A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics

Transcription

1 A non-saionary uniform ension conrolled inerpolaing 4-poin scheme reproducing conics C. Beccari a, G. Casciola b, L. Romani b, a Deparmen of Pure and Applied Mahemaics, Universiy of Padova, Via G. Belzoni 7, 353 Padova, Ialy b Deparmen of Mahemaics, Universiy of Bologna, P.zza di Pora San Donao 5, 407 Bologna, Ialy Absrac In his paper we propose a non-saionary C -coninuous inerpolaing 4-poin scheme which provides users wih a single ension parameer ha can be eiher arbirarily increased, o ighen he limi curve owards he piecewise linear inerpolan beween he daa poins, or appropriaely chosen in order o represen elemens of he linear spaces spanned respecively by he funcions {, x, x, x 3 }, {, x, e sx, e sx } and {, x, e isx, e isx }. As a consequence, for special values of he ension parameer, such a scheme will be capable of reproducing all conic secions exacly. Exploiing he reproducion propery of he scheme, we derive an algorihm ha auomaically provides he iniial ension parameer required o exacly reproduce a curve belonging o one of he previously menioned spaces, whenever he iniial daa are uniformly sampled on i. The performance of he scheme is illusraed by a number of examples ha show he wide variey of effecs we can achieve in correspondence of differen ension values. Key words: Subdivision; Inerpolaion; Curves; Localiy; Tension conrol; Conics reproducion Corresponding auhor. addresses: beccari@dm.unibo.i (C. Beccari), casciola@dm.unibo.i (G. Casciola), romani@dm.unibo.i (L. Romani).

2 Inroducion Subdivision is an efficien mehod for generaing smooh curves and surfaces in Compuer Aided Geomeric Design (CAGD). A univariae subdivision process defines a curve as he limi of a sequence of refinemens performed on an iniial polyline. If he poins generaed a each refinemen level are reained a all successive levels, he scheme is said o be inerpolaory (Dyn, 00). The imporan schemes for applicaions should allow o conrol he shape of he limi curve and be capable of reproducing families of curves widely used in Compuer Graphics, such as conic secions and polynomials. Alhough a wide variey of schemes has been proposed in he lieraure, he difficuly of combining he above requiremens prevened he diffusion of inerpolaory schemes in applicaions. Indeed, he schemes proposed up o now eiher possess a ension parameer o conrol he ighness of he limi curve (Dyn e al., 987; Dyn e al., 005) or are able o reproduce circles (Zhang, 996; Ivrissimzis e al., 00; Jena e al., 003) and generae conic secions (Dyn e al., 003). However, none of hem possesses an inuiive shape parameer ha, gradually increased, allows o increase he ighness of he limi curve and for special values associaed wih he iniial daa, allows o represen all conic secions exacly. The only one which possesses a ension parameer ha, appropriaely chosen, allows o reproduce circles as well, is he non-saionary scheme presened in Morin e al. (00). Anyway, his is no inerpolaory. Aim of his work is herefore o generae an inerpolaory subdivision scheme wih a ension parameer, ha is capable of reproducing circles and all oher conic secions exacly whenever such a parameer has been chosen correcly. The paper is srucured as follows. In Secion we derive a C -coninuous subdivision scheme which allows us o represen cubic polynomials as well as a cerain class of hyperbolic and rigonomeric funcions, relaed o he definiion of conic secions. Nex, in Secion 3, we show is special propery of conics reproducion and we derive an algorihm ha auomaically provides he iniial ension parameer required o exacly reproduce a curve belonging o one of he previously menioned families, whenever he iniial daa are uniformly sampled on i. In Secion 4 we propose an endpoin rule for generaing open curves wih he same regulariy. Finally in Secion 5 we demonsrae he role of he ension parameer by a few examples. Definiion of he scheme In his secion we are going o define an inerpolaing 4-poin scheme ha capures hree differen curve schemes which are capable of represening elemens in he class of cubic polynomials, hyperbolic funcions and rigonomeric funcions. For he sake of conciseness, in he remainder of his paper we will indicae wih V 0, V s and V is he spaces spanned respecively by he funcions {, x, x, x 3 }, {, x, e sx, e sx } and {, x, e isx, e isx }.

