C 1 Interpolation with Cumulative Chord Cubics

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1 Fundamenta Informaticae XXI (2001) IOS Press C 1 Interpolation with Cumulative Chord Cubics Ryszard Kozera School of Computer Science and Software Engineering The University of Western Australia 35 Stirling Highway, Crawley W.A. 6009, Perth, Australia ryszard@csse.uwa.edu.au ryszard Lyle Noakes School of Mathematics and Statistics The University of Western Australia 35 Stirling Highway, Crawley W.A. 6009, Perth, Australia lyle@maths.uwa.edu.au lyle Abstract. Cumulative chord C 1 piecewise-cubics, for unparameterized data from regular curves in R n, are constructed as follows. In the first step derivatives at given ordered interpolation points are estimated from ordinary (non-c 1 ) cumulative chord piecewise-cubics. Then Hermite interpolation is used to generate a C 1 piecewise-cubic interpolant. Theoretical estimates of orders of approximation are established, and their sharpness verified through numerical experiments. Good performance of the interpolant is also confirmed experimentally on sparse data. Keywords: Interpolation, unparameterized data, cumulative chords, curve and length estimation, orders of approximation Address for correspondence: School of Computer Science and Software Engineering, The University of Western Australia, 35 Stirling Highway, Crawley W.A. 6009, Perth, Australia This research was supported by an Alexander von Humboldt Foundation

2 2 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics 1. Introduction Let γ : [0,T] R n be a smooth regular curve, namely γ is C r for some r 1 and γ(t) 0 for all t [0,T]. Our task is to estimate γ from an ordered m + 1-tuple Q m (q 0,q 1,...,q m ) of points in R n, where q i γ(t i ), 0 t 0 < t 1 <... < t i <... < t m T, and the corresponding t i are unknown. In general, depending on what is known about the t i the problem may be straightforward or unsolvable. Let for δ max{t i t i 1 : i 1,2,... m} we have δ 0. (1) In the simpler case with the t i known (then Q m defines the so-called parameterized data) the following standard result holds (see e.g. [9]): Example 1.1. If the t i are given then piecewise Lagrange interpolation through successive k + 1-tuples (q i,q i+1,q i+2,...q i+k ), where k 1 and i 0,k,2k,3k..., approximates γ and d(γ) with uniform errors at most δ k+1 (see Definition (1.1)). Note that for length estimation one needs an extra assumption on sampling i.e. mδ O(1). Without real loss we may suppose m divisible by k. In practice the t i may not be given, so that Q m is the only data available (the so-called unparameterized data). Then γ can at most be approximated up to reparameterizations. Definition 1.1. A family {f δ,δ > 0} of functions f δ : [0,T] R is said to be O(δ p ) when there is a constant K > 0 such that, for some δ 0 > 0, f δ (t) < Kδ p for all δ (0,δ 0 ) and all t [0,T]. In such a case write f δ O(δ p ). For a family of vector-valued functions F δ : [0,T] R n, write F δ O(δ p ) when F δ O(δ p ), where denotes the Euclidean norm. An approximation ˆγ : [0,T] R n to γ determined by Q m is said to have order p when A similar comparison can be made between the length ˆγ γ O(δ p ). (2) d(γ) T 0 γ(t) dt of γ and that of ˆγ. Note that the formula (2) assumes both domains of ˆγ and γ to coincide. If the t i are unknown then ˆγ is generically defined over a different domain i.e. ˆγ : [0, ˆT] R n. Then in order to apply (2), one needs to reparameterize ˆγ to ˆγ ψ for some ψ : [0,T] [0, ˆT] and then compare ˆγ ψ γ accordingly. If we guess the t i blindly, e.g. to be distributed uniformly t i ˆt i i m [0,1], the resulting uniform piecewise-quadratic ˆγ : [0,1] R n is sometimes uninformative. Indeed, for such a guess of t i, piecewise-linear interpolation approximates γ to order 2, but piecewise-quadratic approximations can actually degrade estimates [18], [19]. Thus without knowing the t i it seems difficult to match the order 3 achieved in Example 1.1, but it turns out that higher order approximations are achievable for many special curves γ possibly sampled

