Interpolatory Subdivision Curves with Local Shape Control

Transcription

1 Interpolatory Subdvson Curves wth Local Shape Control Carolna Beccar Dept. of Mathematcs Unversty of Padova va G.Belzon 7, Padova, Italy Gulo Cascola Dept. of Mathematcs Unversty of Bologna P.zza d Porta S. Donato 5, 4016 Bologna, Italy cascola@dm.unbo.t Luca Roman Dept. of Mathematcs Unversty of Bologna P.zza d Porta S. Donato 5, 4016 Bologna, Italy roman@dm.unbo.t ABSTRACT In ths paper we present a novel subdvson scheme that can produce a nce-lookng nterpolaton of the control ponts of the ntal polylne, gvng the possblty of adjustng the local shape of the lmt curve by choosng a set of tenson parameters assocated wth the polylne edges. If compared wth the other exstng methods, the proposed model s the only one that allows to exactly reproduce conc secton arcs of arbtrary length, create a varety of shape effects lke bumps and flat edges, and mx them n the same curve n an unrestrcted way. Whle ths s mpossble usng exstng 4-pont nterpolatory schemes, t can be easly done here, snce the proposed subdvson scheme s non-statonary and non-unform at the same tme. Keywords Non-unform and Non-statonary Subdvson, Curves, Interpolaton, Tenson, Localty. 1 INTRODUCTION Curve-subdvson s an teratve process that defnes a smooth curve as the lmt of a sequence of successve refnements appled to an ntal polylne. If, at each refnement level, new ponts are added nto the exstng polylne and the orgnal ponts reman as members of all subsequent sequences, becomng ponts of the lmt curve tself, the scheme s called nterpolatory. In 1987 Dyn et al. [Dyn87a] ntroduced the frst nterpolatory subdvson scheme for curves, known n the lterature as the "classcal" 4-pont scheme. Ths name derves from the fact that, startng from a coarse control polygon, the new ponts recursvely ntroduced between each par of old ponts are computed by a lnear combnaton of the mmedate four neghborng ponts. The proposed scheme can generate smooth nterpolatory subdvson curves characterzed by a tenson parameter (denoted by ω) that can only be used to control the shape of the whole lmt curve. Another dsadvantage of ths method s that the global parameter ω acts as a tenson parameter only n a very restrcted range. Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. FULL Papers Conference proceedngs, ISBN WSCG 006, January 30 February 3, 006 Plzen, Czech Republc. Copyrght UNION Agency Scence Press More precsely, f 0 ω 1 < ω 16 1, the lmt curve correspondng to ω wll be looser than the one correspondng to ω 1. But outsde of the range [0, 16 1 ] nothng can be predcted. Recent works n nterpolatory subdvson theory [Mar05a, Bec05a, Oht03a] mproved the performance of the "classcal" 4-pont scheme, but, although provdng users many choces to ft the dfferent requrements n applcatons, they stll suffer from some lmtatons to be very useful n future modellng systems. In fact, even though some of them provde degrees of freedom for controllng the shape of the lmt curve, no nterpolatory schemes have been desgned yet to make user manpulatons as ntutve as possble and allow the generaton of a curve whose behavor can be locally modfed n any desrable way. These consderatons motvated the research reported n ths paper, where we propose a novel scheme that contans [Dub86a, Dyn87a] as specal cases and generalzes the non-statonary and unform nterpolatory 4-pont scheme n [Bec05a] to possess local shape desgn parameters whch have a vsual nterpretaton on the screen. Due to ts versatlty, the scheme we are gong to present can be used convenently both for nteractve desgn and for automatc curve fttng. Therefore t s very well-suted for desgn applcatons n varous felds ncludng Image Processng, CAGD and Computer Graphcs. The paper s organzed as follows. In Secton we construct and analyze the novel scheme. In Secton 3 we nvestgate ts propertes and we show ts performance. Next n Secton 4 we conclude the paper