3 Observe ha, depending on he value of (where is eiher a posiive real or imaginary consan), he soluions of he differenial equaion D 4 D = 0 are linear combinaions of he funcions in V 0 (whenever = 0), V s (whenever = s, s > 0) or V is (whenever = is, s > 0). Hence, due o he ideniies cosh(sx) = (esx + e sx ), sinh(sx) = (esx e sx ), cos(sx) = (eisx + e isx ), sin(sx) = i (eisx e isx ), he common inserion rule which unifies he hree cases will be obained by inerpolaion wih a funcion from he linear space spanned by {, x, e x, e x }. In paricular, whenever = 0 such an inserion rule will reproduce cubic polynomials, whenever = s > 0 i will reproduce hyperbolic funcions and whenever = is, s > 0, i will reproduce rigonomeric funcions. In his way, inerpolaing he daa ( k h, p k j+h ), h =, 0,,, by a funcion of he form f(x) = a 0 + a x + a e x + a 3 e x, () we ge he following sysem of equaions f( ) = p k k j f(0) = p k j f( k ) = p k j+ f( k ) = p k j+ from which i follows p k j = a 0 a + a k e k + a 3 e k p k j = a 0 + a + a 3 p k j+ = a 0 + a k + a e k + a 3 e k p k j+ = a 0 + a k + a e k + a 3 e k. () Then, solving () wih respec o a 0, a, a, a 3, we ge for any 0 and, whenever = is, for any s no a muliple of π, he following expression for he coefficiens in (): a 0 = (e k + e k ) p k j pk j pk j+ e k + e k 3

4 a = k a = a 3 = (e ) k + e k + (p k j+ pk j ) + pk j pk j+ (e k + e k e k + e k ) ( p k j + pk j+ pk j+ (e ) + (e ) ( k e k e ) k + e k k ) (p k j p k j + pk j+ ) (e ) k + e k (p k j pk j+ + pk j+ (e ) k ) (p k j p k j + pk j+ ) (e ) ( k e k e ) k + e. k To ge he inserion rule now, we only need o compue he value of he inerpolaing funcion f(x) a he grid poin, defining he new poin p k as a linear k+ j+ combinaion of he four consecuive poins p k j, pk j, pk j+, pk j+ in he las se: ( ) f k+ = + (e k+ + e ) ( k+ e k+ + e ) (p k j + p k j+) k+ + (e k+ + e ) ( (p k+ e k+ + e k+ + ) k j + p k j+) p k j+. In his way we will define he se of poins a he (k + )-h level of refinemen, as: p k+ j = p k j p k+ j+ = pk j+ = ( + wk+ ) (p k j + p k j+) w k+ (p k j + p k j+) (3) where w k+ = (e k+ + e ) ( ). (4) k+ e k+ + e k+ + Remark The subdivision scheme defined in (3)-(4) urns ou o be a special case of he general exponenial reproducing schemes proposed by Dyn e al. (003). Noe ha when = 0 he weigh w k+ is well-defined and urns ou o be 6, in such a way ha he subdivision rules (3) reduce o he 4-poin Dubuc-Deslauriers inserion rules (Dubuc, 986; Deslauriers and Dubuc, 989), which reproduce cubic polynomials. This is due o he fac ha, for = 0, he soluions of he differenial equaion D 4 D = 0 are exacly cubic polynomials. 4

