Point-based methods for estimating the length of a parametric curve

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1 Journl of Computtonl nd Appled Mthemtcs 196 (006) Pont-bsed methods for estmtng the length of prmetrc curve Mchel S. Floter, Atgerr F. Rsmussen CMA, Unversty of Oslo, P.O. Box 1053 Blndern, 0316 Oslo, Norwy Receved 7 Jnury 005; receved n revsed form 4 October 005 Abstrct Ths pper studes generl method for estmtng the length of prmetrc curve usng only smples of ponts. We show tht by mkng specl choce of ponts, nmely the Guss Lobtto nodes, we get hgher orders of pproxmton, smlr to the behvour of Guss qudrture, nd we derve some explct exmples. 005 Elsever B.V. All rghts reserved. Keywords: Curve; Arc length; Polynoml nterpolton 1. Introducton Computng the rc length of prmetrc curve s bsc problem n geometrc modellng nd computer grphcs, nd hs been treted n vrous wys. In [11], Guenter nd Prent use numercl ntegrton on the dervtve of the curve. In [16], Vncent nd Forsey derve method bsed entrely on pont evlutons. Grvesen hs derved method specfclly for Bézer curves [10]. The estmton of rc length s n mportnt ssue n [13,17,18], where pproxmte rc length prmetrztons were sought for splne curves. Ths s necessry, snce prt from trvl cses, polynoml curves never hve unt speed [6]. The rtcle [] trets the ssue of reprmetrzng NURBS curves so tht the resultng curve prmetrzton s close to rc length. The rtcles [3,4] del wth optml,.e., s close to rc length s possble, rtonl reprmetrztons of polynoml curves. In [15], the uthors clculte pproxmte rc length prmetrztons for generl prmetrc curves. Recently, results hve been obtned on pproxmtng the length of curve, gven only s sequence of ponts (wthout prmeter vlues), usng polynomls nd splnes [7,8]. Suppose f :[α, β] R d, d s regulr prmetrc curve, by whch we men contnuously dfferentble functon such tht f (t) = 0 for ll t [α, β], nd denotes the Eucldn norm n R d. Then ts rc length (see [14, Secton 9]) s β L(f) = f (t) dt. (1) α Snce L(f) s smply the ntegrl of the speed functon f, nturl pproch s smply to pply to f some stndrd composte qudrture rule: we splt the prmeter ntervl [α, β] nto smll peces, pply qudrture rule to Correspondng uthor. Tel.: ; fx: E-ml ddresses: (M.S. Floter), (A.F. Rsmussen) /$ - see front mtter 005 Elsever B.V. All rghts reserved. do: /j.cm

2 M.S. Floter, A.F. Rsmussen / Journl of Computtonl nd Appled Mthemtcs 196 (006) f n ech pece, nd dd up the contrbutons. If [,b] s one such pece, wth α <b β, then typcl rule hs the form L(f [,b] ) = f (t) dt w f (q ), () for some qudrture nodes q 0 <q 1 < <q n b, (3) nd weghts w 0,w 1,...,w n. Guenter nd Prent [11] pply such method dptvely. Ths method, however, hs the drwbck tht t nvolves dervtves of f, whch mght be more tme-consumng to evlute thn ponts of f, or mght smply not be vlble. One lterntve s the chord length rule (16), but t only hs second order ccurcy (s wll be shown n 4.1). Ths motvted Vncent nd Forsey [16] to fnd hgher order method usng only pont evlutons (18). In ths pper, we nvestgte the followng much more generl pont-bsed method, whch turns out to nclude these two methods s specl cses. We cn frst nterpolte f wth polynoml p n :[,b] R d, of degree n, t some ponts t 0 <t 1 < <t n b, for some n 1,.e., p n (t ) = f(t ) for = 0, 1,...,n, gvng the pproxmton L(f [,b] ) L(p n [,b] ). (4) We cn then estmte the length of p n by qudrture, gvng the estmte L(p n [,b] ) w j p n (q j ), (5) nd by expressng p n n the Lgrnge form p n (t) = L (t)f(t ), L (t) = n,j = t t j t t j, we get the pont-bsed rule L(f [,b] ) w j L (q j )f(t ). (6) In vew of the defnton of the length L(f [,b] ) n (), t s resonble to expect tht the error n (4) wll be smll due to the well-known fct tht p n s good pproxmton to f when h := b s smll. However, we hve not seen ths method explctly referred to n the lterture, nor re we wre of ny error nlyss. The mn contrbuton of ths pper s to offer thorough nlyss of the pproxmton order of the method, n terms of h, whch depends on the ponts t, nd the qudrture nodes nd weghts q j nd w j s well s the smoothness of f. One result of our nlyss s tht the nterpolton ponts t cn be chosen to mxmze the pproxmton order, nlogously to the use of Guss Legendre ponts for numercl ntegrton.. Error of the dervtve-bsed method For the ske of comprson, we strt wth comment bout the pproxmton order of the dervtve-bsed method (). If the qudrture rule used n () hs degree of precson r then the error wll be of order O(h r+ ) provded the (r + 1)th dervtve of F := f s bounded [1].

