Point-bsed methods for estimting the length of prmetric curve Michel S. Floter, Atgeirr F. Rsmussen CMA, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norwy Abstrct This pper studies generl method for estimting the length of prmetric curve using only smples of points. We show tht by mking specil choice of points, nmely the Guss-Lobtto nodes, we get higher orders of pproximtion, similr to the behviour of Guss qudrture, nd we derive some explicit exmples. Key words: curve, rc length, polynomil interpoltion 1 Introduction Computing the rc length of prmetric curve is bsic problem in geometric modelling nd computer grphics, nd hs been treted in vrious wys. In [11], Guenter nd Prent use numericl integrtion on the derivtive of the curve. In [15], Vincent nd Forsey derive method bsed entirely on point evlutions. Grvesen hs derived method specificlly for Bézier curves [10]. The estimtion of rc length is n importnt issue in [18], [17] nd [16], where pproximte rc length prmetriztions were sought for spline curves. This is necessry, since prt from trivil cses, polynomil curves never hve unit speed [6]. The rticle [2] trets the issue of reprmetrizing NURBS curves so tht the resulting curve prmetriztion is close to rc-length. The rticles [4] nd [3] del with optiml, i.e. s close to rc-length s possible, rtionl reprmetriztions of polynomil curves. In [14], the uthors clculte pproximte rc length prmetriztions for generl prmetric curves. Recently, results hve been obtined on pproximting the length of curve, given only s sequence of points (without prmeter vlues), using polynomils nd splines [7] [8]. Emil ddresses: michelf@ifi.uio.no (Michel S. Floter), tgeirr@mth.uio.no (Atgeirr F. Rsmussen). Preprint 5 October 2005
Suppose f : [α, β] lr d, d 2 is regulr prmetric curve, by which we men continuously differentible function such tht f (t) 0 for ll t [α, β], nd denotes the Euclidin norm in lr d. Then its rc length (see section 9 of [13]) is L(f) = β α f (t) dt. (1) Since L(f) is simply the integrl of the speed function f, nturl pproch is simply to pply to f some stndrd composite qudrture rule: we split the prmeter intervl [α, β] into smll pieces, pply qudrture rule to f in ech piece, nd dd up the contributions. If [, b] is one such piece, with α < b β, then typicl rule hs the form for some qudrture nodes n L(f [,b] ) = f (t) dt w i f (q i ), (2) i=0 q 0 < q 1 < < q n b, (3) nd weights w 0, w 1,..., w n. Guenter nd Prent [11] pply such method dptively. This method, however, hs the drwbck tht it involves derivtives of f, which might be more time-consuming to evlute thn points of f, or might simply not be vilble. One lterntive is the chord length rule (16), but it only hs second order ccurcy (s will be shown in 4.1). This motivted Vincent nd Forsey [15] to find higher order method using only point evlutions (18). In this pper, we investigte the following much more generl point-bsed method, which turns out to include these two methods s specil cses. We cn first interpolte f with polynomil p n : [, b] lr d, of degree n, t some points t 0 < t 1 < < t n b, for some n 1, i.e., p n (t i ) = f(t i ) for i = 0, 1,..., n, giving the pproximtion L(f [,b] ) L(p n [,b] ). (4) We cn then estimte the length of p n by qudrture, giving the estimte nd by expressing p n in the Lgrnge form L(p n [,b] ) w j p n(q j ), (5) n n t t j p n (t) = L i (t)f(t i ), L i (t) =, i=0,j i t i t j 2