5 Proposiion Le k = define k, where whenever = is, we assume s (0, π), and v k = (e k + e k ). (5) Then for any k 0 he parameers v k and v k+ defined as in (5) saisfy he recurrence v k+ + v = k. (6) PROOF. Equaion (6) follows from he fac ha +v k = ) + e k + e k = (e k + e k = ( e k+ + e ) k+ = v k+. Remark 3 Noe ha he recurrence relaion defined in (6) saisfies he propery lim k + vk =. We now express he weighs in (3) in erms of v k+ so ha, given any arbirary ension value v 0 in (, + ), and exploiing (6) o updae i a each refinemen level, we can generae an inerpolaing limi curve whose shape is easily conrolled by he choice of v 0. Definiion 4 Given a se of conrol poins P 0 = {p 0 j j Z} a refinemen level 0 and an arbirary iniial ension parameer v 0 (, + ), we define a subdivision scheme ha generaes a new se of conrol poins P k+ = {p k+ j j Z} k 0 a he (k + )-h level of refinemen, by he subdivision rules (3) wih weigh w k+ = 8v k+ ( + v k+, k 0, (7) ) where for any k 0 he sequence v k+ in (7) is recursively defined hrough equaion (6). Noe ha for any choice of he iniial ension value v 0 in he range (, + ), he recurrence in (6) is always well-defined and v k+ > 0 for any k 0. Furhermore lim k + w k = 6. Remark 5 The subdivision scheme defined in (3)-(7) generaes C -coninuous limi curves for any choice of he iniial ension parameer v 0 (, + ). This follows from he convergence analysis resuls in Dyn and Levin (995), as explained for he general exponenials reproducing schemes proposed in Dyn e al. (003). 5

6 3 Reproducion of conic secions In he firs par of his secion we are going o show ha, choosing correcly he iniial ension parameer v 0, he subdivision scheme defined in (3)-(7) allows o reproduce he classes of cubic polynomials, hyperbolic funcions and rigonomeric funcions idenified respecively by he spaces V 0, V s and V is. Successively, in subsecion 3., we will illusrae a procedure ha allows o auomaically compue he special ension value required o reproduce curves from he above hree classes whenever he iniial poins are uniformly sampled on hem. 3. The iniial ension parameers for conics reproducion Observe ha, defining v 0 as in (5), ha is v 0 = (e + e ), and assuming = 0, = s (wih s > 0) and = is (wih s (0, π)), we ge respecively: v 0 =, hence v k = and w k = 6 for all k ; v 0 = cosh(s) >, hence v k = cosh( s ) and w k = ( ) ( ) k s 6 cosh cosh k s for all k ; k+ v 0 = cos(s) (, ), hence v k = cos( s ) and w k = ( ) ( ) k s 6 cos cos k s for all k (hus he scheme in (3)-(7) coincides wih he inerpolaing 4-poin k+ scheme on he circle presened in Ivrissimzis e al. (00)). In his way, saring from a se of poins uniformly sampled on a funcion in V 0, V s or V is he value of he parameer v 0 idenifies he space o which he limi funcion generaed by he scheme belongs. More precisely, if v 0 =, hen he limi curve belongs o V 0, namely he linear space spanned by cubic polynomials; if v 0 > hen v 0 = cosh(s) for some s R +, hence he limi curve belongs o V s wih s = acosh(v 0 ); if v 0 (, ) hen v 0 = cos(s) for some s (0, π), hence he limi curve belongs o V is wih s = acos(v 0 ). As a consequence, he following resul holds. Proposiion 6 Choosing he iniial ension parameer v 0 = cosh(su), s, u > 0, he subdivision scheme defined in (3)-(7) reproduces exacly he hyperbolic funcions f(x) = cosh(sx) and f(x) = sinh(sx) whenever he given daa poins (ju, p 0 j ), j Z lie on such funcions. Analogously, choosing he iniial ension parameer v 0 = cos(su), su (0, π), he subdivision scheme defined in (3)-(7) reproduces exacly he rigonomeric funcions f(x) = cos(sx) and f(x) = sin(sx) whenever he given daa poins (ju, p 0 j ), j Z 6

