TR/91 January Bézier Polynomials over Triangles and the Construction of Piecewise r C Polynomials G. FARIN

Transcription

1 TR/9 Jauay 98 Bézie Polyomials ove Tiagles ad the Costuctio of Piecewise C Polyomials By G. FARIN

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3 . Itoductio Bézie polyomials ad thei geealizatio to teso-poduct sufaces povide a useful tool i suface desig (Bézie 97, 977; Foest 972). They wee developed as ealy as 959 by de Casteljau at Citoë but owe thei ame to P. Bézie fom Reault who was fist to employ them i ca body desig i the late sixties. De Casteljau 959 also descibes tiagula patches, but these scacely eceived ay attetio util Sabi 977. Fai 979 geealizes ad exteds esults obtaied by de Casteljau ad Sabi, shaig thei estictios to domais that cosist of coguet tiagles oly. The peset pape estates some of the esults of Fai 979, icludig a shot outlie of the uivaiate case, ad the geealizes them to Bezie polyomials defied ove abitay tiagles; fomulas descibig C cotiuity of adjacet tiagula patches ae povided. The last two sectios give applicatios of the theoy: the Clough-Toche scheme is geealized to the the dimesio of the liea space of piecewise ) is deived. 2 C C C case ad a fomula fo polyomials (of degee

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5 2. I Uivaiate Bézie Polyomials. Defiitio A Bézie polyomial Bφ is defied by () [B φ ] (t) = i = b i B i (t) whee the B i ae Bestei polyomials ( 2 ) B i ( t ) = t i ( t ) i ; o i i ad φ is the piecewise liea fuctio joiig the poits (, bi); i. i φ is called the Bézie polygo associated with B *) φ i ; the bi ae called Bézie odiates of Bφ. Sice the Bi satisfy ( 3) B i (t) ; i ; t, ( 4) B i (t) = i=, *) I classical appoximatio theoy, B φ is called the "Bestei appoximat" to φ (Davis 975) the gaph of Bφ, t, lies i the covex hull of the gaph of φ. (Bézie 97, Bézie 976, Foest 97).

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7 2. Degee Elevatio 3. Evey polyomial of degee ca be witte as a polyomial of i degee + ; let E φ be a polygo joiig poits ( +, b * i ), i +. If i ( ) ( i = i b + ) b, i, ( 5) b * i = φ + + i + i + it is easy to show that ( 6) B φ = B+ Eφ 3. Deivatives Fo the -th deivative of B φ we fid ( 7) d [B φ =! Δ ] (t) b dt ( )! i Bi (t ) i= This yields immediately ( 8a) d [B! dt φ] () = Δ b ( )! o ( 8b) d [B! dt φ ] () = Δ b ( )!, i.e. the -th deivative at a edpoit depeds oly o the (+) adjacet Bézie odiates. We also ote that fo ζε [a, b] istead of tε [,], (7) becomes

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9 4. ( 9 ) d [ B! φ ] ( ζ ) = Δ b i B i ( ζ ) d ζ ( )! ( b a ) i = 4. Recusive Algoithm (de Casteljau 959) The Bi satisfy a ecuece elatio () Bi (t) = ( t) B (t) i + t B (t) i ( with Bi (t) = fo i < o i > ). () allows to expad B φ i tems of Bestei polyomials of lowe degee : ( ) [B φ] (t) = bi (t) B (t),, i i = whee the b i (t) ae defied by (2) bi b o i (t) (t) = = ( t) bi. bi + t b i+ Sice ( 3) [ B φ ] (t) = b (t), (2) povides a easy ad stable algoithm fo the umeical evaluatio of [ B φ ](t). This is illustated i fig.

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11 5. Fig. costuctio of [ B 3 φ ] ( 2 ) Oe ca show that (4) b i (t) = j= b i + j j B (t) ; i The b (t) ca also be used to detemie the -th deivative of B ; i φ ( 5) d [B ] (t)! dt φ = ( )! i = b i (t) B i Fo =, (5) states that bo (t) ad b (t) (t) detemie the taget to [ B φ ](t) ad, fo =2, that b 2 (t), b 2 (t), b 2 (t) 2 detemie the osculatig paabola.

