Modelling of curves and surfaces in polar. and Cartesian coordinates. G.Casciola and S.Morigi. Department of Mathematics, University of Bologna, Italy

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1 Modellng of curves and surfaces n polar and Cartesan coordnates G.Cascola and S.Morg Department of Mathematcs, Unversty of Bologna, Italy Abstract A new class of splne curves n polar coordnates has been presented n (Sanchez-Reyes, 992) and ndependently consdered n (de Casteljau, 994). These are ratonal trgonometrc curves n Cartesan coordnates and can be represented as NURBS. From the relatonshp exstng wth the correspondent curves n Cartesan coordnates an alternatve way to derve some useful tools for modellng splnes n polar coordnates has been provded. In partcular an ad hoc algorthm of degree elevaton for splnes n polar coordnates s presented. On the bass of these results we propose a modellng system for NURBS curves and surfaces suppled wth a modellng envronment for splne curves and surfaces n polar, sphercal, and mxed polar-cartesan coordnates. Introducton Recently, n (Sanchez-Reyes, 992) a class of splne curves n polar coordnates was proposed. We refer to these curves as p-splnes. They have proved to be a generalzaton of those consdered n (Sanchez-Reyes, 990), whch we call p-bezer curves. Soon afterwards, classes of splne surfaces n cylndrcal, sphercal and tubular coordnates were proposed n (Sanchez-Reyes, 99, 994a, 994b). Ths research was supported by CNR-Italy, Contract n ct0

2 The p-splnes were ndependently consdered n (de Casteljau, 994), whch he called Focal splnes. These classes of curves and surfaces are nterestng because they allow for modellng and nterpolaton of 2D and 3D free forms n polar coordnates wth the same facltes as Cartesan splnes. Another mportant aspect to be consdered s the possblty of provdng a rapd response to the Pont Membershp Classcaton problem. In all of the cted papers Sanchez-Reyes emphaszes the fact that the p-splne curves and surfaces are pecewse ratonal Bezer n Cartesan coordnates and they are not ratonal splnes. Actually, ths last asserton s nether proved nor supported by any justcaton. In ths paper we wll not only show that a p-splne curve or surface s a NURBS, but we wll also provde the algorthm that leads to a representaton of these curves and surfaces as NURBS. In addton to knot nserton, knot removal and subdvson, another known result from Sanchez-Reyes' papers s the possblty of makng degree elevaton for p-splnes from degree n to degree kn, although an explanaton of how to acheve ths has not been gven. In (Cascola et al., 995) an algorthm for degree elevaton for p-bezer curves has been proposed. In ths paper we wll suggest how to use t for p-splnes, together wth tests on ecency and numercal stablty. These two results, n addton to others wth less theoretcal consequences but essental for practcal purposes, have convnced us of the usefulness of extendng our NURBS-based modellng system by supplyng t wth a modellng envronment for p-splne curves and surfaces n polar, sphercal, and mxed polar-cartesan coordnates n order to manage polar and sphercal models n the best way. Ths envronment makes use both of orgnal tools and of tools already developed for NURBS, and t must be consdered as an added potentalty to generate and model NURBS curves and surfaces. In partcular, at the moment, the polar, sphercal and mxed envronment allows: the modellng of p-splne curves by nterpolaton and control-pont modcaton; the modellng of tensor-product p-splne surfaces by nterpolaton of a mesh of ponts, by nterpolaton of -curves or -curves (sknnng), by revoluton of a curve, or by swngng; the possblty to model product surfaces (see Koparkar and Mudur, 984) n mxed coordnates. As s already known, the latter ncludes sweep and swung surfaces. 2