7 lie on such funcions. PROOF. The resul above follows from he consrucion of he scheme. Corollary 7 By aking as iniial daa he poins p 0 j = (a cosh(ju), b sinh(ju)), j Z, u > 0, equidisan in he parameer u on he parameric represenaion of he hyperbola, and choosing he iniial ension parameer v 0 = cosh(u), he resuling limi curve is he hyperbola iself (see Fig. where a = b = u = ). Analogously, if we ake as iniial daa four poins p 0 j = (a cos(ju), b sin(ju)), j Z, u (0, π), equidisan in he parameer u on he parameric represenaion of he ellipse wih cener 0 and radii a, b, by choosing he iniial ension parameer v 0 = cos(u), he resuling limi curve is exacly he ellipse iself (see Fig. where a = 4, b =, u = π ). In paricular, when a = b, he resuling limi curve is exacly he circle of radius a (see Fig.5 where a = b =, u = π ). Fig.. Reproducion of hyperbolic and rigonomeric funcions: f(x) = cosh(x), f(x) = sinh(x), f(x) = cos(x), f(x) = sin(x). Fig.. Reproducion of conic secions: hyperbola, ellipse, parabola. Remark 8 Figs. - have been obained by exending uniformly on each side he open conrol polygon P 0 = {p 0 j j = 0,..., 4} by wo exra segmens whose endpoins lie on he curve. Whenever wo auxiliary poins have been defined for each endpoin, we can deal wih open polygons leaving he subdivision rules (3)-(7) unmodified. In his way he open curve generaed in he limi hrough (3)-(7) wih j =,..., 6, rivially urns ou o have p 0 0 as firs endpoin and p0 4 as he las one (see Secion 4). 3. Auomaical compuaion of he iniial ension parameer In his subsecion we are going o derive an algorihm ha, saring by a sufficien number of equally-spaced daa, is capable o auomaically compue he iniial en- 7

8 sion parameer v 0 such ha, if he iniial poins are sampled from a curve belonging o one of he spaces V 0, V s or V is, by applying he scheme defined by (3)-(7) wih he so compued ension, we will be able o reproduce he curve from which hose poins are sampled. The following algorihm provides a brief skech of he procedure one could exploi in order o compue he value of he ension parameer ha should be used o reproduce a curve belonging o one of he above hree spaces, wheher he iniial poins belong o i. Algorihm in pseudo-code: [] Consider an iniial se of equally-spaced poins {ju, p j } j=0,...,n (n 4) for some posiive u [] for j =,..., n [.] consider he quadruple of even-indexed poins p j, p j, p j+, p j+4 [.] deermine w j by solving componenwise he equaion given by p j+ = p j + p j+ + w j ( pj + p j+ p ) j + p j+4 (8) [.3] if (8) has no soluion (w j,x w j,y ) sop: he iniial se of poins belongs o none of he specified classes else sore he value of w j [3] compare he values of w j obained for all j in he previous sep: if w j = w, j =,..., n [3.] compue he ension parameer v 0 w(w+) = 4w else he iniial se of poins belongs o none of he specified curve ypes The proposed procedure does no require o know a priori wheher he iniial poins lie on a curve of a prescribed space. Moreover, by sep 3, i is clear ha, if i does no sop before, his mehod can come successfully o an end only when all he coefficiens w j sored a sep.3 have he same value. This is always rue if he iniial poins are sampled from a curve eiher in V 0, V s or V is, and, if his is he case, he algorihm allows o deermine he ension v 0 necessary o reproduce he given curve. In paricular, when he iniial daa belong eiher o V s or V is, such a ension value uniquely idenifies he corresponding space hrough he parameer s respecively. On he oher hand, by applying he proposed algorihm o poins ha belong o a curve of none of hese spaces, he procedure could evenually yield a value of ension, even so, by applying he proposed scheme wih such a ension, i is no defined as s = acosh(v0 ) u or s = acos(v0 ) u 8