3 514 M.S. Floter, A.F. Rsmussen / Journl of Computtonl nd Appled Mthemtcs 196 (006) 51 5 Lemm 1. If f C r+ [α, β], nd f s regulr, then ll the dervtves F, F,...,F (r+1) re bounded n [α, β]. Proof. Let k {1,...,r + 1}. By Lebnz rule, nd so k 1 ( ) FF (k) k + F () F (k ) = (F ) (k) = (f f ) (k) = =1 F (k) 1 F ( k ( ) k f (+1) f (k +1) + k 1 =1 k ( ) k f (+1) f (k +1), ( ) ) k F () F (k ). (7) Now snce f s regulr on the closed ntervl [α, β], F ttns strctly postve mnmum ε > 0. Further, by ssumpton, ll the dervtves f,...,f k+1 re bounded. Therefore, ssumng by nducton tht ll the lower dervtves F,...,F (k 1) re bounded, we see tht F (k) s lso bounded. Ths leds to the pproxmton order of the dervtve-bsed method. Theorem 1. Suppose f C r+ [α, β], f s regulr, nd tht rule () hs degree of precson r. Then L(f [,b] ) w j f (q j ) =O(h r+ ) s h 0. For exmple snce the mdpont rule hs degree of precson r = 1, we get L(f [,b] ) h f (q 0 ) =O(h 3 ), (8) where q 0 = ( + b)/, provded f C 3 [α, β]. Snce Smpson s rule hs degree of precson r = 3, we fnd L(f [,b] ) h( f () +4 f (q 1 ) + f (b) )/6 = O(h 5 ), where q 1 = ( + b)/, provded f C 5 [α, β]. If we tke the q 0,...,q m to be the Guss nodes of order m, then the rule hs degree of precson m + 1 nd so provded f C m+3 [α, β],weget L(f [,b] ) w j f (q j ) =O(h m+3 ). 3. Error of the pont-bsed method There re two contrbutons to the error of the pont-bsed method, nmely the errors n the nterpolton prt (4) nd the qudrture prt (5). We wll tret them both, strtng wth the qudrture error (5). Lettng f nd p n, be the d components of the vector-vlued f nd p n, we recll clsscl result of polynoml nterpolton due to [1, Secton 6.5, p. 90]: f (k) (t) p (k) mx n, (t) hn+1 k s [,b] f (n+1) (s). (9) (n + 1 k)! Ths equton does not hold for vector-vlued functons, but we cn stll use t to derve some error bounds: f (k) (t) p (k) mx n, (t) hn+1 k s [,b] f (n+1) (s). (n + 1 k)!