3 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics 3 according to some restrictive rule. In particular, some of those constraints require convexity of γ, its embedding in Euclidean space either R 2 or R 3, an a priori knowledge of the derivatives of γ at Q m or more-or-less uniformity of t i : i.e. existence of a constant 0 < λ < 1 for which λδ (3) for each m and i 0,...,m 1. Implementations of such schemes, studied in [2], [11], [13], [16], [17], [21], and [23], usually require numerical solutions of systems of nonlinear equations. An alternative, noted in [10] Chap. 11, (or [1] Chap. 16; [4] Chap. 9 or [5] Chap. 7) is interpolation based on cumulative chord length parameterizations [3], [12], used often in computer graphics [20]; Section More precisely, for an integer k 1, set ˆt 0 0 and ˆt j ˆt j 1 + q j q j 1, (4) for j 1,2,...,m. For k dividing m and i 0,k,2k,... m k, let ˆγ be the curve satisfying ˆγ(ˆt j ) q j, (5) for all j 0,1,2,...,m, and whose restriction ˆγ i to each [ˆt i, ˆt i+k ] is a polynomial of degree at most k. Call ˆγ the cumulative chord piecewise degree-k polynomial approximation to γ defined by Q m (q 0,q 1,...,q m ). Then it can be shown [14], [15]: Theorem 1.1. Suppose γ is a regular C r curve in R n, where r k + 1 and k is 2 or 3. Let ˆγ : [0, ˆT] R n be the cumulative chord piecewise degree-k approximation defined by Q m. Then there is a piecewise- C r reparameterization ψ : [0,T] [0, ˆT], with ˆγ ψ γ + O(δ k+1 ). If in addition mδ O(1) then also d(ˆγ) d(γ) + O(δ k+1 ). Note that Theorem 1.1 holds for any sufficiently smooth regular curve γ (not necessarily convex) in Euclidean space R n of arbitrary dimension, and is applicable without extra tight conditions on sampling. Cumulative chord piecewise-cubics (or piecewise-quadratics) approximate at least to order 4 (or 3) which matches the same orders from Example 1.1, where the t i are given [9]. We remark here that cumulative chord piecewise-quartics yield further convergence speed-up greater than 4 only for special subfamilies of samplings (1) (see [7] or [8]). On the other hand, cumulative chord piecewise-polynomials are usually not C 1 at knot points t kj, where j 0 mod k. The purpose of the present paper is to rectify this deficiency for cumulative chord piecewise-cubics (k 3), and propose a new cumulative chord C 1 piecewise-cubic. The scheme reads: 1. First, for each i 0,1,...,m 3, let ˆγ i : [ˆt i, ˆt i+3 ] R n be the cumulative chord cubic interpolating q i,q i+1,q i+2,q i+3 at ˆt i, ˆt i+1, ˆt i+2, ˆt i+3, respectively. Those cumulative chords cubics permit to approximate the derivative of γ. E.g. for each subinterval [ˆt i, ˆt i+3 ] we estimate the velocity v(q i ) of γ at q i as v(q i ) ˆγ i (ˆt i ) - see Figure 1. Note that for the last three points (q m 2,q m 1,q m ) Q the respective derivative estimation is obtained by passing, for each i m,m 1,m 2, the reverse cumulative chord cubic interpolants ˆγ i : [ˆt i, ˆt i 3 ] R n satisfying reverse order interpolation conditions ˆγ i (ˆt i k ) q i k, for k 0,1,2,3, respectively.

4 4 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics Figure 1. Estimation of the velocities of γ at q i by cumulative chord cubics as v(q i ) ˆγ i (ˆt i ). 2. Then let γ i h : [ˆt i, ˆt i+1 ] R n be the Hermite cubic polynomial satisfying γ i h (ˆt i ) q i, γ i h (ˆt i+1 ) q i+1, γ i h (ˆt i ) ˆγ i (ˆt i ), and γ i h (ˆt i+1 ) ˆγ i+1 (ˆt i+1 ) (6) defined by Newton s Interpolation Formula (see [2] Chap. 1) as γ i h (ˆt) γ i h [ˆt i ]+γ i h [ˆt i, ˆt i ](ˆt ˆt i )+γ i h [ˆt i, ˆt i, ˆt i+1 ](ˆt ˆt i ) 2 +γ i h [ˆt i, ˆt i, ˆt i+1, ˆt i+1 ](ˆt ˆt i ) 2 (ˆt ˆt i+1 ). (7) Let γ h : [0, ˆT] R n (called cumulative chord C 1 piecewise-cubic) be the track-sum of the γ i h. Here is our main result: Theorem 1.2. Suppose γ is a regular C 4 curve in R n. Let γ h : [0, ˆT] R n be the cumulative chord C 1 piecewise-cubic defined by Q m as in (6). Then there is a piecewise-c reparameterization φ : [0,T] [0, ˆT], with γ h φ γ + O(δ 4 ), and if in addition mδ O(1) then d(γ h ) d(γ) + O(δ 4 ). There is a version of this construction, and of Theorem 1.2, for cumulative chord piecewise-quadratics, with orders of convergence decreased by 1. After some preliminaries and auxiliary lemmas in Section 2, Theorem 1.2, is proved in Section 3. Then, in Section 4, the sharpness of Theorem 1.2 is illustrated for some planar and space curves (n 2,3). 2. Divided Differences and Cumulative Chords First recall some facts about divided differences [22]: γ[t i ] γ(t i ), the first divided difference of γ for t i t i+1 is γ[t i,t i+1 ] γ(t i+1) γ(t i ),