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13 6. 5. C Cotiuity Suppose we ae give a Bézie polygo φ with Bézie odiates b i ove tε [, ]. We seek a Bézie polygo ψ with Bézie odiates Ci ove ζε [,2] such that the two polyomials defied by φ ad ψ fom a fuctio i C [,2]. Fom (8a) ad (8b) we get the coditios ( 6) Δ ρ b ρ = Δ ρ co ; ρ. (6) implies that, fo fixed ρ, the Bézie polyomials defied by b ρ, b ρ +,..., b ad c o, c,..., c ρ coicide sice all thei deivatives coicide at t=l esp. ς =. Hece ρ i= o ρ ρ ρ b ρ+ i B i (t) = cib i (). i= o sice ς = t. This is tue fo all t, i.e. also fo t=2: ρ i= o ρ ρ ρ b ρ+ i B (2) = i cib (). i i= o The ight-had side equals c ρ, ad we get (7 ) ρ cρ = i = o ρ b ρ + i B i (2) ; ρ. Note that this is equivalet to: ρ (8) cρ = b ρ (2) ; ρ.

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15 7. We ca defie the secod Bézie polyomial ove [, β ] istead of [,2]; i this case, (8) becomes ρ (9) cρ = b ρ (β). Thus we have a coditio fo C cotiuity give by (6) ad a costuctio give by (7), Note that fig. ca be itepeted as the costuctio of the to obtai C b o, b,..., b o fom the cotiuity. We also ote that the two coespodig Bézie polyomials coicide with the oigial polyomial give by o b o, o b,..., b o. b o o, b,..., b o o

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17 8. II Bezie Polyomials ove a Tiagle. Defiitio We coside a tiagle T i the plae with vetices P,P2,P3 ad edges e,e2,e3 i which we assume such that fo each poit P i the plae baycetic coodiates defied P = up + vp2 + wp3 whee ad u, v, w fo all P ε u + v + w =, [P P P 2 ] [P P P 3 ] u = 3, v =, w = [P P 2 P 3 ] [P P 2 P 3 ] T, [P P P 2 ] [P P 2 P 3 ]. Hee, [ P3 P P2] deotes the aea of the tiagle P, P, P2 etc. We defie Bestei polyomials (2) B i (u) =! i! j! k! u i v j w k Sice the B (u ) i ae tems of (2) (u + v + w) =! i! j!k! i + j+ k = i, j, k we have immediately ; B ( u i ) ove T: u + v + w = i + j + k = u i v j w k, u = (u, v, w) i = (i, j, k ) ( 22) B (u ) fo u, v, w i,

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19 9. (23) i B i (u ). The summatio i i (23) is shot fo the oe used i (2). The ( + ) ( 2) 2 + polyomials B fom a basis fo the liea space i of all bivaiate polyomials of degee. A Bézie polyomial ove T is defied by ( 24) [ B ] (u ) b B φ = (u, i ) i i whee φ is the piecewise liea fuctio detemied by the poits j ( i,, k, bi ). The b i ae called Bézie odiates of Bφ ; φ is called the Bézie et of B φ.*) (22) ad (23) imply that the gaph of Bφ; lies i the covex hull of the gaph of stuctue of φ is illustated i fig. 3. φ. The We also ote that the bouday cuves of B φ ae the (uivaiate) Bézie polyomials detemied by the bouday poits of φ. *) This otatio is chose to be like the oe i the uivaiate case to poit out the similaity of both methods. No cofusio should aise, howeve, sice the meaig of B φ, φ, etc. will be clea fom the cotext.

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21 . 2. Degee elevatio Evey bivaiate polyomial of degee ca be witte as a polyomial j of degee + ; let Eφ be a et detemied by poits ( i,, k, b * i ), i+j+k = +. If j ( 25) b * ( i) i i = = bi, j, k bi, j+, k + k b i, j, k ; i + j + k =, it is easy to show that ( 26) B φ = B + E φ. This is illustated i fig. 4 ad the followig example. Example : polyomial: The followig two Bézie ets detemie the same 3. Deivatives Let u = u(s ) be the equatio of a staight lie i tems of the baycetic coodiates of T, e.g. u,. u u (s) = ( s) u + s u with two poits