4 Usng the notton M.S. Floter, A.F. Rsmussen / Journl of Computtonl nd Appled Mthemtcs 196 (006) f (n+1) [,b] := mx s [,b] f(n+1) (s), we explot the fct tht the rght-hnd sde bove does not depend on the component to wrte f (k) p (k) n [,b] C k h n+1 k f (n+1) [α,β], k = 0, 1,...,n, (10) where C k = d/(n + 1 k)!. Lemm. If f C n+1 [α, β] nd f s regulr, then ll dervtves of the functon p n re bounded ndependently of h for smll enough h. Proof. We wll prove ths by showng tht p n s regulr for suffcently smll h, then pply Lemm 1. By the trngle nequlty p n (t) f (t) f (t) p n (t) for ll t. Usng Eq. (10) n the cse k = 1 we then see tht p n (t) f (t) f p n [,b] f (t) C 1 h n f (n+1) [α,β]. Thus, snce f s bounded wy from zero, so wll p n be for suffcently smll h. Then p n s regulr. Snce p n s polynoml, t s n C r+ for ll r nd we cn pply Lemm 1 to show tht ll dervtves of p n re bounded. The pproxmton order of the qudrture prt of the pont-bsed method now mmedtely follows, nlogously to Theorem 1. Provded f C n+1 [α, β], we cn mke the order of ths prt of the error s hgh s we lke smply by usng qudrture rule of hgh enough precson, ndependently of n. Lemm 3. Suppose f C n+1 [α, β], f s regulr, nd tht the qudrture rule n (5) hs degree of precson r for ny r 0. Then L(p n [,b] ) w j p n (q j ) =O(h r+ ). Next we turn to the error n the nterpolton prt of method (4). The pproxmton order of ths prt depends cruclly on the smoothness of f. Agn we wll need to show tht dervtves of certn terms re bounded. Lemm 4. If f C n+1 [α, β] nd f s regulr, then ll dervtves up to order n of the functon g := f /( f + p n ) re bounded ndependently of h for smll enough h. Proof. Clerly g tself s bounded ndependently of h, snce f s regulr. Next let k {1,...,n}. Snce, (( f + p n )g)(k) = f (k+1), Lebnz rule gves g (k) = 1 f + p n ( f (k+1) k =1 ( ) ) k ( f () + p n () )g (k ). By Lemm, p n () s bounded for ech 0 when h s smll enough. By Lemm 1, so s f () for = 0,...,n. Thus, f ll dervtves of g up to order k 1 re bounded, so s g (k). Ths gves us our frst result on the pproxmton order of the pont-bsed method. Lemm 5. If f C n+1 [α, β] nd regulr then, s h 0, L(f [,b] ) L(p n [,b] ) = O(h n+1 ). (11)

5 516 M.S. Floter, A.F. Rsmussen / Journl of Computtonl nd Appled Mthemtcs 196 (006) 51 5 If n ddton t 0 = nd t n = b then L(f [,b] ) L(p n [,b] ) = O(h n+ ). (1) Proof. Lettng e(t) := f(t) p n (t), we use the dentty f p f e e n =e f + p n. Ths gves us where I 1 = f (t) p n (t) dt = I 1 I, (13) e (t) g(t) dt, I = e (t) f (t) + p dt n (t) nd g := f /( f + p n ). Snce e s of order O(h n ) by (10), nd f (t) s bounded wy from zero, we see tht I 1 = O(h n+1 ) nd I = O(h n+1 ), nd snce n + 1 n + 1, ths estblshes (11). If n ddton t 0 = nd t n = b then e() = e(b) = 0, nd so ntegrton by prts mples I 1 = e(t) g (t) dt. (14) Snce e s O(h n+1 ) by (10), nd g (t) s bounded s h 0 by Lemm 4, we now hve I 1 = O(h n+ ). Snce n 1 we lso hve I = O(h n+ ), nd ths estblshes (1). It s nterestng to note tht wthout needng to rse the smoothness