A gentle introduction to rational Bézier curves and NURBS

Transcription

1 A genle inroducion o rionl Bézier curves nd NURBS P.J. Brendrech (p.j.brendrech@suden.ue.nl Eindhoven, June 7, (ls upde: June 5, As you cn see, some prs of his ricle re in need of n upde or furher explnion. Plese send me n e-mil if you noice nyhing h isn correc or jus if you d like o shre your houghs. Thnks! In his ricle I ll inroduce he rionl Bézier curve in (hopefully inuiive wy. The reson for me o wrie his documen ws h, fer reding some secions in books on NURBS nd quie few (inroducory ppers, I sill didn know where he expression for rionl curve cme from. Le lone, how o obin he sndrd form for qudric rionl curve (i.e. he wo ouer weighs being equl o nd he inner weigh being vrible. I men, i s esy o work wih he equion for NURBS, nd wih some plying round i becomes obvious h incresing weigh does nohing more hn mke he corresponding conrol poin more rcive o he curve. Bu how o deduce he expression? How is his ricle srucured? Firs very shor recp of he conceps you should be fmilir wih (Bézier curves. Afer h, few differen wys o ge rionl expression for he uni circle (from he implici form, explici form, nd Möbius rnsformion. The expression for rionl Bézier curve hen follows more or less nurlly, bu i isn ye in unique form. This sndrd form will be derived. Nex, some houghs on oher conceps you ll sumble upon when reserching NURBS, nd finlly word on piecewise rionl Bézier curves. Aside from convenionl lis of references, I dded secion for recommended lierure (nd my houghs on why you should red i. Prerequisies Since here re lredy lo of books nd ricles bou Bézier curves, his isn he plce o explin he heory gin. I only include one picure (see Figure of qudric Bézier curve, where you cn see h he firs nd hird (generlly speking h would be he ls conrol poins re inerpoled. The middle conrol poin (generlly conrol poins, plurl pulls on he curve. P P P Figure : Qudric Bézier curve (ornge nd is conrol polygon (dshed blck If you re no fmilir wih Béziers, you should red up bi before coninuing wih his ricle. An excellen reference re he lecure noes by Sederberg [Sed]. A few good books bou CAGD re he ones by Slomon [Sl6], Mörken nd Lyche [LM] nd of course Frin [Fr]. Be wre h some books jus posule he expression for Bézier curves. They don provide ny explnion bou why nd how o obin hese bsis funcions, he Bernsein polynomils (lhough in rre cses hey menion h he formul ( + b n, wih = nd b = (, cn be used o

2 obin he degree n Bernseins (so Pscl s ringle cn be used o find he coefficiens for, nd nd h i s obvious h hey form priion of uniy. In he lecure noes of Mörken nd Lyche [LM], his is explined nicely (repeed liner inerpolion. The heory behind his mehod is unforunely omied, bu you cn find his in lierure on blossoming (polr forms, like [Rm89] nd [Gl99]. Jus o mke his secion complee, he expression for qudric Bézier curve in R n looks like C( = B (P + B (P + B (P, wih P i he conrol poins in R n, nd B ( = ( B ( = ( B ( = he qudric Bernsein polynomils, for [, ]. Rionl expressions for he uni circle I s well known h i is no possible o describe n exc circle using Bézier curves, whever heir degree (so here is no direc polynomil expression. Alhough very good pproximion cn be obined (less hn.3% difference when using merely cubic Béziers, see [Sl6], someimes perfec circle is needed. There re mny wys o find rionl expression for circle (w.l.o.g. le s look he uni circle. I know of hree pproches, lised below. Choose whever one you like, nd hen coninue o he nex secion.. From he explici form The mos common form o prmerize uni circle is x = cos ϕ y = sin ϕ Wih ϕ [, π]. Wih some knowledge bou rigonomery, we cn rewrie his o rionl form: sin(α + β = sin(α cos(β + cos(α sin(β (see Figure for nice grphicl proof sin(ϕ = sin( ϕ + ϕ ( ϕ ( ϕ ( ϕ ( ϕ ( ϕ ( ϕ = sin cos + cos sin = sin cos Furhermore, sec(ϕ ( cos(ϕ ϕ cos And = sec ( ϕ + n (ϕ = cos (ϕ cos (ϕ + sin (ϕ cos (ϕ = cos (ϕ = sec (ϕ Combine he bove equliies: ( ϕ ( ϕ sin(ϕ = sin cos = sin ( ϕ cos ( ϕ cos ( ϕ Define = n ( ϕ n cos ( ϕ ( = cos ϕ = ( ϕ ( ϕ ( ϕ cos = n sec ( ϕ = ( cos ϕ + n ( ϕ = +