we get the point-bsed rule L(f [,b] ) n w j L i(q j )f(t i ). (6) i=0 In view of the definition of the length L(f [,b] ) in (2), it is resonble to expect tht tht the error in (4) will be smll due the well-known fct tht p n is good pproximtion to f when h := b is smll. However, we hve not seen this method explicitly referred to in the literture, nor re we wre of ny error nlysis. The min contribution of this pper is to offer thorough nlysis of the pproximtion order of the method, in terms of h, which depends on the points t i, nd the qudrture nodes nd weights q j nd w j s well s the smoothness of f. One result of our nlysis is tht the interpoltion points t i cn be chosen to mximize the pproximtion order, nlogously to the use of Guss-Legendre points for numericl integrtion. 2 Error of the derivtive-bsed method For the ske of comprison, we strt with comment bout the pproximtion order of the derivtive-bsed method (2). If the qudrture rule used in (2) hs degree of precision r then the error will be of order O(h r+2 ) provided the (r + 1)-th derivtive of F := f is bounded [12]. Lemm 1 If f C r+2 [α, β], nd f is regulr, then ll the derivtives F, F,..., F (r+1) re bounded in [α, β]. Proof. Let k {1,..., r + 1}. By Leibniz rule, 2F F (k) + k 1 i=1 ( ) k F (i) F (k i) = (F 2 ) (k) = (f f ) (k) = i k i=0 ( ) k f (i+1) f (k i+1), i nd so F (k) 1 2F ( k i=0 ( ) k f (i+1) f (k i+1) + i k 1 i=1 ( ) ) k F (i) F (k i). (7) i Now since f is regulr on the closed intervl [α, β], F ttins strictly positive minimum ɛ > 0. Further, by ssumption, ll the derivtives f,..., f k+1 re bounded. Therefore, ssuming by induction tht ll the lower derivtives F,..., F (k 1) re bounded, we see tht F (k) is lso bounded. 3
This leds to the pproximtion order of the derivtive-bsed method. Theorem 1 Suppose f C r+2 [α, β], f is regulr, nd tht the rule (2) hs degree of precision r. Then L(f [,b] ) w j f (q j ) = O(h r+2 ) s h 0. For exmple since the midpoint rule hs degree of precision r = 1, we get L(f [,b] ) h f (q 0 ) = O(h 3 ), (8) where q 0 = ( + b)/2, provided f C 3 [α, β]. Since Simpson s rule hs degree of precision r = 3, we find L(f [,b] ) h( f () + 4 f (q 1 ) + f (b) )/6 = O(h 5 ), where q 1 = ( + b)/2, provided 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 precision 2m + 1 nd so provided f C 2m+3 [α, β], we get L(f [,b] ) w j f (q j ) = O(h 2m+3 ). 3 Error of the point-bsed method There re two contributions to the error of the point-bsed method, nmely the errors in the interpoltion prt (4) nd the qudrture prt (5). We will tret them both, strting with the qudrture error (5). Letting f i nd p n,i be the d components of the vector-vlued f nd p n, we recll clssicl result of polynomil interpoltion due to [12] (section 6.5, pge 290): f (k) i (t) p (k) n,i(t) h n+1 k mx s [,b] f (n+1) i (s) (n + 1 k)! (9) This eqution does not hold for vector-vlued functions, but we cn still use it to derive some error bounds: f (k) i (t) p (k) n,i(t) h n+1 k mx s [,b] f (n+1) (s). (n + 1 k)! Using the nottion f (n+1) [,b] := mx s [,b] f (n+1) (s), 4
we exploit the fct tht the right hnd side bove does not depend on the component i to write where C k = f (k) p (k) n [,b] C k h n+1 k f (n+1) [α,β], k = 0, 1,..., n. (10) d (n+1 k)!. Lemm 2 If f C n+1 [α, β] nd f is regulr, then ll derivtives of the function p n re bounded independently of h for smll enough h. Proof. We will prove this by showing tht p n is regulr for sufficiently smll h, then pply Lemm 1. By the tringle inequlity p n(t) f (t) f (t) p n(t) for ll t. Using eqution (10) in 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, since