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23 . Hece (27) d ds u (s) = fo (hee, = (,,)). > Fo =, we set u d = u (s) ds Sice u + v + w = we have u + v + w =. The tem u defies a diectio with espect to which we ca take diectioal deivatives. We set D d u. : =. ds Fo the Bestei polyomials B we get i Theoem : Set ( λ,μ, ν). The λ = ( 28 ) D u B i ( u ) =! ( )! λ B λ (u ) B i λ (u ) Remaks: (a) The tem ( u B ) is well-defied eve if the sum λ of the agumets does ot equal. (b) Fo i, λ that do ot satisfy i λ Poof (compoetwise), we set B ( ). u i λ = i) Fo tiple poducts of fuctios of oe vaiable s the Leibiz fomula d f (s). g (s). h (s) ds is tue. = λ! λ! μ! ν! f ( λ)(s). g ( μ) (s). h ( ν) (s)

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25 2 ii) Because of the lieaity of u(s), v(s), w(s) epeated applicatios of the chai ad poduct ules yield: d λ ( ) λ f [u (s)] = f λ (u) u etc. ds λ iii) Settig f (u(s)) = [u (s)] i etc., we obtai u! D B ( d [ u (s) ] i [ v (s) ] j [ w (s) ] k u) i = ds i! j!k! u! λ λ λ! = λ μ ν u.v.w. u i v j w k λ! μ! ν! (iλ)!( jμ)!(k ν)! λ This implies Theoem 2: The -th diectioal deivative wt u of a Bézie polyomial Bφ ove T is give by! ( 29 ) D φ = u [B ] (u ) ( )! b i + λ B λ i λ We ote that this ca be eaaged to (u ) B i ) (u (3) D u [B φ] (u )! ( )! λ B λ (u ) - = i b i + λ B i 4. Recusive Algoithm (de Casteljau 959) (u ). Let us defie a = (,,), = (,,), = (,,). a 2 a 3 Lemma 3: The B satisfy a ecuece elatio i (3) B u B (u ) v. B (u ) w.b (u ), i j k i ( u ) = i a + i a + 2 i a + + = 3 Poof:! i! k! k! Use the idetity = i i j ad the ecusio fomula fo biomial coefficiets.

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27 3 This lemma allows to expad B φ i tems of Bestei polyomials of lowe degee, with polyomial coefficiets b ( u) : i Theoem 4: (32) [B φ] (u ) = - i b (u ) B (u ), i i whee the b i ( u) ae defied by (33) b b i i ( u ) ( u ) = = u b i + a b i (u ) + v. b i + a 2 (u ) + w b i + a 3 (u ) ; i + j + k = - Poof is by iductio o. (32) is tue fo =. Iductio: [B φ] (u ) (3) = = - i b i (u ) B i (u ) b + + i (u ) [u.b (u i a ) v. B i a 2 (u ) w B i a 3 (u ) ] i Sice [B -- = i (33 ) = φ ] (u ) = - - i b [ u b (u) v b (u) w b (u) ] B + + (u) i 2 3 a i a i a i b + i (u ), (u ) B - i (u ) (33) povides a algoithm fo the evaluatio of [B φ ] (u ). This is illustated i figue 5.

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29 4. Figue 5: Costuctio fo [ B ]( 4 φ 3 2, 4, ) The b ( u) have a explicit fom simila to (4): i (34) b ( b B (u ) ; i j k -. u ) i = i + + = λ λ λ To pove this oe checks that (34) is cosistet with the ecusive defiitio (33) of the b ( u). i With (34), we ca simplify (3) to! (35) D [B ] ( u ) b ( u ) B ( u ) u φ = ( )! λ λ λ Hece to take the th (diectioal) deivative of B, we fist pefom - steps of the evaluatio algoithm (33) to obtai the b λ ( u ) ad the evaluate the Bezie polyomial (35) usig the same.. algoithm, but ow with weights u, v., w istead of u, v, w. Fo =,. (35) meas that the b ( u ) i detemie the taget plae to [ Bφ ] ( u) - fo = 2 we see that the osculatig paaboloid is detemied by the b 2 ( u ). i Aothe possibility to compute D u. B is give by! ( 36 ) D [B ) B ] ( u ) b ( u ( u ), ( )! i u i φ = i which is poved fom (29) (Fai 979). We have thus a secod method to compute the th deivative of Bφ : fist, pefom steps of.