3 sin(α sin(β sin(α + β cos(β sin(β α sin(α cos(β cos(α sin(β β α cos(α + β cos(α cos(β Figure : Grphicl proof for he ddiion formuls sin(α + β nd cos(α + β So y = sin(ϕ = + Similrly, i cn be derived h x = cos(ϕ = + This rewriing procedure is lso known s Weiersrss subsiuion. When vries beween nd, ϕ vries beween n ( = (i.e. kπ nd n ( = π (i.e. (4kπ + π. This is he upper righ pr (firs qudrn of he uni circle.. From he implici form Also very ofen used is he implici form for uni circle: x + y =. Using Figure 3, we will derive he rionl expression gin. Apr from he uni circle, here is line (wih slope y x = which goes hrough he poin (,. I s expression herefore is y = x +. Noe h his poin (, is lso on he circle. The line inersecs he circle lso in noher poin i depends on he slope which poin. When he slope is, he line is horizonl nd will inersec he circle in (,. When he slope is, he line is digonl (i.e. 45 nd will inersec he circle in (,. Anywhere in beween, i will inersec he circle somewhere in he firs qudrn (op righ pr. To find his poin, we cn combine he equion for he line (y = x + wih he one for he uni circle (x + y =. Indeed, for poin o be boh on he line nd he circle, i hs o sisfy boh equions. Therefore, x + y = x + (x + = x + x + x + = ( + x + x + ( = 3

4 y (, (, x Figure 3: The line wih slope inersecs he uni circle wice: once in (, nd once in ye unknown poin. We cn solve his for x in differen wys, bu le s jus use he qudric formul x, = b ± b 4c wih = ( +, b = nd c = ( : b ± b 4c = ± 4 4 4( + ( ( + = ± 4 ( + = ± ( + = ± ( + = ± 4 4 4( 4 ( + The firs soluion for x is (when subrcing ( + ( =, ccompnied by y = x + =. This + is of course he poin (,, where our line srs. The oher soluion is our poin of ineres: x = + =, ccompnied by y = x + = + ( + + = ( + + ( + + =, s expeced he sme expressions s derived in he + previous secion..3 From Möbius rnsformion When reding some ricles on rionl funcions, I sumbled upon he menioning of Möbius (someimes wrien s Moebius rnsformions. I relly pprecie complex nlysis, so I looked for some more informion on his subjec in my copy of Snider nd Sff [SS3] relly good inroducory book on complex nlysis! One specific mp is of ineres, nmely he one h mps he rel line o he uni circle. Alhough his migh sound bi srnge, he mp is cully quie simple: T (z = i z z + i Normlly, z C of course, bu since we wn o mp he rel line o circle, we resric z o R, so z R. When z [, ], i is mpped o he op righ pr (firs qudrn of he uni circle. A wy o simplify complex frcions is o muliply boh numeror nd denominor wih he complex conjuge of he denominor, i.e. i z (i z(z i = z + i (z + i(z i = z + zi + z = z + z + zi + z. 4

5 The firs erm on he righ is he rel pr, he second pr he imginry pr. Since his cn be inerpreed s he x-xis nd y-xis, we hve obined he sme rionl expression gin: z + z + zi + z x = +, y = +. 3 From rionl expression o rionl Bézier curve As we ve seen in he previous secion, he rionl prmeric funcion for he firs qudrn of he uni circle is ( x = f( = ( F ( = ( y F ( F ( + And [, ]. The ide is o wrie hese hree polynomil expressions (F ( =, F ( = nd F ( = + s liner combinions of he Bernsein funcions. 3. Finding liner combinions of Bernseins Firs, le s wrie boh he Bernsein funcions B i ( nd our expressions F i ( s vecors in he spce spnned by, nd. For exmple, {, 3, 4} would be he coefficiens for B ( = ( = + + {,, } B ( = ( = + {,, } B ( = {,, } And our hree polynomil expressions F ( = {,, } F ( = {,, } F ( = + {,, } Now find he 3 3 mrix A such h F ( B ( F ( = A B ( F ( B ( Or equivlenly, = A This is possible, since boh B i ( nd F i ( re qudric polynomils. The mrix A cn be obined by invering he upper ringulr mrix (wih Bernsein coefficiens on he righ nd muliplying i wih he mrix on he lef: A = = = 5