5 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics 5 and, for k 2,3,...,m i, the kth divided difference is defined inductively for t i+k t i as γ[t i,t i+1,...,t i+k ] γ[t i+1,t i+2,...,t i+k ] γ[t i,t i+1,...,t i+k 1 ] t i+k t i. In case when the tabular points t i,t i+1,...,t i+k coincide and γ C k then γ[t i,t i+1,...,t i+k ] γ(k) (t i ) k! Newton s Interpolation Formula says γ(t) L + R, where. L γ(t i ) + (t t i )γ[t i,t i+1 ] + (t t i )(t t i+1 )γ[t i,t i+1,t i+2 ] (t t i )(t t i+1 )... (t t i+k 1 )γ[t i,t i+1,...,t i+k ] is the polynomial of degree at most k interpolating γ at t i,t i+1,... t i+k, and R (t t i )(t t i+1 )... (t t i+k )γ[t,t i,t i+1,... t i+k ]. When γ is C k+1 and t i,t i+1,t i+2,...,t i+k+1 (t δ,t+ δ) where δ > 0 then, for j 1,2,...,n, the jth component of the k + 1th divided difference is given by γ[t,t i,t i+1,... t i+k ] j γ(k+1) j ( t j ) (k + 1)!, (8) for some t j (t δ,t + δ). Let γ be C r and regular, where r k + 1 and k is 2 or 3. After a C r reparameterization, as in [6] Chapter 1, Proposition 1.1.5, we can assume for proving purposes that γ is parameterized by arc-length, namely γ is identically 1. Consider the cubic Lagrange interpolants ψ i : [t i,t i+3 ] [ˆt i, ˆt i+3 ] and ψ i+1 : [t i+1,t i+4 ] [ˆt i+1, ˆt i+4 ] satisfying ψ i+k (t i+k+j ) ˆt i+k+j, (9) for each k 0, 1 with j 0, 1, 2, 3. By Newton s Interpolation Formula ψ i+k (t) ψ i+k (t i+k ) + (t t i+k )ψ i+k [t i+k,t i+k+1 ] The following holds [14]: +(t t i+k )(t t i+k+1 )ψ i+k [t i+k,t i+k+1,t i+k+2 ] +(t t i+k )(t t i+k+1 )(t t i+k+2 )ψ i+k [t i+k,t i+k+1,t i+k+2,t i+k+3 ]. Lemma 2.1. If γ is C 4 then, for k 0,1 and t [t i+k,t i+k+3 ] we have ψ i+k 1 + O(δ 2 ), ψi+k O(δ) and d3 ψ i+k dt O(1). 3 In particular, ψ i+k (for k 0,1) is a C diffeomorphism for δ small, which we assume from now on. Similarly for ˆγ defined in (5) we have [14]: Lemma 2.2. If γ is C 4 then, for k 0,1 and s [ˆt i, ˆt i+k+3 ], ˆγ i+k, dˆγi+k ds, d2ˆγ i+k d3ˆγ i+k, ds 2 ds 3 are O(1).

6 6 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics Let ψ : [0,T] [0, ˆT] be the track-sum of the ψ i. The following lemma is used in [14] to prove Theorem 1.1: Lemma 2.3. If γ is C 4 then, for k 0,1 and t [t i,t i+k+3 ], γ(t) (ˆγ i+k ψ i+k )(t) O(δ 4 ) and γ(t) d(ˆγi+k ψ i+k ) (t) O(δ 3 ). (10) dt Let φ i : [t i,t i+1 ] [ˆt i, ˆt i+1 ] be the cubic Hermite polynomial satisfying φ i (t i ) ˆt i, φ i (t i+1 ) ˆt i+1, φi (t i ) ψ i (t i ), and φ i (t i+1 ) ψ i+1 (t i+1 ), (11) given, by Newton s Interpolation Formula, as φ i (t) φ i [t i ]+φ i [t i,t i ](t t i )+φ i [t i,t i,t i+1 ](t t i ) 2 + φ i [t i,t i,t i+1,t i+1 ](t t i ) 2 (t t i+1 ). (12) Then for the track-sum φ : [0, ˆT] R n of the φ i we have: Lemma 2.4. φ i [t i,t i ] 1+O(δ 2 ), φ i [t i,t i,t i+1 ] O(δ), and φ i [t i,t i,t i+1,t i+1 ] O(1)+ O(δ2 ) (t i+1 t i ) 2. Proof: Formulae (8), (11) combined with Lemma 2.1 render and analogously Similarly φ i [t i,t i ] φ(t i ) ψ i (t i ) 1 + O(δ 2 ), φ i [t i,t i,t i+1 ] ψi (t i+1 ) ψ i (t i ) ( ) 2 ψ i (t i ) ψ i [t i,t i,t i+1 ] O( ψ i ) O(δ). φ i [t i,t i,t i+1,t i+1 ] ψ i+1 (t i+1 ) ψ i [t i,t i+1 ] ( ) 2 ψi [t i,t i,t i+1 ]. (13) By Lemma 2.1, ψ i+k (t i+1 ) 1+O(δ 2 ), for k 0,1 and thus we have ψ i+1 (t i+1 ) ψ i (t i+1 )+O(δ 2 ). Hence, (8), (13) and Lemma 2.1 yield φ i [t i,t i,t i+1,t i+1 ] The proof is complete. O(δ 2 ) ( ) 2 + ψ i (t i+1 ) ψ i [t i,t i+1 ] ( ) 2 ψi [t i,t i,t i+1 ] O(δ 2 ) ( ) 2 + ψi [t i,t i,t i+1,t i+1 ] O(δ 2 ) ( ) 2 + ψ i O(d3 dt 3 ) O(δ 2 ) ( ) 2 + O(1).