2 wth some examples and propose some applcatons for whch the novel scheme s very well-suted. DEFINITION OF THE NOVEL IN- TERPOLATORY 4-POINT SCHEME In ths secton we are gong to present a novel nterpolatory 4-pont scheme that allows to adjust the shape of the lmt curve by choosng a set of local tenson parameters that, when ncreased wthn ther span of defnton, allow to apprecate consderable varatons of tenson n the correspondng regons of the lmt curve. Such tenson parameters are assocated wth the edges of the coarsest polylne and are opportunely updated at each refnement level. In ths way, the mask whch provdes a rule to pass successvely from one set of control ponts to the followng, s smultaneously non-statonary (snce t s dfferent for each teraton of the subdvson process) and non-unform (snce t also changes along the curve). However, the scheme s very easy to mplement because the same rule for generatng the new tenson parameters s used at each subdvson step. Consder an orented polylne P 0 = {p 0 } Z (where the upper ndex 0 ndcates that no subdvson steps have been appled yet) and assgn a tenson value v,0 ] 1,+ [ to each edge p 0 p0 +1. The subndex (,0) underlnes that the tenson value v,0 s assocated wth the -th edge of the polylne defned at the 0 th subdvson level. Next, for any k 1, update the tenson parameters v,k 1 nto v,k through the relaton 1 + v,k 1 v,k = (1) and successvely compute the coeffcents w,k = 1 8v,k (1 + v,k ) () that wll be used to map the polygon P k 1 = { } Z to the refned polygon P k = {p k } Z by applyng the followng subdvson rules: p k = (3) p k +1 = w,k ( + ( 1 + w,k 1 + pk 1 + ) ) ( + +1 ). Remark 1. Snce after each round of subdvson one new pont s nserted between two old ones, every edge of the old control polygon P k 1 s splt nto two new edges n the refned one. Thus, denoted by v,k 1 the tenson parameter assocated wth the edge +1, snce such an edge s splt nto the two new edges p k pk +1, pk +1 pk +, accordng to (1) we wll make them nhert respectvely the tenson values v,k that s v,k = v +1,k = v,k. Remark. Note that the ntal tenson values v,0 are updated before computng the coeffcents w,1 used n the frst subdvson step. Hence, for any choce of v,0 ] 1,+ [, (1) mples that v,k ]0,+ [ k 1 and () leads to w,k > 0 k 1..1 Convergence analyss In ths paragraph we are gong to show that our novel nterpolatory subdvson scheme always generates a C 0 -contnuous lmt curve. More precsely, we wll prove that, gven an ntal polylne P 0, the subdvson rules n (3) defne an ncreasngly dense collecton of polylnes P k that converge to a contnuous curve. To show ths, we wll explot two mportant propertes of our scheme descrbed n the followng lemmas. Lemma 1. Gven the ntal parameter v,0 ] 1,+ [, the recurrence relaton n (1) satsfes the property: lm v,k = 1. (4) k + Proof 1. To prove ths we recall that a monotonc and bounded sequence s always convergent and n partcular: f t s non decreasng and upper bounded, then t converges to the upper bound of the values t assumes, whereas f t s non ncreasng and lower bounded, then t converges to the lower bound of the values t assumes. For the