6 So i urns ou h F ( = ( B ( + ( B ( F ( = ( B ( + ( B ( F ( = ( B ( + ( B ( + ( B (. Therefore, our f( cn be wrien s ( ( ( B ( + B ( + B ( f( = ( B ( + ( B ( + ( B ( Noe h we didn provide he coordines of he conrol poins, bu hey followed nurlly from he sep from rionl funcion o rionl Bézier! 3. The core of rionl curves: weighs Le s coninue by muliplying ll erms in he numeror by (he reson will become cler soon: ( ( ( B ( + B ( + B ( f( = (3. B ( + B ( + B ( Now we cn wrie our new bsisfuncions, Q i, s follows: Q i ( = w i B i ( B ( + B ( + B ( Such h when using Q i (, we could rewrie f( s ( f( = Q (P + Q (P + Q (P = Q ( The generl expression for Q i would be ( + Q ( ( + Q ( Q i ( = w i B i ( v B ( + v B ( + v B ( (3. The only wy h hese Q i ( cn form priion of uniy is when he coefficiens (clled weighs re he sme in he numeror (w i nd denominor (v i. So in our cse (see he denominor in Equion (3., v =, v =, v =. Bu s you cn see (Equion (3., he hird weigh in he numeror (w = is differen from he hird weigh in he denominor (v =! This weigh w cn be chnged o, by dividing he conrol poin coordines of P by, resuling in: B ( f( = ( + B ( ( + B ( B ( + B ( + B ( Now he weighs in he numeror re he sme s he ones in he denominor. Since hey re equl, v i cn jus be replced by w i in he generl expression for Q i ( (Equion (3.. I s bou ime o ke look plo of our rionl funcion f(, he qurer circle see Figure 4. We hve jus proved h his expression represens n exc pr of circle. Using Weiersrss subsiuion (or ny oher wy, similr expressions cn be found [Sl6] for generl ellipse nd hyperbol, ogeher wih he prbol (ll ordinry Béziers he hree differen conics. Therefore, wihou furher proof we se h he generl expression for (qudric rionl Béziers is jus ( (3.3 f( = w B (P + w B (P + w B (P w B ( + w B ( + w B ( (3.4 6

7 P = (, P = (, w w = = P = (, w = Figure 4: Our firs rionl Bézier curve: qurer circle 3.3 Weighs re no unique Noe h he se of weighs for curve, in our cse he qurer circle, is no unique! For insnce, ll weighs cn be muliplied by he sme fcor, resuling in n oher se of weighs. Moreover, for qudric rionl Béziers here is his rule (invrin [P85]: w w (w = w w (w Where w i re he weighs for one represenion, nd w i he weighs for noher one. In our cse (he weighs re,, s you cn see in Equion (3.3 he invrin hs vlue =. So oher weighs for he sme curve wih he sme conrol poins! could be (,,, (,, nd mny ohers. Noe: I hve no ye red he proof [P85] in full deil myself, bu here migh be some relion o he concep of cross rios hey re invrin under perspecive projecions (more bou perspecive projecions ler. Finlly, noe h when ll weighs hve he sme vlue (e.g. w i = i, hen he expression for rionl Bézier is gin he expression for n ordinry Bézier curve. This is becuse he denominor will sum o in his cse (priion of uniy of he Bernsein funcions. 4 Obining he sndrd form As menioned in he previous secion, he se of weighs used for rionl curve is no unique. Le s look wy o mke i unique (up o muliplicion by fcor, of course. The resul is clled he sndrd form for conics. The min ide of he sndrd form is h he ouer weighs (w nd w hve vlue, nd h he inner weigh w is kep vrible. This wy, he se of weighs would be unique. Then by looking he vlue of his weigh w, you could deermine [Fr99] wheher he resuling rionl curve is ellipic (w <, prbolic (w =, or hyperbolic (w >. 4. Reprmerize Le s sr by wriing Equion (3.4 wih he B i ( subsiued by he funcions, i.e. ( w P + ( w P + w P ( w + ( w + w (4. We could divide every weigh by one specific weigh, ending up wih one weigh hving vlue. Bu wh hen? Le s firs hve look how o reprmerize his funcion. Reprmerizing is he process of describing he sme curve wih differen vrible (generlly in differen inervl. Noe h I sid sme curve, wih h I men h he curve looks excly he sme bu he derivive is generlly differen! Th mens h he velociy in which he curve is rversed differs from he old prmerizion. 7