f is bounded wy from zero, so will p n be for sufficiently smll h. Then p n is regulr. Since p n is polynomil, it is in C r+2 for ll r nd we cn pply Lemm 1 to show tht ll derivtives of p n re bounded. The pproximtion order of the qudrture prt of the point-bsed method now immeditely follows, nlogously to theorem 1. Provided f C n+1 [α, β], we cn mke the order of this prt of the error s high s we like simply by using qudrture rule of high enough precision, independently of n. Lemm 3 Suppose f C n+1 [α, β], f is regulr, nd tht the qudrture rule in (5) hs degree of precision r for ny r 0. Then L(p n [,b] ) w j p n(q j ) = O(h r+2 ). Next we turn to the error in the interpoltion prt of the method (4). The pproximtion order of this prt depends crucilly on the smoothness of f. Agin we will need to show tht derivtives of certin terms re bounded. Lemm 4 If f C n+1 [α, β] nd f is regulr, then ll derivtives up to order n of the function g := f /( f + p n ) re bounded independently of h for smll enough h. Proof. Clerly g itself is bounded independently of h, since f is regulr. Next let k {1,..., n}. Since, ( ( f + p n )g) (k) = f (k+1), 5
Leibniz rule gives g (k) = ( 1 f (k+1) f + p n k i=1 ( ) k )( f (i) + p n (i) )g (k i) i By lemm 2, p n (i) is bounded for ech i 0 when h is smll enough. By lemm 1, so is f (i) for i = 0,..., n. Thus, if ll derivtives of g up to order k 1 re bounded, so is g (k). This gives us our first result on the pproximtion order of the point-bsed method. Lemm 5 If f C n+1 [α, β] nd regulr then, s h 0, If in ddition t 0 = nd t n = b then L(f [,b] ) L(p n [,b] ) = O(h n+1 ). (11) L(f [,b] ) L(p n [,b] ) = O(h n+2 ). (12) Proof. Letting e(t) := f(t) p n (t), we use the identity This gives us f p n = 2e f e e. f + p n f (t) p n(t) dt = 2I 1 I 2. (13) where I 1 = e e (t) 2 (t) g(t) dt, I 2 = f (t) + p n(t) dt. nd g := f /( f + p n ). Since e is of order O(h n ) by (10), nd f (t) is bounded wy from zero, we see tht I 1 = O(h n+1 ) nd I 2 = O(h 2n+1 ), nd since 2n + 1 n + 1, this estblishes (11). If in ddition t 0 = nd t n = b then e() = e(b) = 0, nd so integrtion by prts implies I 1 = e(t) g (t) dt. (14) Since e is O(h n+1 ) by (10), nd g (t) is bounded s h 0 by Lemm 4, we now hve I 1 = O(h n+2 ). Since n 1 we lso hve I 2 = O(h n+2 ), nd this estblishes (12). It is interesting to note tht without needing to rise the smoothness ssumption on f, we rise the pproximtion order by one simply by including the end points of the intervl [, b] in the interpoltion points t i. Similr observtions were mde in [7] nd [8]. Now the point is tht we cn continue to 6
rise the order of pproximtion by further restricting the loctions of the t i. Notice tht the order of the integrl I 2 in (13) is lredy very high, nmely 2n + 1 which mens tht we cn rise the order of the whole error (13) by mnipulting the first integrl I 1. To do this we borrow from the ide of Guss qudrture. Lemm 6 Suppose f C 2n [α, β] nd regulr, nd tht t 0 =, t n = b nd where ψ n (t) := (t t 0 ) (t t n ). Then ψ n (t)t k dt = 0, k = 0, 1,..., n 2, (15) L(f [,b] ) L(p n [,b] ) = O(h 2n+1 ). Proof. It is enough to show tht I 1 in (13) is of order O(h 2n+1 ). Since e(t) = ψ n (t)[t 0, t 1,..., t n, t]f, where [t 0, t 1,..., t n, t]f denotes the divided difference of f t the points t 0, t 1,..., t n, t, we cn write I 1 in (14) s I 1 = ψ n (t)γ(t) dt, γ(t) := ([t 0, t 1,..., t n, t]f) g (t). Thus if we expnd γ in Tylor series bout, γ(t) = n 2 k=0 1 k! (t )k γ (k) () + 1 (n 1)! (t )n 1 γ (n 1) (ξ t ), with ξ t t, the orthogonlity conditions (15) imply tht Therefore since I 1 = 1 ψ n (t)(t ) n 1 γ (n 1) (ξ t ) dt. (n 1)! ψ n (t)(t ) n 1 h 2n, t b, the lemm will be complete when we hve shown tht γ (n 1) is bounded s h 0. To see this, observe tht Leibniz rule gives γ (n 1) (t) = n 1 (n 1)! (n 1 j)! ([t 0, t 1,..., t n, t,..., t]f) g (n j) (t). }{{} j+1 Since [t 0, t 1,..., t n, t,..., t]f }{{} i = f (n+1+j) i (µ j,i )/(n + 1 + j)! j+1 7