3 2 P-splne curves and surfaces A p-splne curve c(t) of degree n s dened as c(t) = (t) (t)! = P K+n M ;n (t) nt where (t) denotes the polar angle and (t) s the radus. Wthout loss of generalty, we consder t 2 [?; ]. The functons M ;n (t) satsfy the followng recurrence relaton: A and M ;n (t) = sn(t? t ) sn(t +n? t ) M ;n?(t) + sn(t +n+? t) sn(t +n+? t + ) M +;n?(t) () M ;0 (t) = ( f t t < t + 0 otherwse (2) on a non-decreasng knot sequence ft g K+2n+ where? = t 0 = = t n t K+n+ = = t K+2n+ = and the constrant t +n? t <, 8, s satsed. It s easy to recognze that the functons M ;n (t) are the normalzed trgonometrc B-splnes (see Lyche and Wnter, 979). Consderng a support reduced to a sngle nterval, these are dentcal wth Bernsten bass trgonometrc polynomals and span the space ( spanf; cos 2t; sn 2t; cos 4t; sn 4t; :::; cos nt; sn ntg n even T n = spanfcos t; sn t; cos 3t; sn 3t; :::; cos nt; sn ntg n odd wth t 2 [0; ). It s known (see Koch et al., 995) that the control curve of the trgonometrc splne P K+n M ;n (t) s gven by G(t) = g (t) = ; ; K + n t 2 h t ;? t ; t = X +n n t j j=+ where 3

4 g (t) = sn n(t? t?) + sn n(t? t)? sn n(t? t?) nterpolates the ponts t?;?, (t ; ). We note that the t s satsfy n(t?t?) < n vew of the knot constrants; these guarantee that g (t) can be dened everywhere and can nterpolate t?;? and (t ; ). The control curve of c(t) s dened as where F(t) = g (t) nt! = ; ; K + n g = (t) sn n(t?t? ) sn n(t?t? ) +sn n(t?t)? = = sn(?? ) sn(?? ) +sn(?)? havng set = nt. Every -th pece of the control curve F(t) s, n polar coordnates, a straght segment whch nterpolates the ponts? ;?, ;?. Thus, the F(t) s a control polygon, and the coecents? and the Grevlle radal drectons dene, n polar coordnates, the control ponts d = ;? of the p-splne c(t). The knot constrant mples that, n polar coordnates,?? < holds. Fgure llustrates an example of a p-splne curve and relatve control polygon (F(t)), and the assocated trgonometrc splne functon, together wth ts control curve (G(t)). P-splne curves enjoy propertes of local control, lnear precson, convex hull, and varaton dmnshng nherted from splnes n Cartesan coordnates. Moreover, p-splnes of degree 2 are conc sectons wth foc at the orgn of the coordnates. A tensor product p-splne surface s(u; v), n sphercal coordnates, of degree n n u and m n v, s dened as follows: 4

5 Fgure : Example of p-splne curve of degree n = 3 together wth ts control polygon (left) and assocated trgonometrc splne wth ts control curve (rght) - ft g = f0; 0; 0; 0; 0:75; :5; 2:25; 3; 3; 3; 3g, f ;? f(0; 0:3); (0:75; 0:3); (2:25; 0:22); (4:5; 0:3); (6:75; 0:22); (8:25; 0:3); (9; 0:3)g g = s(u; v) = 0 (u; v) (u) (v) 0 C A = P K+n P H+m j=0 j M ;n (u)m j;m (v) nu mv C A where [u; v] 2 [a; b] [c; d]. A product surface of () and d(u) wth scalng factor f(u) n polar- Cartesan mxed coordnates s gven by r(u; ) = 0 f(u) cos () + d (u) f(u) sn () + d 2 (u) d 3 (u) C A wth f(u) = P H+m f R ;m (u) d(u) = P H+m D R ;m (u) () = P K+n M ;n ( n ) 5

6 where f(u) and d(u) are NURBS, wth the R ;m ratonal B-splne functons, and () s a p-splne curve. In the next secton we wll show how all the curves and surfaces consdered n our splne-based modellng system can be represented as NURBS. 3 NURBS representaton of p-splnes It s known (Sanches-Reyes, 992) that the class of p-splne restrcted to a sngle segment (p-bezer curves or Focal Bezer) represents a subclass of ratonal Bezer curves n Cartesan coordnates. Therefore, we can state that p-splnes represent a subclass of NURBS. A rst approach to obtan a NURBS representaton of a p-splne curve has been suggested n (Sanchez-Reyes 992). Gven a p-splne curve over an arbtrary knot sequence, ths can be converted by subdvson nto a pecewse curve whose ndvdual peces are p-bezer curves, so that every p-bezer curve can be represented n terms of ratonal Bezer curves n Cartesan coordnates. In the alternatve approach proposed here, a non-pecewse Bezer representaton of a p-splne c(t) as a NURBS curve q(v) wll be provded. Let c(t) be the p-splne represented as a scalar functon ( n ) = P K+n M ;n ( n ) (3) where 2 [?n; n]. Then the correspondent curve of (3) n Cartesan coordnates wll be obtaned by a smple change of coordnates: Applyng the denttes ( n ) cos sn! cos = P K+n cos M ;n ( ) n sn = P K+n sn M ;n ( ) n (see Goodman and Lee, 984), relaton (4) assumes the followng trgonometrc ratonal form: (4) 6