8 For ordinry Béziers, he vrible could be chnged o new vrible s by sying (for exmple s = This is clled liner reprmerizion. If ws in he inervl [, ], s would be in he new inervl [5, 8]. Noe h when using liner reprmerizion, he inervl for he new vrible (in our cse s is lwys differen from he old one (excep if you sy h s =, bu h s no relly reprmerizion I d sy. If he inervl would chnge, hn so would our bsis funcions, he Bernseins (hey would look he sme bu be defined over differen inervl. Perhps i s bes o hold on o hese fmilir funcions, nd look for noher wy o reprmerize he funcion in (4.. Wh bou rionl reprmerizion? In is generl form, i looks like s = + b c + d So h s jus frcion of he sme ype of funcions we sw he liner reprmerizion. We wn o keep he sme inervl [, ] for s. So when =, s hs o be s well h mens h b =. Furhermore, when =, our s hs o be oo. Looking o (4., we see h his mens h = c + d (or h c = d nd d = c. Rewriing lile: s = c + d = ( d + ( c = ( d = ( ( d + c = And h would mke ( s ( s = α( α( + ( ( d + d + ( c = = ( d ( + c ( α + (α = α + α = (4. = ( ( d + c α( + Since s is defined on he sme inervl s, we cn direcly subsiue by s nd ( by ( s in Equion (4.: ( ( ( α( α( α( + w P + ( α( + w + α( + ( α( + α( + ( α( α( + ( w P + ( w + α( + w P α( + w As you cn see, every erm in boh numeror nd denominor hs he expression (α( +. To ge rid of i, jus muliply boh he numeror nd denominor by his expression. Then rewriing he funcion gin wih he symbolic Bernsein bsis funcions yields: α ( w P + α( w P + w P α ( w + α( w + w = α B (w P + αb (w P + B (w P α B (w + αb (w + B (w Now, s menioned before, divide every weigh w i by w, obining α B ( w w P + αb ( w w P + B (P α B ( w w + αb ( w w + B ( And o ge o our objecive (i.e. he firs nd hird weigh being nd he middle one being vrible, i should be h α w w = α = w w α = w w Assuming h our originl weighs w nd w re posiive (or cully, w nd w should hve he sme sign. If he weighs re indeed posiive, we cn rewrie once more o ge he sndrd form: B (P + B ( w w w w P + B (P B ( + B ( w w w w + B ( = B (P + B ( w w w P + B (P B ( + B ( w w w + B ( Noe h negive weighs re possible (o use in Equion (4. h is, bu cn resul in srnge curves (nd he denominor being so we beer void hem. 8

9 5 Oher ineresing conceps For now only some houghs on perspecive projecions (see he exmple in Figure 5 of perspecive projecion resuling in our discussed qurer circle. Perhps I ll dd pr on cross rios, nd some oher ineresing conceps ler ime. However, i is no he inenion for his ricle o become self-conining resource on NURBS :. 5. Perspecive projecions Rionl qudric Béziers re ofen inroduced s perspecive projecions in R of ordinry Bézier curves in R 3 (using homogeneous coordines. See Figure 5 for n exmple. Noe h rionl curve in R 3 would be perspecive projecion from n ordinry Bézier in R 4! I ve no ye come cross concise proof, when I find one I ll mke sure o dd i o his secion. R P P R P R (, Figure 5: Perspecive projecion of n ordinry Bézier curve (pink one in R 3, o rionl one in R (ornge one in he plne z =. The rionl curve is our discussed qurer circle. 5. Higher order rionl Bézier curves Looking Equion (3.4, you my wonder wheher n even more generl expression exiss for higher order rionl Béziers. Of course here is one, for exmple for rionl cubics: f( = w B 3 (P + w B 3 (P + w B 3 (P + w 3 B 3 3 (P 3 w B 3 ( + w B 3 ( + w B 3 ( + w 3 B 3 3 ( Wih Bi 3 ( he cubic Bernsein funcions. You cn generlize his concep for every order you wn, bu o be hones I don relly see he vlue of nyhing oher hn qudric rionl Béziers. Of course, i gives you he opporuniy o model your curve in differen wy, since you cn now jus chnge weigh o rc he curve more or less o conrol poin insed of moving he conrol poin iself. This ler wy of chnging your curve jus seems more nurl o me. 9