9 possible o predic anyhing abou wha kind of curve we will ge. Remark 9 Taking a look a sep 3. i is clear ha, in order o possess all he iniial daa necessary for he compuaion described a his sage of he procedure, we need o produce a leas wo weighs o compare. To his aim, we need o sar from a minimal number of nine iniial poins. This is due o he fac ha he described algorihm canno be applied o compue he poins p and p n, since we do no possess a well defined wo-neighborhood around hese poins. 4 A subdivision rule for curve endpoins In case of open curves, rules (3)-(7) can be applied only on he inerior of he curve, while for he endpoins we should include an alernaive rule. Since he wo endpoins can be reaed analogously, i will be sufficien o address our aenion only on one side. To his aim we observe ha, if we define jus wo auxiliary poins p 0, p0 in he coarses polygon P 0 = {p 0 j j = 0,..., n}, each new poin pk+ j+ has a welldefined -neighborhood and he open curve generaed in he limi hrough (3)-(7) wih j =,..., n rivially urns ou o have p 0 0 as firs endpoin and o be C - coninuous. However, he exension of his sraegy o surface subdivision, implies he definiion of wo rings of poins around he boundary conrol ne. Thus, o avoid so many compuaions when subdividing he firs edge p k 0, pk, we propose here a special rule for compuing he poin p k+ independenly of he wo auxiliary poins p 0, p0 ; as said above, o work ou he endpoin rule for he las edge, i will be sufficien o proceed analogously. Le p k 0, pk be he firs edge of he non-refined polygon P k = {p k j j = 0,..., k n}. Once defined an auxiliary poin p k, we can compue he poin pk+ hrough (3)-(7) wih j = 0, applying subdivision o he subpolygon p k, pk 0, pk, pk. We choose here he following exrapolaory rule: p k = p k 0 p k (9) since he hree curve schemes (corresponding o he cubic, he hyperbolic and he rigonomeric cases) are all capable of represening linear funcions. In his way he addiional refinemen rule for he endpoin can be expressed as he following saionary linear combinaion of poins from he non-exrapolaed open polygon p k 0, p k, pk : ( ) ( ) p k+ = wk+ p k wk+ p k w k+ p k. (0) Proposiion 0 Rule (0) does no affec he convergence of he original scheme o a coninuously differeniable limi. 9

10 PROOF. I is sufficien o show ha, aken p 0 = p0 0 p0 and p0 = p0 0 p0, and refining he polygon P 0 hrough (3)-(7), afer k rounds of subdivision he expression of he poin p k urns ou o coincide wih (9). 5 Applicaions and examples The following examples show open and closed curves which pass hrough a se of given poins. The conrol polygons (corresponding o he piecewise linear curve beween he given poins) are drawn by a dashed line, and he smooh curves obained by our algorihm by a full line. Fig. 3. Increasing he ighness (ension) of an open curve. Figure 3, depicing he open limi curves obained wih v 0 = 0.4,, 000, demonsraes he increase in he ighness (ension) of he curve wih he increase in v 0. Fig. 4. Increasing he ighness (ension) of a closed curve. Analogously, Figure 4, depicing he closed limi curves obained wih v 0 = 0.5, 0,, 5, 50, 500, demonsraes he increase in he ighness (ension) of he curve wih 0