7 P K+n cos M;n sn ( ) n P K+n M ;n ( ) (5) n In (Koch, 988) the mportant transformaton n : P n? > T n ; ( n f)(x) = cos n x f(tan x), was provded; more precsely, f p 2 P n on [tan ; tan ], then n p 2 T n on [; ] when? < < <. From ths asserton t follows 2 2 that a polynomal B-splne s proportonal to a trgonometrc B-splne. In partcular, the followng mportant relaton can easly be proved: M ;n (t) = Q +n j=+ cosn t cos t j N ;n ('(t)) (6) '(t) = tan t? 2 < t < 2 where the N ;n are the polynomal B-splne functons dened on the knot sequence f'(t )g. In vrtue of (6), relaton (5) becomes where q(v) = P K+n cos P K+n Q +n? cos t j N ;n (v) sn j=+ +n Q? cos t j N ;n (v) j=+ + tan t tan v = '(t) = 2 Thus, we can conclude that a p-splne n Cartesan coordnates has the followng NURBS representaton: (7) wth weghts q(v) = P K+n P w N ;n (v) P K+n w N ;n (v) v 2 [0; ] ; (8) w = +n Q j=+ cos t j (9) 7

8 Fgure 2: NURBS representaton of the p-splne curve shown n Fg.. n = 3, fv g = f0; 0; 0; 0; 0:46; 0:5; 0:53; ; ; ; g, fw g = f; 0:09668; 0:00933; 0:00066; 0:00933; 0:09668; g.! cos and control ponts P =? ; the N sn ;n (v) functons are dened over a knot sequence fv g obtaned applyng relaton (7) to the knots t. Note that the P are gven by the transformaton n Cartesan coordnates of the p-splne control ponts ( ;? ). In Fgure 2 the NURBS representaton of the p-splne curve llustrated n Fgure s shown. If 2, n order to satsfy the applcablty condtons of relaton (6), t wll be necessary to subdvde the p-splne curve nto pecewse p-splnes dened on ntervals whose sze s less than. Also, t should be noted that f the p-splne s dened on a generc nterval [a; b], the translaton, non-translaton, at the orgn of ths nterval nvolves derently parametrzed curves n Cartesan coordnates that can be converted nto each other by means of a sutable lnear ratonal reparametrzaton. 8

9 The exstence of the correspondng Cartesan ratonal splne q(v) of a gven p-splne c(t), allows us to assure that the pecewse ratonal Bezer curve, obtaned followng the approach suggested by Sanchez-Reyes, can be transformed nto q(v) by a sutable lnear ratonal reparametrzaton of each pece, followed by consecutve knot removal untl the partton fv g s acheved. 4 Tools for p-splnes Of the many tools that play an mportant role n a splne-based modellng system, we report knot nserton, subdvson, knot removal, and degree elevaton. Algorthms for knot nserton, subdvson, and knot removal for p- splnes can be obtaned from analogous algorthms for trgonometrc splnes, as can be easly deduced from the denton of c(t). Alternatve algorthms can be obtaned usng relaton (6) and the analogous algorthms for polynomal splnes. For example, the knot nserton algorthm for p-splnes may be schematzed through the followng steps: Let t` < ^t t`+ be the knot to be nserted.. Compute c = +n Q j=+ cos t j = `? n; ; ` ; 2. Insert knot '(^t) by means of the polynomal splne algorthm on c coef- cents to acheve ^c over the f ^v g knot partton; 3. Compute ^ = +n Q j=+ cos ^t j ^c = `? n; ; ` +. In fact, applyng (6) to c(t), we have M ;n (t) = cos n t P K+n c N ;n (v) P K+n executng knot nserton for polynomal splnes, (0) becomes (0) = cos n t P K+n+ ^c ^N;n (v) 9

