelng, nclung especally nqury, analyss an verfcaton of the moele curves/surfaces. Compare to hgh egrees, hgh menson poses a much bgger challenge on any B-splne multplcaton algorthms. There are both tme effcency an memory effcency ssues relate to hgh mensons, an typcally there has to be a trae-off between these two. In the slng wnows algorthm, tme effcency s acheve at the expense of preconstructng an storng a par of n-mensonal blossom meshes. However, all the elements,.e., blossom values, n the constructe blossom meshes are gong to be use rectly for the computaton of some control ponts of the prouct B-splne; whle all those blossom values whch are use nrectly from recursve evaluaton are kept only for a certan stage n the teratve process of constructng blossom meshes (cf. Algorthm (4, specfcally ther memory usage wll be e-allocate when the teratve constructon goes to the next menson. It shoul also be note that the par of blossom meshes has the szes n the same orer as that of the control mesh of prouct B-splne. Therefore, the slng wnows algorthm oes not ncur any unnecessary space requrement, or any hgher orer space complexty other than that s ntrnsc to the nvolve B-splnes. Other than egrees an mensons as scusse n ths paper, there s actually another ssue that also affects the effcency of B-splne multplcaton. That s the number of separate polynomal peces, or the sze of the knot vector, n each menson. For example, conser a cubc 6- mensonal B-splne, wth 7 peces n each menson, then t has 0 control ponts along each recton, an there wll be 64 mllon total control ponts to be store, a farly large memory requrement. However, Ths s a relatvely margnal ssue because, unlke the ever ncreasng egree an mensons from complex geometrcal formulaton, any CAD applcaton typcally has fxe knot vector sze for all ts nvolve B-splnes. We are currently workng on etale effcency an numercal stablty comparson of the varous B-splne multplcaton algorthms especally nclung the one presente n ths paper, an s also conserng any possble harware acceleraton strateges for the actual mplementaton of the slng wnows algorthm. 0. REFERENCES [] X. Chen. Dynamc geometrc computaton by sngularty etecton an shape analyss. Ph.D. Thess Manuscrpt, 006. [] X. Chen, E. Cohen, J. Damon, an E. Cohen. Trackng ntersecton curves of two eformng surfaces. Sprnger-Verlag Lecture Notes n Computer Scence 4077 (GMP 006: 0-4, 006. [3] X. Chen, R. Resenfel, an E. Cohen. Degree reucton strateges for nurbs symbolc computaton. Proceengs of IEEE Shape Moelng an Applcatons 006: 8-93, 006. [4] E. Cohen, R. F. Resenfel, an G. Elber. Geometrc Moelng wth Splnes:An Introucton. A K Peters, eton, 00. [5] C. e Boor. A Practcal Gue to Splnes. Sprnger-Verlag, eton, 978. [6] T. DeRose an R. Golman. A tutoral ntroucton to blossomng. In H. Hagen an D. Roller, etors, Geometrc Moelng: Methos an Applcatons, pages Sprnger-Verlag, 99. [7] T. DeRose, R. N. Golman, H. Hagen, an S. Mann. Functonal composton algorthms va blossomng. ACM Trans. Graph., (:3 35, 993. [8] G. Elber. Free form surface analyss usng a hybr of symbolc an numerc computaton. Ph.D. thess, Unversty of Utah, Computer Scence Department, 99. [9] G. Elber, X. Chen, an E. Cohen. Mol accessblty va gauss map analyss. ASME Transactons, Journal of Computng & Informaton Scence n Engneerng, June 005:79-85, 005. [0] G. Elber an E. Cohen. Secon orer surface analyss usng hybr symbolc an numerc operators. ACM Transactons on Graphcs, (:60 78, Aprl 993. [] G. Elber an E. Cohen. A unfe approach to verfcaton n 5-axs freeform mllng envronments. Computer-Ae Desgn, 3(3: , 999.

13 [] G. Elber an M.-S. Km. A computatonal moel for nonratonal bsector surfaces: Curve-surface an surface-surface bsectors. GMP, pages , 000. [3] G. Elber an M.-S. Km. Geometrc constrant solver usng multvarate ratonal splne functons. ACM Symposum on Sol Moelng an Applcatons, pages 0, 00. [4] G. Farn. Curves an Surfaces for CAGD: A Practcal Gue. Acaemc Press, 5 eton, 00. [5] R. T. Farouk an V. T. Raan. Algorthms for polynomals n bernsten form. Computer Ae Geometrc Desgn, 5(: 6, 988. [6] J. Galler. Curves an Surfaces n Geometrc Moelng: Theory an Algorthms. Morgan Kaufman, eton, 998. [7] D. Knuth. The Art of Computer Programmng, Volume : Funamental Algorthms. Ason-Wesley, 3 eton, 997. [8] E. Lee. Computng a chan of blossoms, wth applcaton to proucts of splnes. Computer Ae Geometrc Desgn, (6:597 60, 994. [9] K. M. Morken. Some enttes for proucts an egree rasng of splnes. Constructve Approxmaton, 7:95 08, 99. [0] L. A. Pegl an W. Tller. Symbolc operators for nurbs. Computer-Ae Desgn, 9(5:36 368, 997. [] L. Ramshaw. Blossomng: A connect-the-ots approach to splnes. System Research Center, DEC, 987. [] L. Ramshaw. Blossoms are polar forms. Computer Ae Geometrc Desgn, 6(4:33 359, 989. [3] H.-P. Seel. An ntroucton to polar forms. IEEE Computer Graphcs an Applcatons, 3:38 46, 993. [4] J.-K. Seong, G. Elber, an M.-S. Km. Trmmng local an global self-ntersectons n offset curves/surfaces usng stance maps. Computer-Ae Desgn, 38(3:83 93, March 006. [5] J.-K. Seong, K.-J. Km, M.-S. Km, an G. Elber. Perspectve slhouette of a general swept volume. The Vsual Computer, (:09 6, 006. [6] E. C. Sherbrooke an N. M. Patrkalaks. Computaton of the solutons of nonlnear polynomal systems. Computer Ae Geometrc Desgn, 0(5: , 993. [7] K. Uea. Multplcaton as a general operaton for splnes. Curves an Surfaces n Geometrc Desgn, pages ,