11 he increase in v 0. The following figures show he effec of he ension parameer v 0 when our algorihm is applied on a regular N-sided conrol polygon inscribed in he uni circle. Fig. 5. Inerpolaion of he verices of a square wih he uniform ension conrolled inerpolaing 4-poin scheme defined by he following values of he parameer v 0 : -0.95, -0.75, -0.5, 0,, 5, 5, 500. While choosing v 0 < he parameer acs as a looseness and, smaller i is, looser he limi curve is, for high values of he ension parameer v 0, he limi curve ends o shrink o he iniial conrol polygon. In addiion, whenever we choose v 0 = cos( π N ), in he limi we obain exacly he uni circle (see Figs. 5, 6). 6 Conclusions and Fuure Work This paper describes a simple and efficien non-saionary subdivision scheme for curve inerpolaion depending on a single ension parameer, ha is capable of reproducing conic secions exacly whenever such a parameer is chosen correcly. The subdivision algorihm (3)-(7) is acually an inserion algorihm since all he poins a sage k are carried over o sage k + and new poins are insered in beween he old ones. Evidenly, he resuling limi curve inerpolaes he iniial poins. The local naure of he scheme, he possibiliy of reproducing cubic polynomials as well as cerain classes of hyperbolic and rigonomeric funcions, and he conrol of he ension by he associaed parameer, are imporan feaures for curve design. The algorihm proposed here is unique in combining hese five ingrediens: subdivision, localiy, inerpolaion, global ension conrol, reproducion of conic secions.

12 Fig. 6. Inerpolaion of he verices of a regular penagon wih he uniform ension conrolled inerpolaing 4-poin scheme defined by he following values of he parameer v 0 : -0.5, -0.5, 0, cos( π 5 ),, 5, 5, 500. An ineresing generalizaion of his proposal could include he possibiliy of working wih a differen ension parameer v 0 for each segmen of he iniial polygon P 0. In his way, since during each subdivision sep each segmen is spli ino wo new segmens, hese wo will inheri a new ension via equaion (6). The resuling subdivision scheme will allow herefore differen ensions on disinc curve segmens. The curve scheme proposed can also be naurally exended o ensor-produc surfaces. Nex sep will be herefore generalizing he univariae scheme o a surface scheme over arbirary quadrilaeral meshes. Acknowledgemens This research was suppored by MIUR-PRIN 004 and by Universiy of Bologna Funds for seleced research opics. Many hanks go o he anonymous reviewers for heir helpful commens. The auhors are also graeful o Nira Dyn for her precious suggesions. References Deslauriers, G., Dubuc, S., 989. Symmeric ieraive inerpolaion processes. Consr. Approx. 5, Dubuc, S., 986. Inerpolaion hrough an ieraive scheme. J. Mah. Anal. Appl. 4, Dyn N., Levin D., Gregory J.A., 987. A 4-poin inerpolaory subdivision scheme

13 for curve design. Compuer Aided Geomeric Design 4, Dyn, N., Levin, D., 995. Analysis of asympoically equivalen binary subdivision schemes. J. Mah. Anal. Appl. 93, Dyn, N., 00. Inerpolaory subdivision schemes. In: Iske, A., Quak, E., Floaer, M.S. (Eds.), Tuorials on Muliresoluion in Geomeric Modelling. Springer-Verlag, Dyn, N., Levin, D., Luzzao, A., 003. Exponenials Reproducing Subdivision Schemes. Found. Compu. Mah. 3, Dyn, N., Floaer, M.S., Hormann, K., 005. A C Four-Poin Subdivision Scheme wih Fourh Order Accuracy and is Exensions. In: Dæhlen, M., Mørken, K., Schumaker, L.L. (Eds.), Mahemaical Mehods for Curves and Surfaces: Tromsø 004. Nashboro Press, Ivrissimzis, I.P., Dodgson, N.A., Hassan, M.F., Sabin, M.A., 00. On he geomery of recursive subdivision. Inern. J. Shape Modeling 8(), 3-4. Jena, M.K., Shunmugaraj, P., Das, P.C., 003. A non-saionary subdivision scheme for curve inerpolaion. Anziam J. 44(E), Morin, G., Warren, J., Weimer, H., 00. A subdivision scheme for surfaces of revoluion. Compuer Aided Geomeric Design 8, Zhang, J., 996. C-curves: an exension of cubic curves. Compuer Aided Geomeric Design 3,