7 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics 7 Lemma 2.5. For t [t i,t i+1 ] φ i 1 + O(δ 2 ), φ i O(δ) + O(δ2 ) t i+1 t i, and d3 φ i dt 3 O(1) + O(δ2 ) (t i+1 t i ) 2. Proof: Differentiating (12) accordingly together with Lemma 2.4 and (t t i+k )(t i+1 t i ) 1 1 (for k 0,1) completes the proof. Thus, asymptotically, each φ i is a diffeomorphism. Similarly, the following holds for γ i h : Lemma 2.6. γ i h [ˆt i, ˆt i ] O(1), γ i h [ˆt i, ˆt i, ˆt i+1 ] O(1), and γ i h [ˆt i, ˆt i, ˆt i+1, ˆt i+1 ] O(1) + O(δ2 ) (ˆt i+1 ˆt i ) 2. Proof: By (6), (8), and Lemma 2.2 γ i h [ˆt i, ˆt i ] γ i h (ˆt i ) ˆγ i (ˆt i ) O(1), and similarly for each j-th component of γ i h and ˆγi (where 1 j n) and some t i j [t i,t i+1 ] we have γh i [ˆt i, ˆt i, ˆt i+1 ] j ˆγi j (ˆt i+1 ) ˆγ j i(ˆt i ) ˆγi j (ˆt i ) ˆγ (ˆt i+1 ˆt i ) 2 ˆt i+1 ˆt i j [ˆt i, ˆt i, ˆt i+1 ] j O(ˆγi ( t i j ) ) O(1). i 2 Evidently the latter extends to the vector form Analogously, (6) yields γ i h [ˆt i, ˆt i, ˆt i+1 ] O(1). γh i [ˆt i, ˆt i, ˆt i+1, ˆt i+1 ] ˆγi+1 (ˆt i+1 ) (ˆt i+1 ˆt i ) ˆγi (ˆt i+1 ) ˆγ i (ˆt i ) ˆγi (ˆt i+1 ) ˆγ i (ˆt i ) + ˆγi (ˆt i ) 2 (ˆt i+1 ˆt i ) 3 (ˆt i+1 ˆt i ) 3 (ˆt i+1 ˆt i ). 2 (14) Taylor s Theorem combined with Lemma 2.2 yield ˆγ i (ˆt i ) ˆγ i (ˆt i+1 ) ˆγ i (ˆt i+1 )(ˆt i+1 ˆt i ) + ˆγi Thus the first two terms in (14) read Furthermore, by (10) ˆγ i+1 (ˆt i+1 ) ˆγ i (ˆt i+1 ) (ˆt i+1 ˆt i ) 2 (ˆt i+1 ) (ˆt i+1 ˆt i ) 2 + O((ˆt i+1 ˆt i ) 3 ). 2 + ˆγi (ˆt i+1 ) + O(1). (15) 2(ˆt i+1 ˆt i ) ˆγ i+1 (ˆt i+1 ) ψ i+1 (t i+1 ) γ(t i+1 ) O(δ 3 ) and ˆγ i (ˆt i+1 ) ψ i (t i+1 ) γ(t i+1 ) O(δ 3 ) which combined with ψ i+k (t i+1 ) 1 + O(δ 2 ), for k 0,1 (see Lemma 2.1) yields ˆγ i+k (ˆt i+1 ) γ(t i+1 ) ˆγ i+k (ˆt i+1 )O(δ 2 ) + O(δ 3 ).