10 a b c d Fgure 3: Degree elevaton steps; (a) orgnal cubc p-splne curve, (b) subdvson n 3 p-bezer curves, (c) degree elevaton of each p-bezer curve, (d) degree-elevated curve after the knot removal step; the degree s rased to 6. and applyng relaton (6) once agan, we obtan = P K+n+ ^ ^M;n (t) wth c, ^c and ^ as ndcated n.,2. and 3. Analogously, relaton (6) can be used n order to evaluate the p-splne c(t), referrng the evaluaton of a trgonometrc splne to a polynomal splne. It should be noted that these tps can mprove the ecency of a p-splne-based modellng system. Unlke the above-consdered tools, the algorthm for the degree elevaton of a p-splne s not achevable ether from the trgonometrc splne degree elevaton algorthm consdered n (see Alfeld et al., 995), or from the degree elevaton algorthm for polynomal splnes. In fact, the applcaton of such algorthms does not modfy the parametrc nterval sze, as results from the 0

11 denton of c(t). From ths the need emerges for an ad hoc algorthm to determne the degree elevated p-splne curve. 4. Degree elevaton for p-splnes From the expresson of a p-splne n terms of the Fourer bass, one can deduce that ths subset of curves s closed wth respect to degree elevaton from degree n to degree kn, for any natural value k. Assume, for example, n = 2; the p-splne can be represented, for each segment, n the Fourer h bass f; cos 2t; sn 2tg, for t 2 [?; ], and also f; cos 4s; sn 4sg s 2? ; 2 2. Therefore, t can be expressed by the Fourer bass of elevated degree kn = 4, f; cos 2s; sn 2s; cos 4s; sn 4sg. Followng the dea n (Pegl and Tller, 994) for polynomal splnes, we provde a degree elevaton technque for p-splnes that conssts n the followng steps:. decompose the p-splne nto pecewse p-bezer curves (subdvson); 2. apply degree elevaton to each p-bezer curve; 3. remove unnecessary knots untl the contnuty of the orgnal curve s guaranteed (knot removal). In order to realze step 2, the followng result s exploted (Cascola et al., 995). j=0 Degree elevaton formula for p-bezer curves P Let p(t) = n c j A j;n (t); t 2 [?; ] be a generc trgonometrc polynomal of degree n n the Bernsten trgonometrc bass, then p(t) = knx r=0 c r A r;kn (s); s = t=k; s 2? k ; k () where c r = n! sn n (2) kn r? n X j=0 c j X? j j j;r

12 and j = DY d = d odd (?) ddv2 k d d!j d! d +j d tan I+J 2 k j;r = mn(k(n?j)?i;r?j) X h=max(0;r?kj) k(n? j)? I D = max odd less than or equal to k, h kj? J r? J? h cos k(n?j)+r?2h ( 2 k )? j = f( ; 3 ; ::; D ; j ; j 3 ; ::; j D ) ; ; 3 ; ::; D ; j ; j 3 ; ::; j D 0 ; :: + D = n? j; j + j 3 + :: + j D = jg I = ::: + D D and J = j + 3j 3 + ::: + Dj D Although the formula may appear complex and ts mplementaton may not be mmedately comprehensble, n (Cascola et al., 995) useful detals and tps to produce an ecent and stable algorthm are provded. At rst sght, t s easy to realze that the crtcal pont of the formula conssts n the cycle n? j ; ths s due both to the dculty n determnng the ordered sequences ( ; 3 ; ::; D ; j ; j 3 ; ::; j D ), and to ther large quantty that makes the cycle expensve. For ths purpose, t should be ponted out that only ordered sequences of (k+)dv2 tems ; 3 ; ::; D satsfyng + 3 +::+ D = n?j wth j = 0; ; n needed to be determned, because the j ; j 3 ; ::; j D tems are the same. The ordered sequences n? j are obtaned by makng sutable combnatons of the ; 3 ; ::; D tems. Moreover, we can observe that the number of ordered sequences n? j for k = 2 s reduced to one, and, for small k and n, the number of ordered sequences remans lmted. Consderng that, generally, n a splne modellng system, the most frequently used maxmum degrees are 2,3 or 4, t s easy to see how ths formula can produce good results. The proposed algorthm provdes a preprocessng phase that performs the enttes that recur many tmes n the gven formula; n partcular, the coecents j are precomputed. Note that, beng j = n?j, the number of 2