for ech component f i of f nd f C 2n [α, β], nd since ll the derivtives g..., g (n) re bounded by lemm 4, this shows tht γ (n 1) is bounded s climed. Thus in order to increse the pproximtion order we cn choose the t i so tht both t 0 = nd t n = b nd ψ n is orthogonl to π n 2 (the spce of polynomils of degree t most n 2) on [, b]. This cn be done by choosing ψ n (t) = (t )(t b)p n(t) where P n is the Legendre polynomil of degree n on the intervl [, b]. A short clcultion yields ψ n (t)t k dt = + ) P n (t) ((t )(t b)kt k 1 + (2t b)t k dt [ P n (t)(t )(t b)t k ] b. For k = 0,..., n 2 this is zero, since P n is orthogonl to π n 1. The interpoltion nodes we chieve in this mnner re known in numericl integrtion s Guss-Lobtto qudrture nodes. A tble of nodes cn be found in [1]. We re now ble to give our min result. Theorem 2 Suppose tht f C 2n [α, β], f is regulr, nd tht {t i } n i=0 re the Guss-Lobtto points in [, b]. Suppose lso tht {q j } m nd {w j } m re the nodes nd weights, respectively, of qudrture rule with degree of ccurcy 2n 1 on [, b]. Then L(f [,b] ) w j p n(q j ) = O(h 2n+1 ). Proof. This follows from the tringle inequlity 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. Using our nlysis, we now see tht the point-bsed method is more robust thn the derivtive-bsed method from the point of view of the smoothness 8
of f. Given desired locl order of pproximtion, sy 2n + 1, the point-bsed method of theorem 2 only requires f C 2n [α, β], while the derivtive-bsed method of theorem 1 requires f C 2n+1 [α, β]. 4 Exmples 4.1 Second order method For n = 1 the only choice of interpoltion points stisfying lemm 5 is t 0 = nd t 1 = b. Computing the length of liner curve does not cll for qudrture, nd we re left with the fmilir chord length rule: L(f [,b] ) f(b) f(). (16) By theorem 2, this rule hs locl error of O(h 3 ), so when used s composite rule, it hs globl error of O(h 2 ). We hve thus proved tht the chord length rule hs order of ccurcy 2. According to theorem 2, the required smoothness is tht f C 2 [α, β]. If we compre this to the midpoint method (8), we see tht we hve the sme order of ccurcy, but the midpoint rule requires f C 3 [α, β]. 4.2 Fourth order methods For n = 2 there is precisely one choice of the points t 0, t 1, t 2 which stisfies the condition of lemm 6. We must set t 0 = nd t 2 = b. Then we must choose t 1 in order to mke ψ 2 orthogonl with π 0, i.e., with the constnt function 1. The only wy this cn be chieved is by the symmetric solution t 1 = (+b)/2. With this choice, if f C 4 [α, β] then L(f [,b] ) L(p 2 ) = O(h 5 ) s h 0. Now we consider three choices of qudrture rule for p 2 in order to chieve n O(h 5 ) rule for L(f [,b] ). All methods presented in this subsection will thus hve locl pproximtion order 5, nd globl order 4 (when used s composite method). 9