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31 5. algoithm (33) with weights u, v, w to obtai the b (u), the i evaluate the Bézie polyomial (36) usig (33) with weights u, v, w. Actually, oe ca switch fom oe method to the othe at each step. * Let us ow evaluate deivatives acoss a bouday of T, say e 3, this implies w =. Fom (36) we see that the th deivative of B φ is a Bézie polyomial of degee - with Bézie odiates b i (u ). O the bouday e 3, this Bézie polyomial will oly deped o those b i (u ) fo which k =. Theefoe D. u [Bφ] e3 depeds oly o the + paallels (of Bézie odiates) to. e 3 Note also that i v: ( 37 ) D & [B φ ] e 3 u D u& [Bφ] e is a (-)th degee uivaiate polyomial 3! = ( )! j = b i ( u) & 3 B (v) j whee i 3 is shot fo (--j, j,). * The elatioship of this statemet with the uivaiate case becomes clea if we view tems i. u as geealizatios of the diffeece opeato Δ.

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33 6. III Composite Sufaces. C -Cotiuity betwee adjacet tiagles Let a Bézie polyomial Let a secod tiagle P + T 2 = B φ be defied ove a tiagle T = P P 2 P 3. P P 4 P 2 4 = u P + v P 2 w P3 with be give. We seek a Bézie polyomial Bφ 2 defied ove T 2 that has C -cotiuity with B φ alog the commo edge P P 2. Let the baycetic coodiates i xists a liea tasfomatio T i be u i, i =,2. The thee.. u 2 = u.a, u 2 u. A ( 38) = with a osigula matix A such that a = u. A, a 2 = a 2. A, a 3 = a. A (see also Fig. 6). We fid fo A: w w ; w = v u A = w

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35 7. Let the Bézie odiates of Bφ, be b i, those of B φ be c. i The th coss-bouday deivative with espect to some diectio.. u (esp. u 2 ) of B φ i is detemied by the (+) ows of Bezie odiates i Ti paallel to the edge e3. The ext theoem gives a simple method to compute the elevat c i fom the elevat b i. Theoem 5: With the above otatios Bφ ad Bφ 2 have alog e 3 if ad oly if C -cotiuity ( 39) c, j, j b ρ ρ ρ = ρj, j, (u o ) ; ρ j ρ Example 2: Fo =, (39) becomes fo ρ = : c, j, j = bj, j, ;, j ad fo ρ = : c b, j, j = j, j, (u o) j = u b j, j, + v b j, j+, + w b j, j,. The fist equatio esues that Bφ ad Bφ 2 have a commo bouday cuve. The secod equatio states that evey shaded quadilateial i fig, 7 is plae. (Figue 7 shows the pojectio ito the plae oly). Poof of theoem 5: Let i be of the fom (, j, k) ad i 3 of the fom (i, j, ). (37) gives the C -coditio

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37 8. ρ j= b ρ ρ ρ ρ ρ = i 3 ( j 2 j i (. ) B u. ) B (v) c u (v); ρ. j= Compaiso of coefficiets yields ρ. ρ. (4) b 3 (u ) = c (Au) ; i i ρ j ρ ρ. The tem b (u ) i 3 ca be viewed as the ρ -th diectioal deivative of the Bézie polyomial defied by b (u ) i 3 ; the same is tue fo ρ. (Au ). These two polyomials coicide i thei deivatives up to c i ode ρ : hece they ae equal: ρ b ρ (. u ) c (Au) i ρ 3 = i ; ρ j ρ. This is tue fo all u, e.g. also fo u = u : ρ i ρ b 3 ( u ) = c (Au ) i = = ρ c i c (,, ) ρ,j,ρ j ρ j ρ Example 3: Coside the two tiagles below. cetoid of P, P 4, P 3, such that P 4, = 3P 2 P P 3. Let P 2 be the P 4 has baycetic coodiates u = (-, 3, -) with espect to T.