8 8 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics The latter combined with Lemma 2.2 reduces (15) (and thus the first two terms of (14)) to ˆγ i+1 (ˆt i+1 ) (ˆt i+1 ˆt i ) ˆγi (ˆt i+1 ) ˆγ i (ˆt i ) ˆγi (t i+1 )O(δ 2 ) ˆγ i+1 (t i+1 )O(δ 2 ) + O(δ 3 ) 2 (ˆt i+1 ˆt i ) 3 (ˆt i+1 ˆt i ) 2 On the other hand, Taylor s Theorem and Lemma 2.2 yield + ˆγi (ˆt i+1 ) + O(1). (16) 2(ˆt i+1 ˆt i ) ˆγ i (ˆt i+1 ) ˆγ i (ˆt i ) + ˆγ i (ˆt i )(ˆt i+1 ˆt i ) + ˆγi (ˆt i ) (ˆt i+1 ˆt i ) 2 + O((ˆt i+1 ˆt i ) 3 ) 2 and hence the last two terms in (14) read Adding (16) and (17) reduces (14) to ˆγ i (ˆt i ) (ˆt i+1 ˆt i ) ˆγi (ˆt i+1 ) ˆγ i (ˆt i ) O(1) ˆγi (ˆt i ) 2 (ˆt i+1 ˆt i ) 3 2(ˆt i+1 ˆt i ). (17) γh i [ˆt i, ˆt i, ˆt i+1, ˆt i+1 ] ˆγi (ˆt i+1 ) ˆγ i (ˆt i ) + O(1) + ˆγi (ˆt i+1 )O(δ 2 ) ˆγ i+1 (ˆt i+1 )O(δ 2 ) + O(δ 3 ), 2(ˆt i+1 ˆt i ) (ˆt i+1 ˆt i ) 2 which by Taylor s Theorem combined with Lemma 2.2 renders γ i h [ˆt i, ˆt i, ˆt i+1, ˆt i+1 ] O( d3ˆγ i dˆt ) + O(1) + O(1)O(δ2 ) + O(1)O(δ 2 ) + O(δ 3 ) 3 (ˆt i+1 ˆt i ) 2 O(1) + O(δ2 ) (ˆt i+1 ˆt i ) 2. The proof is complete. Lemma 2.7. For ˆt [ˆt i, ˆt i+1 ] γ i h O(1), γi h O(1) + O(δ2 ) t i+1 t i, and d3 γ i h dˆt 3 O(1) + O(δ2 ) (t i+1 t i ) 2. Proof: The Mean Value Theorem coupled with the definition of cumulative chords ˆt i, render for each ˆt [ˆt i, ˆt i+1 ] ˆt i+1 ˆt i O( ) O(δ) and thus ˆt ˆt i+k O( ) O(δ), (18) with k 0,1. Upon differentiating (7), Lemma 2.6 and (18) yield γ i h (ˆt) γ i h [ˆt i, ˆt i ] + 2γ i h [ˆt i, ˆt i, ˆt i+1 ](ˆt ˆt i ) + γ i h [ˆt i, ˆt i, ˆt i+1, ˆt i+1 ](2(ˆt ˆt i )(ˆt ˆt i+1 ) + (ˆt ˆt i ) 2 ) O(1) + O(1)O(δ) + O(δ 2 ) O(1),

9 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics 9 and similarly γ i h (ˆt) 2γ i h [ˆt i, ˆt i, ˆt i+1 ] + 2γ i h [ˆt i, ˆt i, ˆt i+1, ˆt i+1 ](2(ˆt ˆt i ) + (ˆt ˆt i+1 )) O(1) + O(δ2 ) ˆt i+1 ˆt i. (19) As by (4) ˆt i+1 ˆt i γ(t i+1 ) γ(t i ) Taylor s Theorem combined with γ 1, the Binomial Theorem and geometric expansion yield (ˆt i+1 ˆt i ) 1 (< γ(t i )( ) + O(( ) 2 ) γ(t i )( ) + O(( ) 2 ) >) 1/2 1 ( ) 1 + O( ) 1 ( )(1 + O( )) 1 ( ) (1 + O(δ) + O(δ2 ) + + O(δ l ) +...) O(1), where l N. Coupling the latter with (19) renders Analogously, by (7) and Lemma 2.6 γ i h (ˆt) O(1) + O(δ2 ). The proof is complete. d 3 γ i h dˆt 3 (ˆt) 6γ i h [ˆt i, ˆt i, ˆt i+1, ˆt i+1 ] O(1) + O(δ2 ) ( ) Proof of Theorem 1.2 Proof: To prove Theorem 1.2 we compare γ with γ h φ and γ with (γ h φ) (1) γ h φ over each subinterval [t i,t i+1 ]. We first show that over [0,T] we have: γ(t) (γ i h φi )(t) O(δ 4 ). (20) Indeed, for γ h reparameterized with φ and for ˆγ reparameterized with ψ, by Lemma 2.3 we have γ(t) (γ i h φi )(t) γ(t) (ˆγ i ψ i )(t) + (ˆγ i ψ i )(t) (γ i h φi )(t) O(δ 4 ) + (ˆγ i ψ i )(t) (γ i h φi )(t), (21) over t [t i,t i+1 ]. Newton s Interpolation Formula for the C function ρ i ˆγ i ψ i γ i h φi (22)