13 k n n Table : Convergence of control ponts to the curve; each entry has to be multpled by 0?3. these coecents actually computed s reduced to half of the cardnalty of? j. Only one preprocessng phase s requred for the degree elevaton of all the p-bezer curves, and ths also contrbutes to the ecency of the degree elevaton algorthm for p-splnes. In Fgure 3, the three man algorthm steps are tested on an ntal p- splne curve of degree n = 3 wth 2 sngle nteror knots, to obtan a p-splne of degree kn = 6 wth 2 nteror knots, both havng a multeplcty of 4. Computatonal results The algorthm has been mplemented n Pascal (BORLAND 7.0), carred out n double precson (5-6 sgncant gures), and tested on a Pentum 90 PC. The test curves consdered, wthout loss of generalty, have been chosen wth 2 [0; ], the coecents? =, = 0; ; n, and randomly dstrbuted knots. Tests were performed on the followng aspects:. numercal stablty of the algorthm; evaluated by means of a convergency test on the control ponts of the degree elevated curve to the curve tself, and by means of a maxmum devaton test between the orgnal p-splne and the assocated degree-elevated p-splne; 3

14 k n n (a) k n n (b) k n n (2a) k n n (2b) k n n (3a) k n n (3b) Table 2: Executon tme (0?2 sec) results of degree elevaton ;(a) our mplementaton, (b) nterpolaton technque; (), (2) and (3) respectvely wth, 3 and 5 nternal knots. 4

15 Fgure 4: Degree elevaton results shown n the second column n Table 2a 2. executon tme of the algorthm; evaluated as a functon of the start degree n, the ncrement factor k, and the number L of nteror knots. Table summarzes convergency results of control ponts to the curve, for p-splne test curves of degrees n = ; 2; 3; 4 wth 2 nteror knots and ncreasng k. The values reported were evaluated by computng Numercal stablty was assessed by max jc( )?? j (2) ;;K+kn+L(kn?n) MAXERR := kc(t)? c(s)k D (3) on a unformly-spaced set of ponts, where c(s) denotes the degree-elevated curve, resultng n 0?6 MAXERR 0?5 (4) Ths reveals the accuracy of our results. In order to evaluate performance, the algorthm for the degree elevaton of p-splnes was compared wth the nterpolaton technque, the only means at our dsposal for degree-elevatng a p-splne. Table 2 reports a comparson of executon tmes requred by our algorthm and by the nterpolaton technque for,3 and 5 nteror knots, wth n < 5 and k < 6. 5

16 Fgure 5: Degree elevaton results as a functon of the number of nteror knots The results emphasze the performance of our algorthm when compared wth the nterpolaton technque; Fgures 4 and 5 provde a clearer understandng of these results. In Fgure 4 tmngs relatng to an ntal curve of degree n = 2 wth 3 nteror knots are reported for ncreasng k values. Whle the executon tmes of our algorthm were slghtly worse than the lnear growth rate, the nterpolaton has an exponental growth rate. The tests n Fgure 5 llustrate the executon tmes as functons of the number of nternal knots, whle the degree kn remans unchanged at value 9. From Fgure 5 we can observe that the performance of our algorthm for p-splnes takes full advantage of the preprocessng phase, makng the growth rate less than the lnear one. All of the tests consdered used p-splne curves wth sngle nteror knots. It s clear that our algorthm performs better, compared wth the nterpolaton technque, when the knot multeplcty s ncreased. 5 Modellng examples In ths secton we show some examples of surfaces obtaned usng our NURBSbased modellng system. Ths system s suppled wth a modellng envronment for p-splne curves and surfaces n polar, sphercal and mxed polar- Cartesan coordnates. 6