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39 9. Let φ be defied ove T by the Bézie odiates We seek φ2, defied ove T2, such that B 3 φ ad B 3 φ 2 have commo fist ad secod coss-bouday deivatives alog P, P2. Theoem 5 suggests the followig costuctio of the Bézie odiates ci of φ 2 : Step : c b, j, 2 j = 2 j, j, (, 3, ). The scheme of the b i is give by ad hece the udelied umbes ae the desied c. Step 2: 2 c 2, j, j = b j, j, o (, 3, )., j, 2 j The scheme of the 2 b i is give by The Bézie odiates of ad φ 2 φ ae theefoe

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41 2. Example 4: Suppose we ae give φ ad φ 2 fom the pevious example ad wat to veify that the two coespodig Bézie polyomials joi i C 2. This is easily accomplished by meas of. (4): Choosig u to be a diectio pepedicula to P P2, we get... u. = [, 3, 2]; u 2 = u A = [2, 3, ]. The C coditio becomes (fo j = O,, 2) (4) 3 b j, j+, = b j, j, + c o, j, j + b j, j, ad fo the C 2 coditio we get (fo j =,) b j, j, 3 b 2 j, j+, + b 2 j, j, 2 (42) = c, j, j 3 c, j+, j + b 2, j, 2 j.

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43 2. 2. Degees of Feedom Theoem 5 eables us to costuct a Bφ 2 that jois a give B φ i C. Thee ae *) Bezie odiates i φ 2 that we ca specify abitaily, the emaiig oes beig fixed by C -cotiuity. Sice we could specify all + Bézie odiates of φ abitaily, the piecewise suface give by Bφ ad Bφ 2 has of feedom (d.o.f's). + + degees This costuctio may be epeated, thus addig ew B φ. d.o.f's fo each We may evetually ecoute a situatio whee a ew added i this fashio because two of its edges ae shaed by peviously costucted Bézie polyomials. give examples whee such a costuctio must fail. B φ caot be Oe ca i fact easily It is always possible, howeve, to add two Bézie polyomials simultaeously as is show i figue 8. Ou poblem is ow to detemie how may Bézie odiates i these two tiagles ca be abitaily specified; we call this umbe of degees of feedom ρ(,). *) We defie k = k(k + ) = k. 2

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45 22. Theoem 6 2 = + ( 2)( ) ( ) 2 (43) ρ(, ) = 2 = The method used fo the poof is illustated i example 5: seach evey quadilateal of "side legth" + esposible fo the umbe of d.o.f.'s it offes, poceedig fom left (close to pedetemied poits) to ight. pocedue is give i Fai 979. C fo A moe detailed desciptio of this Remak: The two vetices shaed with peviously detemied poyomias must ot fom a staight ie. Example 5. (see Fig. 8 ad the C coditios i Fig. 7). Let the " " be detemied by C, the " " is fixed because the quadilateal must be plae (this justifies the above emak). We ca specify the " " abitaily; togethe with " ", they will detemie "O". Hece, ρ(3,l) = 2. Coside a tiagle T that is subdivided ito thee subtiagles T i by its cetoid (see Fig. 9). Defie τ to be the liea space of -th degee polyomials defied ove each T i ad joiig i C. Its dimesio is give by (44) dim τ = + + ρ(,) This leads to a somewhat supisig esult:

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47 23. Theoem 7 (45) τ = τ Poof: Combiig (44) with (43), we get dim τ dim = τ. (45) follows sice is a subspace of τ. A simple cosequece is Coollay 8: Evey elemet of τ has cotiuous deivatives of ode O,,..., + at the cetoid of T. Poof: A elemet of τ cotais a subtiagle that ca be cosideed a elemet of τ +. This subtiagle is esposible fo the deivatives at the cetitoid, ad a applicatio of Theoe 7 completes the poof. IV Itepoatio i τ `.. The case τ 3 We defie Q k j j i, j = P i + ( ) P k + k + i +, *) m = - 2. Let D f(p i ); i =,2,3; deote all + patials of ode,,... of f at P i ; we shall always equie that the thee D f (P ) cosistet with each othe (this is tivially the case if 2 < ) Let u. i deote a diectio ot paallel to the edge Pi P i+. *) vetices ae couted mod (3). i be

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49 24. Ou itepolatio poblem will be: Fid a elemet f i τ that assumes the followi ( V ) D f ( P i ) ; i =, 2, 3 (E ) ρ D ui f (Q m + ρ i, j ) ; i =,2,3 ρ =,,..., j =,,..., m + ρ <. Note that ( E) is void fo m+ρ<l. The solutio to the itepolatio poblem give by (V ) ad (E ) fo 3 τ is kow as the C clough-toche scheme (Stag/Fix 973). This solutio ca easily be costucted usig Bézie polyomials; it cosists of thee steps, see also fig. 9. Fig. 9; Costuctig the C solutio.. The Bézie odiates " " ae give by (V ); the " " stem fom (E ). 2. C acoss the iteio vetices detemies the "o", cf. (4) 3. C at the ceto id detemies " ": it has to be the cetoid of the "o". Note that this costuctio also implies the uiqueess of the