4.2.1 Simpson-bsed rule Simpson s rule pplied to p 2 gives L(f [,b] ) (b ) ( p 6 2(q 0 ) + 4 p 2(q 1 ) + p 2(q 2 ) ), where t i = q i. Writing out the rule with f i := f(t i ), we get L(f [,b] ) 1 6 ( ) 3f 0 + 4f 1 f 2 + 4 f 2 f 0 + f 0 4f 1 + 3f 2. 4.2.2 Guss-bsed ( 3 ) rule The two-point Guss rule gives L(f [,b] ) (b ) ( p 2 2(q 0 ) + p 2(q 1 ) ), where q 0, q 1 re +b b 2 6 3. Writing out this rule gives where L(f [,b] ) r f 0 + f 2 r, (17) r = 1 2 (f 0 + f 2 ) + 1 3 3( f0 + 2f 1 f 2 ). This rule my be the one best suited for implementtion, s it requires the computtion of only two Euclidin norms, i.e., squre roots. The other fourth order methods require three such computtions. 4.2.3 The Vincent-Forsey rule A third choice gives very simple rule in terms of the points f i, i = 0, 1, 2. The open Newton-Cotes rule with three nodes hs degree of precision 3, nd gives (b ) L(f [,b] ) (2 p 3 2(q 0 ) p 2(q 1 ) + 2 p 2(q 2 ) ), where q 0 = (3 + b)/4, q 1 = ( + b)/2, q 2 = ( + 3b)/4. This cn be written s L(f [,b] ) 4 3 ( ) f(q 1 ) f() + f(b) f(q 1 ) 1 f(b) f(), (18) 3 which is the method of Vincent nd Forsey proposed in [15]. Their resoning ws bsed on pproximting circulr segment, however nd not polynomils. Since the method stisfies the conditions of theorem 2, we hve proved tht the Vincent-Forsey method hs locl error O(h 5 ), nd globl error O(h 4 ). Therefore it hs fourth order of ccurcy when used s composite method. 10
4.3 Sixth order method We now derive sixth order method, by tking n = 3 nd choosing interpoltion nodes fulfilling the conditions of lemm 6. To do this, we must tke the interpoltion nodes to be the nodes of the four-node Guss-Lobtto scheme (see for instnce [1]): t 0 =, t 1 = + b 2 (1 α), t 2 = + b 2 (1 + α), t 3 = b where α = 1 5 5. In order to get optimum order, we must pick qudrture method with locl error O(h 7 ). If we use the three-point Guss method with the nodes L(f [,b] ) q 0 = + b 2 (b ) (5 p 18 3(q 0 ) + 8 p 3(q 1 ) + 5 p 3(q 2 ) ), (1 β), q 1 = + b 2, q 2 = + b (1 + β) 2 where β = 1 5 15, then we get the formul L(f [,b] ) r 1 f 0 + r 2 r 1 + f 3 r 2, (19) 3 3 r 1 = η i f i, r 2 = η 3 i f i, i=0 i=0 where the coefficients η i re given by η = 1 ( 16 5 15, 10 5 + 5 15, 10 5 + 5 15, 20 5 ) 15. 36 In figure 1, we hve results from evluting the length of smple curve (in this cse circulr segment) with composite rules built on vrious bsic rules. We cn see tht we get the expected slope of 6 for the order 6 method until roundoff error becomes dominnt. For the other methods, we lso get the expected pproximtion order. 5 Geometric properties As we hve seen, the pproximtions of the 3 method (17) nd the 6th order method (19) cn be written s the lengths of certin polygons. This 11
5 0 chord length Vincent-Forsey sqrt(3) order 6-5 -10 ln(error) -15-20 -25-30 -35 0 1 2 3 4 5 6 7 8 ln(num_intervls) Fig. 1. Method error comprison geometric interprettion of the point-bsed method turns out to hold under firly generl conditions. Theorem 3 Suppose the qudrture weights w j of the rule (6) re positive, tht the rule hs precision of degree n 1, nd tht t 0 = nd t n = b. Then the length estimte of (6) is equl to the length of polygon with end points f() nd f(b). Proof. We strt from (6) nd compute n w j L m i(q j )f(t i ) = n w j L i(q j )f(t i ) = m j = r j+1 r j i=0 i=0 where r 0 = f() nd r j = f() + 0 +... + j 1. This is the length of the polygon with vertices r 0,..., r m+1. It remins to show tht r m+1 = f(b). This follows from n r m+1 = f() + j = f() + w j L i(q j )f(t i ) i=0 n n = f() + f(t i ) w j L i(q j ) = f() + f(t i ) L i(t) dt i=0 i=0 n = f() + f(t i ) (L i (b) L i ()) = f(b). i=0 12