sequence defned by v,0 ] 1,+ [ (5) 1+v,k 1 v,k = k 1 t holds: - f v,0 = 1, then the sequence {v,k } k 1 s statonary; - f v,0 ] 1,1[, then the sequence {v,k } k 1 s non decreasng; - f v,0 ]1,+ [, then the sequence {v,k } k 1 s non ncreasng. Therefore, n all these cases v,k s convergent and converges to 1. In fact, called l ts lmt, we have ( ) 1 + l = lm v v,k l,k = lm = k + k +. Thus, solvng the last equaton wth respect to l we get l = 1. Lemma. Due to the recurrence (1), the parameters v,k 1 and v,k satsfy the relaton 1 v,k 1 v,k 1 < 1 for any v,k ]0,+ [, k 1. (6)

3 Proof. Explotng recurrence (1) we can wrte the rato 1 v,k 1 v n the followng way,k 1 1+v,k 1 1 v,k = 1 1 = 1 v,k 1 1 v,k v,k 1 and observe that snce 1 + v,k 1 > 0 k 1, thus 1 v,k 1 v,k 1 < 1. Now, explotng the well-known theorem n [Dyn95a], that relates the convergence of a non-statonary scheme to ts asymptotcally equvalent statonary scheme, we are gong to show the followng result. Proposton 3. The locally controlled scheme n (3) converges and generates C 0 -contnuous lmt curves. Proof 3. To prove our thess we need to show that, gven an ntal polylne, the novel locally controlled scheme defned by the followng mask [ m k 1 1 = 8v 1,k (1 + v 1,k ), 0, (v,k + 1) 8v,k (1 + v,k ), 1, (v +1,k + 1) 8v +1,k (1 + v +1,k ), 0, 1 8v +,k (1 + v +,k ) converges to a C 0 -contnuous lmt curve. Snce from (4) t follows that m lm k + mk 1 = [ 1 16, 0, 9 16, 1, 9 16, 0, 1 16 ] (7) then the non-statonary subdvson scheme assocated wth (7) converges to the statonary scheme n [Dub86a], that s exactly the "classcal" 4-pont scheme n [Dyn87a] obtaned wth ω = Next, by computng m k 1 m we have that m k 1 m = 1 v 1,k ( + v 1,k ) 16v 1,k (1 + v 1,k ) + 1 v,k ( + v,k ) 16v,k (1 + v,k ) + 1 v +1,k ( + v +1,k ) 16v +1,k (1 + v +1,k ) + 1 v +,k ( + v +,k ) 16v +,k (1 + v +,k ) where v 1,k, v,k, v +1,k, v +,k ]0,+ [, k 1. Now, by the well-known theorem n [Dyn95a], f ], (8) + m k 1 m <, (9) k=1 then the two schemes are asymptotcally equvalent, and snce m s C 0, we can conclude that the scheme assocated wth m k 1 s C 0 too. Thus, to show our thess we only have to prove that relaton (9) holds as the parameters v 1,k, v,k, v +1,k, v +,k vary n the nterval ]0,+ [. For space lmtatons, we just gve a bref sketch of the proof here. Observe that, from the analytc propertes of the seres, once we have proved the convergence of each term n (8), relaton (9) wll mmedately follow. To ths am, thanks to the results n lemma 1 and, we can repeat, for each term of the sum n (8), the calculatons presented n [Bec05a] and clam that the schemes m k 1 and m are asymptotcally equvalent, whch proves our thess. Remark 3. Note that the proposed scheme s asymptotcally equvalent to the "classcal" nterpolatory 4- pont scheme n [Dyn87a] wth parameter ω = 16 1, that s C 1. Hence, snce accordng to [Dyn95a], the contnuty level of a non-statonary scheme s, n general, the same as that of the statonary scheme to whch ths scheme converges, we expect the lmt curve generated by the scheme n (3) to be C 1. 3 THE LOCAL TENSION PARAME- TERS The novel scheme descrbed by (3) has a very powerful and ntutve geometrc nterpretaton. If we wrte the second lne of (3) n the form p k +1 = pk ( p k 1 + w,k + +1 pk pk 1 + (10) ) t s clear that the new pont p k +1 turns out to be the md-pont of the edge +1, corrected by a vector w,k e, where e denotes the vector connectng the md-ponts of the edges +1 and pk 1 1 pk 1 + (see Fgure 1). 1 p k +1 w e,k e +1 + Fgure 1: Geometrc nterpretaton of the local parameter w,k.