ssumpton on f, we rse the pproxmton order by one smply by ncludng the end ponts of the ntervl [,b] n the nterpolton ponts t. Smlr observtons were mde n [7,8]. Now the pont s tht we cn contnue to rse the order of pproxmton by further restrctng the loctons of the t. Notce tht the order of the ntegrl I n (13) s lredy very hgh, nmely n + 1 whch mens tht we cn rse the order of the whole error (13) by mnpultng the frst ntegrl I 1. To do ths we borrow from the de of Guss qudrture. Lemm 6. Suppose f C n [α, β] nd regulr, nd tht t 0 =, t n = b nd ψ n (t)t k dt = 0, k = 0, 1,...,n, (15) where ψ n (t) := (t t 0 ) (t t n ). Then L(f [,b] ) L(p n [,b] ) = O(h n+1 ). Proof. It s enough to show tht I 1 n (13) s of order O(h n+1 ). Snce e(t) = ψ n (t)[t 0,t 1,...,t n,t]f, where [t 0,t 1,...,t n,t]f denotes the dvded dfference of f t the ponts t 0,t 1,...,t n,t, we cn wrte I 1 n (14) s I 1 = ψ n (t)γ(t) dt, Thus f we expnd γ n Tylor seres bout, n γ(t) = k=0 1 k! (t )k γ (k) () + γ(t) := ([t 0,t 1,...,t n,t]f) g (t). 1 (n 1)! (t )n 1 γ (n 1) (ξ t ),

6 M.S. Floter, A.F. Rsmussen / Journl of Computtonl nd Appled Mthemtcs 196 (006) wth ξ t t, the orthogonlty condtons (15) mply tht I 1 = 1 ψ (n 1)! n (t)(t ) n 1 γ (n 1) (ξ t ) dt. Therefore snce ψ n (t)(t ) n 1 h n, t b, the lemm wll be complete when we hve shown tht γ (n 1) s bounded s h 0. To see ths, observe tht Lebnz rule gves n 1 γ (n 1) (n 1)! (t) = (n 1 j)! ([t 0,t 1,...,t n,t,...,t]f) g }{{} (n j) (t). j+1 Snce t 0,t 1,...,t n,t,...,t }{{} f = f (n+1+j) (μ j, )/(n j)! j+1 for ech component f of f nd f C n [α, β], nd snce ll the dervtves g,...,g (n) re bounded by Lemm 4, ths shows tht γ (n 1) s bounded s clmed. Thus n order to ncrese the pproxmton order we cn choose the t so tht both t 0 = nd t n = b nd ψ n s orthogonl to π n (the spce of polynomls of degree t most n ) on [,b]. Ths cn be done by choosng ψ n (t) = (t )(t b)p n (t), where P n s the Legendre polynoml of degree n on the ntervl [,b]. A short clculton yelds ψ n (t)t k dt = P n (t)((t )(t b)kt k 1 + (t b)t k ) dt +[P n (t)(t )(t b)t k ] b. For k = 0,...,n ths s zero, snce P n s orthogonl to π n 1. The nterpolton nodes we cheve n ths mnner re known n numercl ntegrton s Guss Lobtto qudrture nodes. A tble of nodes cn be found n [1]. We re now ble to gve our mn result. Theorem. Suppose tht f C n [α, β], f s regulr, nd tht {t } n re the Guss Lobtto ponts n [,b]. Suppose lso tht {q j } m nd {w j } m re the nodes nd weghts, respectvely, of qudrture rule wth degree of ccurcy n 1 on [,b]. Then L(f [,b] ) w j p n (q j ) =O(h n+1 ). Proof. Ths follows from the trngle nequlty L(f [,b]) w j p n (q j ) L(f [,b]) L(p n [,b] ) + L(p n [,b] ) w j p n (q j ). nd Lemms 6 nd 3. Usng our nlyss, we now see tht the pont-bsed method s more robust thn the dervtve-bsed method from the pont of vew of the smoothness of f. Gven desred locl order of pproxmton, sy n + 1, the pont-bsed method of Theorem only requres f C n [α, β], whle the dervtve-bsed method of Theorem 1 requres f C n+1 [α, β].