10 6 Piecewise rionl Béziers: NURBS Perhps you know h B-Spline curves (boh uniform nd non-uniform ones cn be splied ino Bézier curves so B-Spline curve is relly nohing more hn piecewise Bézier, where only he conrol poins re plced in more efficien wy. The blossoming ool cn be used o spli B-Spline curve (mind you, B-Splines re he bsis funcions, B-Spline curves re he cul curves ino he Bézier segmens. For n exmple of qudric B-Spline curve (nd is Bézier segmens, see Figure 6. P 3 P 5 P P P P 4 P P P 4 P 3 P 6 P Figure 6: Qudric uniform B-Spline curve (wih kno vecor e.g. Ξ = [,,, 3, 4, 5, 6, 7] nd is hree Bézier segmens. When combining his knowledge wih he definiion of rionl Béziers being perspecive projecions from ordinry Béziers, we cn se h NURBS (Non-Uniform Rionl B-Splines curves re jus concened rionl Bézier curves, bu gin wih more efficienly plced conrol poins. Why? Becuse if B-Spline curves re piecewise Bézier curves, nd ech Bézier is projeced individully, hen he resul would be piecewise rionl Bézier curve (concened rionl Béziers. The rel definiion of NURBS curve is perspecive projecion of B-Spline curve nd since he perspecive projecion of his B-Spline curve would (of course resul in excly he sme curve s he concened rionl Béziers, NURBS curve nd piecewise rionl Bézier curve is he sme hing. 7 Recommended liure On my wy o undersnding Bézier curves, B-Splines, NURBS, nd reled subjecs, I sumbled upon mny useful ricles, lecure noes nd books (nd sill do, becuse I cerinly don know ll here is o know ye will I ever. Below I ried o summrize he mos useful ones, rrnged by heir opic. 7. CAGD in generl The lecure noes of Sederberg [Sed] re one of he bes I ve seen. In my opinion i s jus he righ blnce beween mh, exmples nd how o pply he mer. Check i ou! The lecure noes by Lyche nd Mörken [LM] re lso relly good, in priculr he inroducory chper. Subsequen chpers re more mhemiclly involved, proving lo of properies for B-Splines in comprehensive wy. One book I cn cerinly recommend is he one by Dvid Slomon [Sl6]. I s very redble, hs lo of exmples, nd for ll exercises here is soluion in he bck in he book. Frin s book [Fr] is he one referenced mos ofen for CAGD. Alhough I cerinly gree h i s good book, some prs migh ke some ime o undersnd. I hs very exensive lis of references. The somewh older book by Brels e l. [BBB95] describes B-Splines using he clssicl pproch, divided differences. Definiely worh o ke look. Of course, mny more books nd ricles re vilble on CAGD. If you know one h should be in his lis, plese le me know!

11 7. Rionl Béziers nd NURBS The NURBS book by Frin [Fr99] is in my opinion he bes. I srs wih he bsics, grdully builds up o rionl Béziers nd hen o NURBS curves. The books by Piegl nd Tiller [PT96] nd Rogers [Rog] re somewh more dvnced. Especilly he former book hs lo of heories nd proofs. The mny ppers by Jvier Sánchez-Reyes re definiely worh reding. He wroe lo on rionl Béziers even bou complex rionl Béziers! 7.3 Blossoming (or polr forms Rmshw wroe couple of bookles on his subjec, [Rm89] [Rm87] [Rm88]. There re lso lecure noes from his hnd, wih some geomeric inuiive houghs on blossoming. The book by Gllier [Gl99] (disconinued, now vilble on his webpge is very exensive source on ffine spces, nd he derivion of blossoms. From ime o ime mny exmples, someimes lile difficul o ge hrough. The book by Mnn [Mn6] is more concise, bu quie redble. lyou, beuiful grphics. I lso looks very good (nice I should menion h Pul de Cselju lso wroe bou polr forms (he sme s blossoming, Rmshw cme up wih his erm, nd h he ppers of Seidel [Sei93] nd de Boor (B-Splines wihou divided differences re lso worh reding. Noe: I pln on wriing similr ricle for blossoming, bu firs I wn o finish one bou subdivision surfces nd noher one bou Lindenmyer sysems. Boh very ineresing subjecs, nd srngely enough conneced o ech oher : A MATLAB Scrips Even hough i is quie esy o plo Bézier curve (eiher ordinry or rionl in Mlb, I hough i would be useful o supply he code o do so. A. Qudric Bézier curve The Mlb code o plo qudric Bézier curve is displyed in Lising A.. Noe h pr from he curve iself, he conrol polygon nd he conrol poins re ploed s wel. % Iniilise cler ll; 3 close ll; 4 clc; 5 Lising A.: Mlb code for ploing qudric Bézier curve. 6 % Priion inervl [,] o ge good resoluion for ploing 7 = :.5:; 8 % Define he qudric Bernsein polynomils 9 Bernsein = [ (-.ˆ; *.*(-;.ˆ ]; % Define hree conrol poins (X,Y P = [ ; 3 3,4; 4 5,]; 5 6 % Beziers for he X nd Y coordines 7 DX = Bernsein(,: * P(, + Bernsein(,: * P(, + Bernsein(3,: * P(3,;