4 The value of w,k n (10) clearly depends on the choce of the ntal tenson parameter v,0. For dfferent values of v,0 rangng from -1 to +, we wll have dfferent values of w,1 n ]0,+ [. The mportant cases to be mentoned are v,0 = 1 (w,1 = 1 16 ) and v,0 + (w,1 0). In the frst case, n fact, the subdvson scheme becomes statonary and the lmt curve we obtan concdes wth the one generated by [Dub86a] and by [Dyn87a] choosng the parameter ω equal to In the second case, nstead, the porton of curve confned to the endponts of the -th edge s pulled towards the lnear nterpolant of those two ponts, snce when w,1 s exactly zero, the scheme exactly generates the lnear nterpolant to the ntal control ponts. In partcular, progressvely ncreasng the value of v,0, the porton of curve confned to the endponts of the - th edge wll become progressvely tghter and tghter. It s nterestng to note also that, by settng all ntal tenson parameters equal to the same value v 0, we get the unform tenson-controlled nterpolatory scheme defned n [Bec05a]. 3.1 Local tenson parameters provded by the user The local tenson parameters v,0 that defne the nterpolatory scheme presented n Secton, can be ether arbtrarly provded by the user to ntutvely model the lmt shape, or automatcally set to let the lmt curve ft specal requrements n applcatons. In Fgure we show an example of nteractve desgn where the local tenson parameters play an mportant role n the overall dsplay of the fnal shape. The top left curve has been obtaned by settng all the parameters {v,0 } =0,...,6 to γ = cos( π 7 ). As explaned n [Bec05a], such a confguraton of tenson values, appled to the regular heptagon denoted by the dashed lne, ( generates the exact) crcle nterpolatng ts vertces cos( jπ jπ 7 ),sn( 7 ), j = 0,...,6. The consecutve fve nterpolants demonstrate the role of the local tenson parameters specfed n the array v, by showng ther nfluence on the resultng shapes. Comparng the sx fgures, we can see that, by opportunely modfyng the tenson values γ, we can obtan dfferent ways of achevng local shape control on the lmt curve. These results and many others, obtaned usng dfferent ntal polylnes and varous local tenson parameters, confrm our conjecture about the C 1 -smoothness of the curve generated by the refnement scheme n (3). Addtonally, they pont out that the model we have proposed exhbts many propertes that a subdvson scheme should nclude to become useful for geometrc modellng applcatons. Indeed, t allows: Intutve shape deformatons Among all the shape parameters that we can fnd n v=[γ, γ, γ, γ, γ, γ, γ] v=[1000, 1000, -0.8, 1000, -0.95, 1000, -0.8] v=[γ, γ, γ, 1000, γ, γ, γ] v=[γ, γ, γ, -0.85, γ, γ, γ] v=[γ, γ, 1000, 1000, 1000, γ, γ] v=[γ, γ, -0.85, 1000, -0.85, γ, γ] Fgure : Closed subdvson curves produced by nterpolaton of the vertces of a regular heptagon usng the tenson parameters n the ndcated array v where γ = cos( π 7 ). exstng subdvson models, the ones that provde a tenson effect ndeed appear totally ntutve and possess a drect vsual nterpretaton on the screen (see Fgure ). Local control Each tenson parameter, assocated wth a specfed edge of the ntal polylne, nfluences the shape of the lmt curve only n a restrcted zone, correspondng to the related edge and the neghborng parts of ts two adjacent edges. Ths allows a very powerful local shape control (see Fgure ). Warpng, flattenng and mxed effects Warpng s a tool whch can be used to deform a local segment of a curve pullng out a bump; flattenng s a tool whch allows to easly ntroduce straght lne segments nto curves. The local tenson parameters can reproduce both warpng and