7 518 M.S. Floter, A.F. Rsmussen / Journl of Computtonl nd Appled Mthemtcs 196 (006) Exmples 4.1. Second order method For n = 1 the only choce of nterpolton ponts stsfyng Lemm 5 s t 0 = nd t 1 = b. Computng the length of lner curve does not cll for qudrture, nd we re left wth the fmlr chord length rule: L(f [,b] ) f(b) f(). (16) By Theorem, ths rule hs locl error of O(h 3 ), so when used s composte rule, t hs globl error of O(h ).We hve thus proved tht the chord length rule hs order of ccurcy. Accordng to Theorem, the requred smoothness s tht f C [α, β]. If we compre ths to the mdpont method (8), we see tht we hve the sme order of ccurcy, but the mdpont rule requres f C 3 [α, β]. 4.. Fourth order methods For n = there s precsely one choce of the ponts t 0, t 1, t whch stsfes the condton of Lemm 6. We must set t 0 = nd t = b. Then we must choose t 1 n order to mke ψ orthogonl wth π 0,.e., wth the constnt functon 1. The only wy ths cn be cheved s by the symmetrc soluton t 1 = ( + b)/. Wth ths choce, f f C 4 [α, β] then L(f [,b] ) L(p ) = O(h 5 ) s h 0. Now we consder three choces of qudrture rule for p n order to cheve n O(h5 ) rule for L(f [,b] ). All methods presented n ths subsecton wll thus hve locl pproxmton order 5, nd globl order 4 (when used s composte method) Smpson-bsed rule Smpson s rule ppled to p gves L(f [,b] ) (b ) ( p 6 (q 0) +4 p (q 1) + p (q ) ), where t = q. Wrtng out the rule wth f := f(t ),weget L(f [,b] ) 1 6 ( 3f 0 + 4f 1 f +4 f f 0 + f 0 4f 1 + 3f ) Guss-bsed ( 3 ) rule The two-pont Guss rule gves L(f [,b] ) (b ) ( p (q 0) + p (q 1) ), where q 0,q 1 re ( + b)/ ((b )/6) 3. Wrtng out ths rule gves where L(f [,b] ) r f 0 + f r, (17) r = 1 (f 0 + f ) ( f0 + f 1 f ).

8 M.S. Floter, A.F. Rsmussen / Journl of Computtonl nd Appled Mthemtcs 196 (006) Ths rule my be the one best suted for mplementton, s t requres the computton of only two Eucldn norms,.e., squre roots. The other fourth order methods requre three such computtons The Vncent Forsey rule A thrd choce gves very smple rule n terms of the ponts f, = 0, 1,. The open Newton Cotes rule wth three nodes hs degree of precson 3, nd gves (b ) L(f [,b] ) ( p 3 (q 0) p (q 1) + p (q ) ), where q 0 = (3 + b)/4, q 1 = ( + b)/, q = ( + 3b)/4. Ths cn be wrtten s L(f [,b] ) 4 3 ( f(q 1) f() + f(b) f(q 1 ) ) 1 3 f(b) f(), (18) whch s the method of Vncent nd Forsey proposed n [16]. Ther resonng ws bsed on pproxmtng crculr segment, however, nd not polynomls. Snce the method stsfes the condtons of Theorem, we hve proved tht the Vncent Forsey method hs locl error O(h 5 ), nd globl error O(h 4 ). Therefore t hs fourth order of ccurcy when used s composte method Sxth order method We now derve sxth order method, by tkng n = 3 nd choosng nterpolton nodes fulflng the condtons of Lemm 6. To do ths, we must tke the nterpolton nodes to be the nodes of the four-node Guss Lobtto scheme (see for nstnce [1]): t 0 =, t 1 = + b (1 α), t = + b (1 + α), t 3 = b, where α = In order to get optmum order, we must pck qudrture method wth locl error O(h 7 ). If we use the three-pont Guss method L(f [,b] ) (b ) (5 p 3 18 (q 0) +8 p 3 (q 1) +5 p 3 (q ) ), Fg. 1. Method error comprson.