12 8 DY = Bernsein(,: * P(, + Bernsein(,: * P(, + Bernsein(3,: * P(3,; 9 % Iniilise figure figure xis equl 3 hold on 4 5 % Plo conrol polygon 6 plo( P(:,, P(:,, 'k.--', 'LineWidh',, 'MrkerSize', 3 ; 7 % Plo Bezier curve 8 plo( DX, DY, 'Color', [55/55,7/55,4/55], 'LineWidh', ; 9 3 % Add lbels for conrol poins P i 3 for i=:lengh(p 3 ex( P(i,, P(i,, [' P ', numsr(i ], 'FonSize', 6, 'Color', [,9/55,95/55] ; 33 end Resul (see Figure 7: P P 3 P Figure 7: Plo resul from Mlb (version R A. Qudric Rionl Bézier curve Replce lines 6-8 in Lising A. by he 5 lines in Lising A. o plo qudric rionl Béziers: Lising A.: Mlb code for ploing qudric rionl Bézier curve. Weighs = [,.75, ]; 3 % Rionl Beziers for he X nd Y coordines 4 DX = ( Weighs(*Bernsein(,: * P(, + Weighs(*Bernsein(,: * P(, + Weighs(3*Bernsein(3,: * P(3,./ ( Weighs(*Bernsein(,: + Weighs (*Bernsein(,: + Weighs(3*Bernsein(3,: ; 5 DY = ( Weighs(*Bernsein(,: * P(, + Weighs(*Bernsein(,: * P(, + Weighs(3*Bernsein(3,: * P(3,./ ( Weighs(*Bernsein(,: + Weighs (*Bernsein(,: + Weighs(3*Bernsein(3,: ; References [BBB95] Richrd H. Brels, John C. Bey, nd Brin A. Brsky. An Inroducion o Splines for Use in Compuer Grphics And Geomeric Modeling. Morgn Kufmnn, 995. [Fr99] Gerld E. Frin. NURBS: from projecive geomery o prcicl use. A.K. Peers, 999. [Fr] Gerld E. Frin. Curves nd Surfces for Cgd: A Prcicl Guide. Morgn Kufmnn,.

13 [Gl99] Jen Gllier. Curves nd Surfces In Geomeric Modeling: Theory And Algorihms. Morgn Kufmnn, 999. [LM] Tom Lyche nd Knu Mörken. Spline mehods drf. Lecure Noes,. [Mn6] Sephen Mnn. A Blossoming Developmen of Splines. Morgn & Clypool Publishers, 6. [P85] Richrd R. Person. Projecive rnsformions of he prmeer of bernsein-bezier curve. ACM Trnscions on Grphics, Vol. 4:Pges 76 9, 985. [PT96] Les A. Piegl nd Wyne Tiller. The NURBS Book. Springer, 996. [Rm87] Lyle Rmshw. Blossoming: connec-he-dos pproch o splines. Digil Equipmen Corporion. Sysems Reserch Cener, 987. [Rm88] Lyle Rmshw. Theoreicl Foundions of Compuer Grphics nd CAD, chper Béziers nd B-splines s muliffine mps, pges Springer, 988. [Rm89] Lyle Rmshw. Blossoms Are Polr Forms. Digil Sysems Reserch Cener, 989. [Rog] Dvid F. Rogers. An Inroducion o Nurbs: Wih Hisoricl Perspecive. Morgn Kufmnn,. [Sl6] Dvid Slomon. Curves And Surfces for Compuer Grphics. Springer, 6. [Sed] Thoms W. Sederberg. Compuer ided geomeric design. Lecure Noes,. [Sei93] Hns-Peer Seidel. An inroducion o polr forms [SS3] E. B. Sff nd Arhur Dvid Snider. Fundmenls of complex nlysis wih pplicions o engineering nd science. Prenice Hll, 3. 3