5 flattenng behavors as well as provde a varety of nterestng shape effects. The subdvson curves that we can generate, n fact, are able to ncorporate both round shapes and flat shapes, lke bumps and localzed flat edges. Addtonally such effects can be mxed n the same curve n an unrestrcted way, always allowng soft transtons between them (see Fgure ). 3. Local tenson parameters automatcally set In ths subsecton we wll show the capablty of the proposed subdvson scheme to ft certan classes of curves wdely used n CAGD applcatons, lke polynomals, conc sectons and pecewse cubc Bézer curves. Snce an nterpolatory subdvson process defnes a smooth curve as the lmt of a sequence of successve refnements appled to an ntal polylne, the approxmatng algorthm we are gong to descrbe requres to select an adequate set of ponts on the reference curve C to start the locally controlled refnement procedure descrbed n Secton. Note that, the greater the number of selected ponts s, the more the approxmatng shape tends towards the orgnal curve C, but the more the shape descrptor sze and the number of computatons ncrease. To balance these two extreme scenaros, the strategy adopted for determnng the ntal polylne to whch apply the refnement process should select the mnmum number of ponts needed to get a vsually precse reconstructon. By the term vsually precse we mean that, f you compare our nterpolatory subdvson curve wth the orgnal one, you can hardly spot the dfferences between them (see Fgures 4, 5, 6, 7). Let P 0 = {p 0 } Z be the ntal sequence of ponts opportunely sampled on C. Due to (10), the ponts p 1 +1 computed at the frst subdvson step, wll be placed on the lne r passng through the md-ponts of the edges p 0 p0 +1 and p0 1 p0 +, at a dstance from p 0 +p0 +1 nversely proportonal to the tenson value v,0. Therefore, n order to acheve a vsually precse fttng, each pont p 1 +1, defned n correspondence of the - th edge p 0 p0 +1 of the coarsest polygon P0, should be placed exactly on the curve C. Accordng to ths request, the ntal tenson value v,0 wll be determned n such a way the pont p 1 +1 turns out to be the ntersecton pont between the lne r and the curve C. Repeatng ths procedure for each curve segment, we wll be able to automatcally determne a tenson value v,0 that, used n the refnement process (3), generates the nterpolatory subdvson curve that approxmates the gven curve C. Although we have no guarantee that n the followng refnement levels the subdvson process wll nsert all the new ponts on the curve C, whenever we start from a sequence of ponts P 0 = {p 0 } Z sutably sampled on C, all our experments showed vsually precse reconstructons. The followng algorthm mplements an automatcal procedure to get an nterpolatory subdvson curve that turns out to be vsually precse n respect to the orgnal curve C. Algorthm 1 1. Determne the ntal polylne P 0 = {p 0 } Z;. compute the tenson parameters v,0 assocated wth the edges p 0 p0 +1, n such a way the ponts p1 +1, nserted through (3), le on C (for each edge of the ntal polylne, ths requres to solve the system of equatons dervng by forcng the pont p 1 +1 to be placed on C); 3. for k = 1 to the desred refnement level 3.1 compute the tenson values v,k through (1); 3. compute the coeffcents w,k through (); 3.3 apply the refnement rules (3); 4. stop. Remark 4. Note that, due to the rule n the second lne of equaton (3), to nsert a new pont p k +1 n the refned polylne P k, t s necessary to possess a well defned two-neghborhood (gven by two ponts on ts left and rght sde n the coarsest polylne P k 1 ). Hence, f C s an open curve, the ntal polylne P 0 should be extended to contan two more sample ponts both at the begnnng and at the end of the ntal sequence P 0 = {p 0 } Z. In the followng we wll show how the proposed algorthm leads to some useful applcatons of our locally controlled nterpolatory 4-pont scheme. The frst one s related to two desrable propertes of subdvson schemes, namely polynomal precson and concs reproducton. These two terms usually refer to the capablty of exactly reproducng a specfed curve by applyng the subdvson scheme to a set of control ponts unformly sampled on t. As far as we know, exstng nterpolatory subdvson schemes are able to reproduce nether polynomals nor conc sectons startng from unevenly sampled ponts. Explotng Algorthm 1 we can show that our nterpolatory scheme turns out to be exact for lnear polynomals even f the ntal data are unevenly sampled on them. Addtonally, a vsually precse ft to ponts unevenly sampled on any quadratc or cubc polynomal and any arbtrary conc secton, can be acheved too (see Fgure 3).