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51 25. itepolat. costucted i cetoid. Moeove, Coollay 8 implies that it has (though C cotext) cotiuous secod deivatives at the This itepolatio scheme has cubic pecisio. The above case = caot be geealized: Theoem 9: The itepolatio poblem fo τ give by ( V ) ad ( E ) is ovedetemied fo 2, + 2. Poof: I fig., let " " deote Bezie odiates detemied by ( V ) ad ( E ) (fo (, ) = (5,2)). Figue : Icompatibility i itepolatio poblem. The "fat buttefly" - which is esposible fo C 2 cotiuity, see (42) - is detemied by six idepedet pieces of ifomatio, five of which ae aleady fixed; i.e. oe of the two Bézie odiates " " detemies the buttefly completely (by equatios (42) ad (4). Sice ( E ) pescibes both of them abitaily, the poblem is ovedetemied.

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53 2. 2 C Itepolatio i τ Theoem 9 suggests to coside the followig C 2 -itepolatio poblem: Fid a elemet i τ 6 that satisfies (E2) ad ' (V3 ). V ' 3 demads that thid deivatives paallel to edges ae ogaized as to detemie seve (istead of eight) Bézie odiates pe bouday cuve. This does ot maitai C 3 cotiuity at the vetices ay moe, by (a) etais C 2 cotiuity thee ad (b) elimiates the icompatibility that caused the failue of schemes usig (E 2 ) ad (V 2 ). The choice of τ 2 6 is suggested by the followig easoig: We wat to be able to solve the C 2 poblem i adjacet tiagles. If we wee wokig i, say, τ (defied fo each tiagle), C ifomatio alog the commo edge ad (cosistet!) C 3 ifomatio at the coespodig vetices would ot guaatee C 2 cotiuity betwee the two itepolats; but it is guaateed usig τ 2 6 We shall ow tu to the solutio of the C 2 poblem. 2 Sice dim τ 6 = 37 ad (E2) ad (V ' 3 ) povide 33 costaits, we may specify fou additioal Bézie odiates. Agai, the costuctio of the solutio cosists of thee steps, see Figue : Step : The Bezie odiates "* " ad the ai, i=l,..., 5, ae give by (V 3 ) ad (E 2 ).

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55 27. Figue : Costuctig the C 2 solutio. Step 2: We specify the Bezie odiates s, s 2, s 3 i ode to detemie the g i. This is doe by solvig a 2 x 2 liea system fo each of the "fat butteflies", e.g. (4') a 5 + g 5 + g 4 = 3s 3. (42') a l + a - 3g 5 = a 7 + a 8-3g 4. These two equatios ae eadily solved fo g 4 ad g 5. Step 3: We specify s 4 ad detemie the x i by solvig a 6 x 6 liea system, the fist fou equatios beig applicatios of (4), the last two coespodig to (42).

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57 28. The system is (46) g g g g g g g g 3s s s s x x x x x x = ad has the solutio (47) g g g g g g g g 3s s s s x x x x x x + + = The above choice of the is ot the oly oe possible; but it s i miimizes the sizes of the liea systems that have to be solved. Coollay 8 ad the deivatio of ρ(,) yield Theoem The above scheme has sextic pecisio. Ay itepolat costucted by it has cotiuous thid deivatives at the cetoid.

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59 29. V. The Dimesio of τ (i, b) Let τ (i,b) be a simply-coected (but ot ecessaily covex) tiagulatio with i iteio vetices ad b bouday vetices, such that the piecewise liea) bouday cuve is a simply closed cuve. We exclude tiagulatios that cotai vetices whose sta is a covex quadilateal with the diagoals daw i.*) Let τ (i, b) be the liea space of C piecewise polyomials of degee ove τ (i,b). Theoem (48) dim τ (i,b) = + + i.ρ(,) + (b 3) Poof: We use iductio o the umbe of tiagles i τ (i,b) ( τ (i,b) cosists of b+2i-2 tiagles).. If τ (i,b) cosists of oe tiagle oly, dim τ (,3) = Suppose (48) holds fo a simply-coected subtiagulatio τ (j, c) of τ (i,b). We add a ew tiagle to τ (j',c'). We have to coside two cases. τ (j, c), thus obtaiig Case a: j' = j, c' = c+ dim τ (j, c + ) = = dim τ + + (j,c) + j. ρ(, ) + (c 2). *) This estictio is a cosequece of the emak afte theoem 6.