10 10 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics yields ρ i (t) ρ i [t i ] + ρ i [t i,t i ](t t i ) + ρ i [t i,t i,t i+1 ](t t i ) 2 + ρ i [t i,t i,t i+1,t i+1 ](t t i ) 2 (t t i+1 ) +(t t i ) 2 (t t i+1 ) 2 ρ i [t i,t i,t i+1,t i+1,t], (23) for t [t i,t i+1 ]. We prove now that ρ i (t) O(δ 4 ), (24) which combined with (21) is sufficient to prove the first claim of Theorem 1.2. Indeed, note that by (9), (11) combined with (5) and (6) the divided differences of ρ i appearing in (23) satisfy ρ i [t i ] ˆγ i (ψ i (t i )) γ i h (φi (t i )) ˆγ i (ˆt i ) γ i h (ˆt i ) q i q i 0, (25) ρ i [t i,t i ] ρ i (t i ) ˆγ i (ˆt i ) ψ i (t i ) γ i h (ˆt i ) φ i (t i ) ˆγ i (ˆt i ) ψ i (t i ) ˆγ i (ˆt i ) ψ i (t i ) 0, (26) and hence as for (25) we have ρ i (t i+1 ) 0 and thus and furthermore by (10) ρ i [t i,t i,t i+1 ] ρi (t i+1 ) ρ i (t i ) ( ) 2 ρi [t i,t i ] 0, (27) ρ i [t i,t i,t i+1,t i+1 ] ρi [t i,t i+1,t i+1 ] ρi (t i+1 ) ( ) 2 ˆγi (ˆt i+1 ) ψ i (t i+1 ) γ(t i+1 ) + γ(t i+1 ) γh i (ˆt i+1 ) φ i (t i+1 ) ( ) 2 O(δ 3 ) ( ) 2 + γ(t i+1) ˆγ i+1 (ˆt i+1 ) ψ i+1 (t i+1 ) ( ) 2 O(δ 3 ) ( ) 2 + O(δ3 ) ( ) 2 O(δ 3 ) ( ) 2. (28) Combining (23) with (25), (26), (27) and (28) yields, for t [t i,t i+1 ], To prove (24) it suffices now to show ρ i (t) O(δ 4 ) + (t t i ) 2 (t t i+1 ) 2 ρ i [t i,t i,t i+1,t i+1,t]. (29) (t t i ) 2 (t t i+1 ) 2 ρ i [t i,t i,t i+1,t i+1,t] O(δ 4 ), for t [t i,t i+1 ]. In doing so recall that by (8) for each component ρ i j, where 1 j n ρ i [t i,t i,t i+1,t i+1,t] j 1 4! d 4 ρ i j dt 4 ( t i j ), (30)

11 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics 11 for some t i j [t i,t i+1 ]. As the degrees of ˆγ i, ψ i, γh i and φi do not exceed 3, the Chain Rule combined with (30) and Lemmas 2.1, 2.2, 2.5, and 2.7 yield for all derivatives of ψ i, φ i evaluated at t i j and for all derivatives of ˆγ j i and γ h i j evaluated at ψi ( t i j ) and at φi ( t i j ), respectively ρ i [t i,t i,t i+1,t i+1,t] j 6ˆγ i j 6γ h i j ψ i2 ψi + 4ˆγ i j φ i2 φi i 4γ h ψ id3 ψ i dt 3 + 3ˆγi j ψ i2 j φ id3 φ i dt 3 3γ h i j φ i2 O(1) + O(δ2 ) + O(δ3 ) ( ) 2 + O(δ4 ) ( ) 3. (31) The latter extends to the vector form of ρ i [t i,t i,t i+1,t i+1,t]. Consequently, for each t [t i,t i+1 ], as the inequality (t t i+k )( ) 1 1 holds with k 0,1 we have (t t i ) 2 (t t i+1 ) 2 ρ i [t i,t i,t i+1,t i+1,t] (t t i ) 2 (t t i+1 ) 2 O(1) + (t t i ) 2 (t t i+1 )O(δ 2 ) +(t t i ) 2 O(δ 3 ) + (t t i )O(δ 4 ) O(δ 4 ). This together with (29), (23) and (21) proves the first claim of Theorem 1.2 i.e. the formula (20). To prove the second claim of Theorem 1.2 we first justify the following Indeed by Lemma 2.3 and (22), for t [t i,t i+1 ], we have γ(t) d(γi h φi ) dt By (23), (25), (26), and (27) we obtain γ(t) d(γi h φi ) (t) O(δ 3 ). (32) dt (t) γ(t) d(ˆγi ψ i ) (t) + d(ˆγi ψ i ) (t) d(γi h φi ) (t) dt dt dt O(δ 3 ) + ρ i (t). (33) ρ i (t) ρ i [t i,t i,t i+1,t i+1 ](2(t t i )(t t i+1 ) + (t t i ) 2 ) +2ρ i [t i,t i,t i+1,t i+1,t]((t t i )(t t i+1 ) 2 + (t t i ) 2 (t t i+1 )) + dρi [t i,t i,t i+1,t i+1,t] (t t i ) 2 (t t i+1 ) 2, dt and furthermore by (28), (31), the symmetry and the continuity of divided differences for t [t i,t i+1 ] (see [1] Chap.1): ρ i (t) O(δ 3 ρ i [t i,t i,t i+1,t i+1,t + h] ρ i [t,t i,t i,t i+1,t i+1 ] ) + lim (t t i ) 2 (t t i+1 ) 2 h 0 h O(δ 3 ) + ρ i [t i,t i,t i+1,t i+1,t,t](t t i ) 2 (t t i+1 ) 2. (34) As previously, for each 1 j n, we have ρ i [t i,t i,t i+1,t i+1,t,t] j 1 5! d 5 ρ i j dt 5 ( t i j ), (35)