6 the subsequence consstng exactly of all the juncton ponts of the sngle Bézer segments. Next, to defne the ntal polylne P 0 for generatng the nterpolatory subdvson curve, we add supplementary ponts wherever two consecutve selected ponts (so far computed) turn out to be too dstant. Accordng to our experments ths strategy always ensures a vsually precse reconstructon. Fgure 3: Representaton of polynomals and conc sectons va the nterpolatory subdvson curve wth local shape control (dashed lnes: polylnes; sold lnes: subdvson curves). On the other hand, t can be easly verfed that, when we consder sample ponts unformly spaced, the proposed scheme s exact for cubc polynomals. In fact, performng the computatons descrbed n Algorthm 1, we get that the same tenson parameter v,0 = 1 should be appled on the whole curve. Ths value concdes wth the global tenson parameter ntroduced n [Bec05a] to exactly reproduce cubc polynomals. Analogous results can be found repeatng the same computatons when the ntal data are evenly sampled on a conc secton [Bec05a]. Therefore, when startng from unformly dstrbuted ponts, we can conclude that by applyng Algorthm 1 we can work out the ntal tenson parameters that allow the subdvson scheme n (3) to be exact for cubc polynomals and to exactly reproduce all conc sectons. Snce n CAGD applcatons curves are frequently represented n the pecewse cubc Bézer form, we want to show now an addtonal applcaton of our algorthm whch allows to obtan a very good representaton of a curve C n the pecewse cubc Bézer form. Denoted by Q 0 the sequence of control ponts defnng the pecewse cubc Bézer curve C, we select from Q 0 4 APPLICATIONS The proposed subdvson scheme was desgned to make our model a canddate of choce for many applcatons. An example conssts n proposng ths novel class of nterpolatory subdvson curves as an effcent and economcal representaton of the outlne curves used to descrbe fonts as well as output mages of tracng algorthms [Sel03a]. In general, a vector outlne descrbes a dgtalzed mage va a famly of pecewse cubc Bézer curves that may jon ether C 0 or C 1 contnuously. Our subdvson scheme provdes a mathematcal descrpton of vector outlnes that turns out to be smpler and more effcent than the one currently used n Postscrpt fles and tracng algorthms. In fact, although n vectoral outlnes one swtches frequently between round shapes and flat shapes, by usng only one subdvson curve wth local shape control, we wll be able to make a good job (see Fgure 4). Fgure 4: Two pecewse cubc Bézer curves descrbng the outlne of the character "@" - 73 cubc segments (left); two sngle-pece nterpolatory subdvson curves - 75 ponts, 73 tenson parameters (rght). Remark 5. Note that, due to the smoothness propertes of the scheme, to make the job perfectly, and then allowng the possblty of reproducng sharp corners between flat regons and round ones, we need to generate dstnct subdvson curves and sttch them together n the corners (see Fgure 5). Now we pont to a number of advantages that could make ths novel class of nterpolatory subdvson curves the standard representaton of vectoral outlnes. The prmary advantage of nterpolatory subdvson curves wth local shape control over pecewse cubc Bézer curves has to do wth the physcal storage

7 Fgure 5: Two pecewse cubc Bézer curves descrbng the outlne of the character "d" - 4 cubc segments (left); the correspondng pecewse nterpolatory subdvson curves - 9 sngle-pece subdvson curves, 33 ponts, 4 tenson parameters (rght). of vector outlnes n fles. In fact, as the set of local tenson parameters act as handles to model the contour that best fts the orgnal mage, explotng a small number of ponts per contour, we can obtan a very compact numercal format for representng outlne curves. Snce the necessary nformaton to represent the outlne curve are 1 endpont and 1 tenson parameter per segment, we manage to remarkably compress the data to be stored n the fle and optmze them for speed and for handlng contour mages wth large character sets. Another consstent advantage s that our technque turns out to be also very convenent for computng the ponts requred for the rasterzaton of the fnal curve, whch conssts n convertng the outlne to a pattern of dots (whether t s screen pxels or the dots of a laser, nkjet or wre-pn prnter) on the grd of the output devce. All these observatons let us notce that ndeed there