17 Fgure 6: p-splne prole curves n polar coordnates for a half face In the rst example we have modelled a face n polar-sphercal coordnates by the sknnng technque. In Fgure 6 the prole curves for half face are shown. These p-splne curves have derent degree 2 and 3, and derent knot partton. They are obtaned by nterpolaton n polar coordnates. Before applyng the sknnng technque n sphercal coordnates, n order to obtan a tensor product p- splne surface, all the prole curves have been maked compatble by usng our degree elevaton algorthm and knot nserton technque. In Fgure 7 the resultng modellng face s shown. The second example, n Fgure 8, shows a tensor product p-splne surface modelled by changng ts control ponts. Modellng ths knd of surfaces s complex n Cartesan coordnates but becomes easy n sphercal coordnates because s sucent to modfy the coecents accordng to a specc pattern. The last example shows a swngng surface as a partcular case of a product surface n mxed polar-cartesan coordnates. Fgure 9 shows the p-splne trajectory curve modelled n polar coordnates 7

18 Fgure 7: p-splne surface n sphercal coordnates obtaned by sknnng Fgure 8: p-splne surface n sphercal coordnates obtaned by modellng the control grd 8

19 Fgure 9: left: p-splne trajectory; rght: NURBS prole Fgure 0: swngng surface n mxed polar-cartesan coordnates 9

20 and the NURBS prole curve n Cartesan coordnates. The resultng surface s shown n Fgure 0. Once the p-splne surface has been modelled, t s possble to represent t as a NURBS surface, generalzng the result n secton 3. Ths turns out to be mportant both for havng the model n a standard NURBS form, and to explot for p-splnes the note technques for NURBS as, for example, the renderng algorthms. References Alfeld P.,Neamtu M., Schumaker L.L. (995), Crcular Bernsten-Bezer Polynomals, n: Daehlen M., Lyche T., Schumaker L.L. eds., Mathematcal Methods n CAGD. Cascola G., Lacchn M. and Morg S. (995), Degree elevaton for snglevalued curves n polar coordnates, submtted to CAGD. de Casteljau P. (994), Splnes Focales, n: Laurent P.J. et al.eds., Curves and Surfaces n Geometrc Desgn, Peters A.K. Wellesley, Goodman T.N.T. and Lee S.L. (984), B-Splnes on the crcle and trgonometrc B-Splnes, n: Sngh S.P. et al.eds., Approxmaton theory and splne functon, Redel D. Publshng Company, Koch P.E. (988), Multvarate Trgonometrc B-splnes, Journal of Approxmaton Theory, 54, Koch P.E., Lyche T., Neamtu M. and Schumaker L. (995), Control curves and knot nserton for trgonometrc splnes, Advances n Computatonal Mathematcs, 3, Koparkar P.A. and Mudur S.P. (984), Computatonal Technques for Processng Parametrc Surfaces, Computer Vson, Graphcs, and Image Processng, 28, Lyche T. and Wnther R. (979), A Stable Recurrence Relaton for Trgonometrc B-Splnes, Journal of Approxmaton theory, 25, Pegl L. and Tller W. (994), Software-engneerng approach to degree elevaton of B-Splne curves, Computer Aded Desgn, 26, Sanchez-Reyes J. (990), Sngle-valued curves n polar coordnates, Computer Aded Desgn, 22, Sanchez-Reyes J. (99), Sngle-valued surfaces n cylndrcal coordnates, Computer Aded Desgn, 23, Sanchez-Reyes J. (992), Sngle-valued splne curves n polar coordnates, Computer Aded Desgn, 24,

21 Sanchez-Reyes J. (994a), Sngle-valued surfaces n sphercal coordnates, Computer Aded Geometrc Desgn,, Sanchez-Reyes J. (994b), Sngle-valued tubular patches, Computer Aded Geometrc Desgn,,

Spline curves. in polar and Cartesian coordinates. Giulio Casciola and Serena Morigi. Department of Mathematics, University of Bologna, Italy

Spline curves. in polar and Cartesian coordinates. Giulio Casciola and Serena Morigi. Department of Mathematics, University of Bologna, Italy Spline curves in polar and Cartesian coordinates Giulio Casciola and Serena Morigi Department of Mathematics, University of Bologna, Italy Abstract. A new class of spline curves in polar coordinates has