9 50 M.S. Floter, A.F. Rsmussen / Journl of Computtonl nd Appled Mthemtcs 196 (006) 51 5 wth the nodes q 0 = + b (1 β), q 1 = + b where β = , then we get the formul L(f [,b] ) r 1 f 0 + r r 1 + f 3 r, r 1 = 3 η f, r =, q = + b (1 + β), 3 η 3 f, (19) where the coeffcents η re gven by η = 1 36 ( , , , 0 5 ) 15. In Fg. 1, we hve results from evlutng the length of smple curve (n ths cse crculr segment) wth composte rules bult on vrous bsc rules. We cn see tht we get the expected slope of 6 for the order 6 method untl roundoff error becomes domnnt. For the other methods, we lso get the expected pproxmton order. 5. Geometrc propertes As we hve seen, the pproxmtons of the 3 method (17) nd the sxth order method (19) cn be wrtten s the lengths of certn polygons. Ths geometrc nterpretton of the pont-bsed method turns out to hold under frly generl condtons. Theorem 3. Suppose the qudrture weghts w j of rule (6) re postve, tht the rule hs precson of degree n 1, nd tht t 0 = nd t n = b. Then the length estmte of (6) s equl to the length of polygon wth end ponts f() nd f(b). Proof. We strt from (6) nd compute w j L (q m j )f(t ) = w j L (q m j )f(t ) = j = r j+1 r j, where r 0 = f() nd r j = f() j 1. Ths s the length of the polygon wth vertces r 0,...,r m+1.it remns to show tht r m+1 = f(b). Ths follows from r m+1 = f() + j = f() + = f() + = f() + w j L (q j )f(t ) f(t ) w j L (q j ) = f() + f(t )(L (b) L ()) = f(b). Now, we know tht for ny (contnuous) curve f, L(f [,b] ) f(b) f(). f(t ) L (t) dt

10 M.S. Floter, A.F. Rsmussen / Journl of Computtonl nd Appled Mthemtcs 196 (006) It turns out tht the estmted curve length gven by the pont-bsed rule (6) hs the sme property: Corollry 1. Under the ssumptons of Theorem 3, the length estmte of (6) hs the chord length s lower bound: w j p n (q j ) f(b) f(). Proof. The length of ny polygon from f() to f(b) s greter thn or equl to the length of the strght lne from f() to f(b) by the trngle nequlty. Note tht the condtons of the theorem re suffcent, but not necessry. For exmple, the Vncent Forsey rule (18) s bounded below by chord length, n spte of not fulflng the condtons of the theorem. 6. PH curve exctness For generl curves, t s not possble to fnd n nlytc form for the rc length. However, there re clsses of curves for whch the rc length ndeed hs n nlytc form. Exmples of ths nclude the pythgoren hodogrph (PH) curves of Frouk [5], nd the curve fmly ntroduced by Gl nd Keren [9]. In ths secton we show tht some of the pont-bsed methods constructed re exct for PH curves. The PH curves re plnr polynoml curves f :[α, β] R wth the property tht f s lso polynoml. One of the smplest exmples s the curve f(t) = (x(t), y(t)) where Snce x(t) = t t 3 /3, y(t) = t. (x (t)) + (y (t)) = (1 + t ), t follows tht f (t) =1 + t. Thus f s PH cubc. In generl PH curve s ny plnr polynoml curve of degree k +1 such tht f s polynoml of degree k. If we pply the dervtve-bsed method () to estmte the length of curve f over n ntervl [,b], s long s we use qudrture rule wth degree of precson k, the method wll clerly be exct when f s PH curve of degree k + 1. Thus for exmple, f we pply Smpson s rule or the two-pont Guss rule to estmte the length of PH cubc, the error wll be zero. Next consder the pont-bsed method (6). Clerly, f f s polynoml of degree n then p n = f nd so p n = f. In ths cse the pont-bsed method reduces to the dervtve-bsed one. Thus, for exmple, pont-bsed method wth n 3 (t lest four ponts) wll be exct for PH cubcs f. An nterestng stuton s the cse tht f s polynoml of exct degree n + 1, one hgher thn p n. Ths s the cse when f s for exmple PH cubc nd we use the Guss-bsed 3 rule (17). Recll tht f(t) p n (t) = ψ n (t)[t 0,t 1,...,t n,t]f. Therefore f f s polynoml of degree n + 1, f (t) p n (t) = ψ n (t)[t 0,t 1,...,t n,t]f. Thus we gn fnd p n (q )=f (q ) t certn ponts q, nmely those for whch ψ n (q )=0. Now f the ponts t 0,t 1,...,t n re the Guss Lobtto ponts then one cn show tht the ponts q 1,...,q n for whch ψ n (q )=0 re precsely the Guss ponts. Thus f we use Guss Lobtto ponts n the frst prt nd Guss ponts n the second, we get exctness for PH curves f of degree n+1. Ths s precsely wht hppens n the 3 rule when ppled to PH cubc. TheVncent Forsey rule on the other hnd does not shre ths property.