are several key mprovements whch may swng the future n curve-subdvson s favor. Whle subdvson surfaces have not been wdely adopted by the CAGD communty [Gon01], maybe some day subdvson curves could ntellgently substtute the use of pecewse cubc Bézer curves n descrbng Postscrpt fonts as well as the output mages of tracng algorthms. Explotng the procedure descrbed n Algorthm 1, very good results have been obtaned over a range of outlne mages. We show here the reconstructons we got by testng the algorthm on the data obtaned from the outlne of a Type1, cmr10 Postscrpt font (Fgures 4, 5), an hand-drawn sketch (Fgure 6) and a detal of a unversty logo (Fgure 7). As t appears, f you compare Fgures 4, 5, 6, 7 (rght) wth the correspondng outlnes of pecewse cubc Bézer curves n Fgures 4, 5, 6, 7 (left), you can hardly spot the dfferences between them. 5 CONCLUSIONS AND FUTURE WORK Stmulated by the observaton that a "good" local nterpolatory scheme does not seem to be presently avalable, we have proposed a novel subdvson scheme for nterpolatory curves whch, n contrast to exstng models, gves the possblty of generatng a bg varety of good qualty curves modfyng the tenson parameters assocated wth the edges of the ntal polylne. To our knowledge, ths s the frst nterpolatory subdvson scheme, easy to mplement and computatonally economcal, that ncludes many classcal propertes as well as several orgnal features consdered vtal or smply desrable n applcatons. For example, the proposed scheme can generate a lmt curve that can be locally modfed wth the help of shape control parameters. Namely, t s able to produce a locally controlled nterpolatory curve whch can ncorporate a varety of shape effects lke bumps and flat edges, and can mx them n an unrestrcted way, always allowng a soft transton between them. Addtonally, nstead of beng altered by the user for nteractve desgn purposes, the tenson parameters can be automatcally set n such a way the lmt curve generated can elegantly ft curves extremely used n geometrc modellng systems, such as conc secton arcs and pecewse cubc Bézer curves. As an applcaton, we have proposed our nterpolatory subdvson curves wth local shape control as an effcent and economcal representaton of the vector outlne of a scanned mage or a letter-form shape. Ths makes t possble to ntroduce subdvson technques n document descrpton languages as Postscrpt and encourages the CAGD communty to swng n curve-subdvson s favor. The authors are lookng, as a future work, to extend the curve subdvson scheme to surfaces. Ths could be qute useful for varous applcatons n dfferent felds of studes. 6 ACKNOWLEDGEMENTS Ths research was supported by MIUR-PRIN 004 and by Unversty of Bologna "Funds for selected research topcs". 7 REFERENCES [Bec05a] Beccar C., Cascola G., Roman L., A nonstatonary unform tenson controlled nterpolatng 4- pont scheme reproducng concs. Submtted to Comput. Aded Geom. Desgn, July 005, avalable at cascola/html/publcatons.html. [Dub86a] Dubuc S., Interpolaton through an teratve scheme. J. Math. Anal. Appl. 114 (1986),

8 Fgure 6: Outlne of Loxe and Zoot cartoon: 36 pecewse cubc Bézer curves cubc segments (left); reconstructon va nterpolatory subdvson curves wth local control - 36 sngle-pece subdvson curves, 4003 ponts, 3767 tenson parameters (rght). Fgure 7: Outlne of a detal of the Unversty of Bologna logo: 561 pecewse cubc Bézer curves cubc segments (left); reconstructon va nterpolatory subdvson curves wth local control snglepece subdvson curves, 740 ponts, 6679 tenson parameters (rght). [Dyn87a] Dyn N., Levn D., Gregory J.A., A 4-pont nterpolatory scheme for curve desgn. Comput. Aded Geom. Desgn 4 (1987), [Dyn95a] Dyn N., Levn D., Analyss of asymptotcally equvalent bnary subdvson schemes. Math. Anal. Appl. 193 (1995), [Gon01] Gonsor D., Neamtu N., Can subdvson be useful for geometrc modelng applcatons? Boeng Techncal Report [Mar05a] Marnov M., Dyn N., Levn D., Geometrcally controlled 4-pont nterpolatory schemes. Advances n Multresoluton for Geometrc Modellng, Dodgson N.A., Floater M.S. and Sabn M.A. (eds), Sprnger- Verlag 005, [Oht03a] Ohtake Y., Belyaev A., Sedel H.-P., Interpolatory subdvson curves va dffuson of normals. Proceedngs of the Computer Graphcs Internatonal (CGI 03) IEEE Computer Socety (003), 1-6. [Sel03a] Selnger P., Potrace: a polygon-based tracng algorthm, September 003, unpublshed, avalable at