Now, we know tht for ny (continuous) curve f, L(f [,b] ) f(b) f(). It turns out tht the estimted curve length given by the point-bsed rule (6) hs the sme property: Corollry 1 Under the ssumptions of theorem 3, the length estimte 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) is greter thn or equl to the length of the stright line from f() to f(b) by the tringle inequlity. Note tht the conditions of the theorem re sufficient, but not necessry. For exmple, the Vincent-Forsey rule (18) is bounded below by chord length, in spite of not fulfilling the conditions of the theorem. 6 PH curve exctness For generl curves, it is not possible to find n nlytic form for the rc length. However, there re clsses of curves for which the rc length indeed hs n nlytic form. Exmples of this include the pythgoren hodogrph (PH) curves of Frouki [5], nd the curve fmily introduced by Gil nd Keren [9]. In this section we show tht some of the point-bsed methods constructed re exct for PH curves. The PH curves re plnr polynomil curves f : [α, β] lr 2 with the property tht f is lso polynomil. One of the simplest exmples is the curve f(t) = (x(t), y(t)) where x(t) = t t 3 /3, y(t) = t 2. Since it follows tht (x (t)) 2 + (y (t)) 2 = (1 + t 2 ) 2, f (t) = 1 + t 2. Thus f is PH cubic. In generl PH curve is ny plnr polynomil curve of degree 2k + 1 such tht f is polynomil of degree 2k. If we pply the derivtive-bsed method (2) to estimte the length of curve f over n intervl [, b], s long s we use qudrture rule with degree of 13
precision 2k, the method will clerly be exct when f is PH curve of degree 2k + 1. Thus for exmple, if we pply Simpson s rule or the two-point Guss rule to estimte the length of PH cubic, the error will be zero. Next consider the point-bsed method (6). Clerly, if f is polynomil of degree n then p n = f nd so p n = f. In this cse the point-bsed method reduces to the derivtive-bsed one. Thus, for exmple, point-bsed method with n 3 (t lest four points) will be exct for PH cubics f. An interesting sitution is the cse tht f is polynomil of exct degree n+1, one higher thn p n. This is the cse when f is for exmple PH cubic 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 if f is polynomil of degree n + 1, f (t) p n(t) = ψ n(t)[t 0, t 1,..., t n, t]f. Thus we gin find p n(q i ) = f (q i ) t certin points q i, nmely those for which ψ n(q i ) = 0. Now if the points t 0, t 1,..., t n re the Guss-Lobtto points then one cn show tht the points q 1,..., q n for which ψ n(q i ) = 0 re precisely the Guss points. Thus if we use Guss-Lobtto points in the first prt nd Guss points in the second, we get exctness for PH curves f of degree n + 1. This is precisely wht hppens in the 3 rule when pplied to PH cubic. The Vincent-Forsey rule on the other hnd does not shre this property. More generlly, if f is ny cubic polynomil curve then the 3 rule is the sme s pplying 2-point Guss integrtion to the speed function f. 7 Concluding remrks We hve mde frmework for computing lengths of curves with only point evlutions, nd shown tht we don t lose ccurcy compred to methods bsed on evluting derivtives. We hve lso observed tht the methods re robust, requiring one less order of smoothness thn derivtive-bsed methods with the sme order of ccurcy. We hve shown tht some previously investigted methods fit in the frmework, nd thereby been ble to give proofs of their pproximtion order. In future rticle we will investigte evluting the res of surfces with only point evlutions. 14
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