12 12 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics for some t i j [t i,t i+1 ]. The Chain Rule combined with (35) and Lemmas 2.1, 2.2, 2.5, and 2.7 yield for all derivatives of ψ i, φ i evaluated at t i j and for all derivatives of ˆγi j and γ h i j evaluated at ψi ( t i j ) and at φ i ( t i j ), respectively ρ i [t i,t i,t i+1,t i+1,t,t] j 15ˆγ i j 15γ h i j ψ i ψ i2 + 10ˆγ j i ψ i2d3 ψ i dt ˆγi j φ i φ i2 i 10γ h j φ i2d3 φ i dt 3 10γ h i j ψ id3 ψ i dt 3 φ id3 φ i dt 3 O(1) + O(δ3 ) + O(δ2 ) ( ) 2 + O(δ4 ) ( ) 3 + O(δ4 ) ( ) 4. The latter extends to the vector form of ρ i [t i,t i,t i+1,t i+1,t]. Consequently for t [t i,t i+1 ] (t t i ) 2 (t t i+1 ) 2 ρ i [t i,t i,t i+1,t i+1,t,t] O(δ 4 ). which combined with (34) yields ρ i O(δ 3 ) and thus by (33) proving (32). Define now f γ h φ γ and let v(t) be the projection of f onto the orthogonal space γ(t) to γ(t). As γ(t) 1 for each f i γh i φi γ : [t i,t i+1 ] R n f i (t) f i (t), γ(t) γ(t) + v(t), where by (32) and the latter v(t) O(δ 3 ). Thus for reparameterized γ i h as γi h γi h φi we have γ i h(t) (1 + f i (t), γ(t) ) γ(t) + v(t). Thus as γ(t),v(t) 0, γ 1 and (1 + f i (t), γ(t) ) 2 v(t) 2 > 1 + ε (for some fixed ε > 0): γ i h(t) γ(t) f i (t), γ(t) + O(δ 6 ). (36) As f i (t i ) f i (t i+1 ) 0, the integration by parts, (20) and (36) yield ti+1 ( γ h i (t) γ(t) )dt t i ti+1 t i ti+1 f i (t), γ(t) dt + O(δ 7 ) f(t), γ(t) dt + O(δ 7 ) t i O(δ 5 ). Thus, over each T i [t i,t i+1 ] as d(γh i ) d( γi h ) (by Lemma 2.5 φi : [t i,t i+1 ] [ˆt i, ˆt i+1 ] is a reparameterization as φ i 1 + O(δ 2 ) > 0, asymptotically) and therefore d(γh i ) d(γ i) O(δ 5 ), where d(γ i ) t i+1 t i γ(t) dt. Hence as mδ O(1) m 1 d(γ h ) d(γ) (d(γh i ) d(γ i)) O(δ 4 ). i0 The proof is complete.

13 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics Figure 2. 7 points on the spiral γ sp (dashed) sampled as in (37), interpolated by a cumulative chord C 1 piecewisecubic γ sph, with d(γ sph ) d(γ sp ) Remark 3.1. Recall that for the last three points (q m 2,q m 1,q m ) Q the respective derivative estimation is performed by passing reverse cumulative chord cubic interpolants ˆγ i introduced in Section 1. For completeness, we also need to use reverse Lagrange cubics (see (9)) ψ i : [t i,t i 3 ] [ˆt i, ˆt i 3 ] satisfying ψ i (t i k) ˆt i k, for k 0,1,2,3, respectively. These cubics reparameterize ˆγ i exactly as ψ i reparameterize ˆγ i. The proof of Theorem 1.2 applies also to this non-generic case. 4. Numerical Experiments We experiment here with sampling points obtained from smooth regular curves not necessarily parameterized by arc-length. The arc-length reparameterization is only needed as a technical tool substantially simplifying the proof of the Theorem 1.2. The testing is performed in Mathematica. We first verify the claim of Theorem 1.2 for some planar curves. Example 4.1. Consider a spiral γ sp : [0,1] R 2 as γ sp (t) (t + 0.2)(cos(π(1 t)),sin(π(1 t))) with length d(γ sp ) Cumulative chord C 1 piecewise-cubic based on the 7-tuple Q m with t i i m + ( 1)i+1 3m, t 0 0, t m 1, (37) is shown in Figure 2 for which a good performance in trajectory and length estimation is confirmed on such sporadic data. The respective absolute errors in length estimates for m 48, 90, 150, and 198 are , , and The plot for cumulative chord C 1 piecewisecubic interpolation of log d(γ sph ) d(γ sp ) against log m in Figure 3, for m 18,24,30,...,198, appears almost linear, with least squares estimate of slope Note that since m 1 i0 ( ) T, by (1) we obtain mδ T. Thus to examine or test the orders of convergence in O(δ α ) rates (where α > 0), it suffices to show or verify the corresponding rates in O(1/m α ) asymptotics. The next example tests the scheme in question for a non-convex cubic curve γ c R 2 having one inflection point at (0,0).