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61 3. Case b: j' = j +, c' = c- by ρ(,) whe two tiagles ae added simultaeously (see figue 8), q(,) deotes the chage if oly oe tiagle is added: dim τ (j +,c ) = dim τ (j,c) + q(, )*) = + + (j = ) p (,) + (c 4) Remaks:. Evey simply-coected tiagulatio ca be costucted by usig steps a) ad b) fom the above poof. 2. If a optimizatio pocedue is applied to a tiagulatio (Bahill 977), the dimesio of the coespodig liea spaces does ot chage. 3. The above poof ca be used to costuct a basis fo τ (i,b) i tems of Bezie polyomials. 4. Fo 4, =, theoem coicides with a esult obtaied by Moga/Scott, 975. The poof of theoem ca easily be adapted to tiagulatios τ '(i,b) that have a hole, whee the vetices aoud the hole ae cosideed bouday vetices: Thus addig a tiagle to τ (i,b) may decease dim τ (i,b)!

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63 3. Coollay 2 (49) dim τ ' (i,b) = + + (i + )ρ (,) + (b -3) - Theoem implies that uivaiate B-splies caot be geealized : Theoem 3 No τ (i,b) ca cotai a o-zeo elemet f such that i) f is idetically zeo outside τ (i,b) ii) f ε C (IR x IR ) Poof: Suppose such a f existed. Costuct a tiagulatio τ ( i,3) that cotais τ(i,b). (This is tivially possible sice τ (i,b) is fiite.) i) ad ii) imply f ε τ (i,3). Sice dim have τ (i',3) = +, we τ (i',3) = P (the liea space of bivaiate polyomials of degee ), i.e. f is a (global) polyomial But o o-zeo polyomial ca satisfy i).

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65 REFERENCES. Bahill R.E. 'Repesetatio ad Appoximatio of Sufaces'. Mathematical Softwae III. J. Rice (ed.), Academic Pess, New Yok (977). 2. Bézie P. 'Emploi des Machies a Commade Numéique'. Masso & Cie, Pais (97). Taslatio ito Eglish by R. Foest: Numeical Cotol - Mathematics & Applicatios. Wiley & Sos, Lodo (972). 3. Bézie P. 'Essai de Défiitio Numéique des Coubes et des Sufaces Expéimetales.' Dissetatio, Pais (977) 4. de Casteljau 'Coubes et Sufaces à Pôles'. "Eveloppe 4.4", Istitut Natioal de la Popiété Idustielle. Pais (959) 5. Davis P. 'Itepolatio & Appoximatio'. Dove, New Yok (975). 6. Fai G. 'Subsplies uebe Deiecke.' Dissetatio. Bauschweig (979). 7. Foest A.R. 'iteactive Itepolatio ad Appoximatio by Bézie Polyomials'. The Compute Joual 5, S.7-79, (972). 8. J. Moga/R.Scott. 'A Nodal Basis fo C' Piecewise Polyomials of Degee > 5. Math, of Comp. 29, pp , (975). 9. Sabi M.A. 'The Use of Piecewise Foms fo the Numeical Repesetatio of Shape'. Dissetatio, Budapest (977).. Stak E. 'Mehfach diffeeziebae Béziekuve ud BézieflSche.' Dissetatio, Bauschweig (976).. Stag G. ad Fix G. 'A Aalysis of the Fiite Elemet Method'. Petice-Hall, New Yok (973). ACKNOWLEDGMENTS wish to thak R. E. Bahill (Utah/Buei), J.A. Gegoy (Buel), ad F. F. Little (Utah) fo may helpful discussios. This eseach was suppoted by the Sciece Reseach Coucil with Gat GR/A ad by the Natioal Sciece Foudatio with Gat MCS to the Uivesity of Utah.

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