11 5 M.S. Floter, A.F. Rsmussen / Journl of Computtonl nd Appled Mthemtcs 196 (006) 51 5 More generlly, f f s ny cubc polynoml curve then the 3 rule s the sme s pplyng two-pont Guss ntegrton to the speed functon f. 7. Concludng remrks We hve mde frmework for computng lengths of curves wth only pont evlutons, nd shown tht we do not lose ccurcy compred to methods bsed on evlutng dervtves. We hve lso observed tht the methods re robust, requrng one less order of smoothness thn dervtve-bsed methods wth the sme order of ccurcy. We hve shown tht some prevously nvestgted methods ft n the frmework, nd thereby been ble to gve proofs of ther pproxmton order. In future rtcle we wll nvestgte evlutng the res of surfces wth only pont evlutons. References [1] M. Abrmowtz, I.A. Stegun, Hndbook of Mthemtcl Functons wth Formuls, Grphs, nd Mthemtcl Tbles, nnth ed., Dover, New York, 197. [] G. Cscol, S. Morg, Reprmetrzton of nurbs curves, Internt. J. Shpe Modellng (1996) [3] P. Constntn, R.T. Frouk, C. Mnn, A. Sestn, Computton of optml composte re-prmetrztons, Comput. Aded Geom. Desgn 18 (001) [4] R.T. Frouk, Optml prmetrztons, Comput. Aded Geom. Desgn 8 (1997) [5] R.T. Frouk, Pythgoren-hodogrph curves, n: Hndbook of Computer Aded Geometrc Desgn, 00, pp [6] R.T. Frouk, T. Skkls, Rel rtonl curves re not unt speed, Comput. Aded Geom. Desgn 8 (1991) [7] M.S. Floter, Arc length estmton nd the convergence of polynoml curve nterpolton, BIT, to pper. [8] M.S. Floter, Chordl cubc splne nterpolton s fourth order ccurte, IMA J. Numer. Anl. (005), to pper. [9] J. Gl, D. Keren, New pproch to the rc length prmeterzton problem, n: 13th Sprng Conference on Computer Grphcs, 1997, pp [10] J. Grvesen, Adptve subdvson nd the length nd energy of Bézer curves, Comput. Geom. 8 (1997) [11] B. Guenter, R. Prent, Computng the rc length of prmetrc curves, IEEE Comp. Grph. Appl. 5 (1990) [1] E. Iscson, H.B. Keller, Anlyss of Numercl Methods, Wley, New York, [13] J. Kerney, H. Wng, K. Atknson, Arc-length prmeterzed splne curves for rel-tme smulton, n: Curve nd Surfce Desgn, Snt-Mlo, 00, pp [14] E. Kreyszg, Dfferentl Geometry, Dover, New York, [15] R.J. Shrpe, R.W. Thorne, Numercl method for extrctng n rc length prmeterzton from prmetrc curves, Comput. Aded Desgn 14 () (198) [16] S. Vncent, D. Forsey, Fst nd ccurte prmetrc curve length computton, J. Grph. Tools 6 (4) (00) [17] M. Wlter, A. Fourner, Approxmte rc length prmeterzton, n: Proceedngs of the Nnth Brzln Symposum on Computer Grphcs nd Imge Processng, 1996, pp [18] F.-C. Wng, D.C.H. Yng, Nerly rc-length prmeterzed quntc-splne nterpolton for precson mchnng, Comput. Aded Desgn 5 (5) (1993)