14 14 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics Figure 3. Plot of -log d(γ sph ) d(γ sp ) against log m for a spiral γ sp sampled irregularly, as in Example 4.1. Example 4.2. Consider the following regular cubic curve γ c : [0,1] R 2 defined as γ c (t) (2t 1,(2t 1) 3 ). Cumulative chord C 1 piecewise-cubic based on sampling (37) yields the respective absolute errors in length estimates for m 48,90,150, and 198 are , , and As previously, linear regression applied to the set of points log d(γ ch ) d(γ c ) against log m, for m 180,186,...,300, yields the estimate for convergence order of length approximation equal to Again, cumulative chord C 1 piecewise-cubic based on the 7-tuple Q m gives an excellent estimate on sporadic data for length d(γ ch ) d(γ c ) , where d(γ c ) In the next step, we verify that Theorem 1.2 holds for not necessarily more-or-less uniform samplings (see (3)) which are required to prove some of the existing convergence results mentioned in the Introduction. Example 4.3. Apply cumulative chord C 1 piecewise-cubic for the following regular quartic curve γ q : [0,1] R 2 defined as γ q (t) (t,(t + 1) 4 /8) with d(γ q ) and sampled according to: t i i m, for i even ; t i i m + 1 m 1 m 2, for i odd ; t m 1. (38) Clearly condition (3) does not hold for sampling (38). The respective absolute errors in length estimates for m 6,48,90,150, and 198 are , , , and As previously, linear regression applied to the set of points log d(γ qh ) d(γ q ) against log m, for m 480,486,...,600, yields the estimate for convergence order of length approximation equal to Finally, we experiment with a regular elliptical helix space curve. Example 4.4. Figure 4 shows a cumulative chord C 1 piecewise-cubic interpolant γ ehh of 7 points on the elliptical helix γ eh : [0,2π] R 3, given by γ eh (t) (1.5cos t,sin t,t/4) and sampled with t i equal to either 2πi π(2i 1) m or m according as i is even or odd. Although sampling is uneven, sparse, and not available for interpolation, γ ehh seems very close to γ eh : d(γ eh ) and d(γ ehh ) As previously, linear regression applied to the set of points log d(γ ehh ) d(γ eh ) against log m, for

15 R. Kozera, L. Noakes /C 1 Interpolation with Cumulative Chord Cubics Figure 4. 7 points on the elliptical helix γ eh (dashed) sampled as in Example 4.4, interpolated by a cumulative chord C 1 piecewise-cubic γ ehh, with d(γ ehh ) d(γ eh ) m 90, 96,..., 300, yields the estimate for convergence order of length approximation equal to The respective absolute errors in length estimates for m 6,48,90,150, and 198 are , , , and Similarly, for sampling (38) linear regression applied to the set of points log d(γ ehh ) d(γ eh ) against log m, for m 240,246,...,300, yields the estimate for convergence order of length approximation equal to So the orders of convergence for length estimates given in Theorem 1.2 for cumulative chord C 1 piecewise-cubics are sharp. Curves need not be planar nor convex and sampling apart from natural condition (1) need not be more-or-less uniform for estimating orders of approximation. For length estimation weak condition on sampling mδ O(1) needs however to be imposed. The scheme in question performs well also on sporadic data. References [1] de Boor, C.: A Practical Guide to Spline, Revised edition, Springer-Verlag, New York Heidelberg Berlin, [2] de Boor, C., Höllig, K., Sabin, M.: High accuracy geometric Hermite interpolation, Computer Aided Geom. Design, 4, 1987, [3] Epstein, M. P.: On the influence of parameterization in parametric interpolation, SIAM J. Numer. Anal., 13, 1976, [4] Farin, G.: Curves and Surfaces for Computer Aided Geometric Design. A Practical Guide, Academic Press, San Diego, 1993.

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