key: cord-0045735-yqackjjy authors: Kozera, Ryszard; Noakes, Lyle; Wilkołazka, Magdalena title: Parameterizations and Lagrange Cubics for Fitting Multidimensional Data date: 2020-06-15 journal: Computational Science - ICCS 2020 DOI: 10.1007/978-3-030-50417-5_10 sha: 5876c39328c11d16e3bdd6e531927addc07ae6d9 doc_id: 45735 cord_uid: yqackjjy This paper discusses the issue of interpolating data points in arbitrary Euclidean space with the aid of Lagrange cubics [Formula: see text] and exponential parameterization. The latter is commonly used to either fit the so-called reduced data [Formula: see text] for which the associated exact interpolation knots remain unknown or to model the trajectory of the curve [Formula: see text] passing through [Formula: see text]. The exponential parameterization governed by a single parameter [Formula: see text] replaces such discrete set of unavailable knots [Formula: see text] ([Formula: see text] - an internal clock) with some new values [Formula: see text] ([Formula: see text] - an external clock). In order to compare [Formula: see text] with [Formula: see text] the selection of some [Formula: see text] should be predetermined. For some applications and theoretical considerations the function [Formula: see text] needs to form an injective mapping (e.g. in length estimation of [Formula: see text] with any [Formula: see text] fitting [Formula: see text]). We formulate and prove two sufficient conditions yielding [Formula: see text] as injective for given [Formula: see text] and analyze their asymptotic character which forms an important question for [Formula: see text] getting sufficiently dense. The algebraic conditions established herein are also geometrically visualized in 3D plots with the aid of Mathematica. This work is supplemented with illustrative examples including numerical testing of the underpinning convergence rate in length estimation [Formula: see text] by [Formula: see text] (once [Formula: see text]). The reparameterization has potential ramifications in computer graphics and robot navigation for trajectory planning e.g. to construct a new curve [Formula: see text] controlled by the appropriate choice of interpolation knots and of mapping [Formula: see text] (and/or possibly [Formula: see text]). Abstract. This paper discusses the issue of interpolating data points in arbitrary Euclidean space with the aid of Lagrange cubicsγ L and exponential parameterization. The latter is commonly used to either fit the so-called reduced data Qm = {qi} m i=0 for which the associated exact interpolation knots remain unknown or to model the trajectory of the curve γ passing through Qm. The exponential parameterization governed by a single parameter λ ∈ [0, 1] replaces such discrete set of unavailable knots {ti} m i=0 (ti ∈ I -an internal clock) with some new values {ti} m i=0 (ti ∈Î -an external clock). In order to compare γ withγ L the selection of some φ : I →Î should be predetermined. For some applications and theoretical considerations the function φ : I →Î needs to form an injective mapping (e.g. in length estimation of γ with anyγ fitting Qm). We formulate and prove two sufficient conditions yielding φ as injective for given Qm and analyze their asymptotic character which forms an important question for Qm getting sufficiently dense. The algebraic conditions established herein are also geometrically visualized in 3D plots with the aid of Mathematica. This work is supplemented with illustrative examples including numerical testing of the underpinning convergence rate in length estimation d(γ) by d(γ) (once m → ∞). The reparameterization has potential ramifications in computer graphics and robot navigation for trajectory planning e.g. to construct a new curveγ =γ • φ controlled by the appropriate choice of interpolation knots and of mapping φ (and/or possibly Qm). (forming the so-called reduced data Q m ) belong to an arbitrary Euclidean space E n . Here T = {t i } m i=0 is not given (here t i < t i+1 ). We introduce now (see e.g. [1, 7, 12] or [19] ) some preliminary notions (applicable for m → ∞). (1) Definition 1.2. The interpolation knots T are more-or-less uniform if there exist constants 0 < K l ≤ K u such that: for all i = 1, 2, . . . , m and any m ∈ N. Alternatively, more-or-less uniformity amounts to the existence of some constant 0 < β ≤ 1 such that βδ m ≤ t i −t i−1 ≤ δ m for all i = 1, 2, . . . , m and arbitrary m ∈ N. Lastly, the subfamily T β0 of moreor-less uniform samplings represents a set of β 0 -more-or-less uniform samplings if each of its representatives satisfies β 0 ≤ β ≤ 1, for some 0 < β 0 ≤ 1 fixed. Having selected the fitting schemeγ of Q m the unknown knots T for the interpolantγ must somehow be replaced by estimatesT = {t i } m i=0 subject tô γ(t i ) = q i . We use here the so-called exponential parameterization (see e.g. [17] ) which depends on a single parameter λ ∈ [0, 1] according to: for i = 1, 2, . . . , m. It is also assumed here that q i = q i+1 so that the extra conditiont i 0 such that, for someδ > 0 the inequality F δm (t) < Kδ α m holds for all δ m ∈ (0,δ), uniformly over I. For a givenγ fitting dense data Q m based onT ≈ T (and some a priori selected mapping φ : I →Î) the natural question arises about the distance measurement F δm = γ−γ • φ|| tending to 0 (uniformly over I), while m → ∞. Of course, by (1) proving F δm = γ −γ • φ = O(δ α m ) not only guarantees the latter but also establishes lower bound on convergence speed (if α > 0). The coefficient α > 0 appearing in Definition 1.3 is called the convergence rate in approximating γ byγ • φ uniformly over [0, T ]. If additionally such α cannot be improved (once γ and T are given) then α is sharp. The latter analogously extends to the length estimation (with n = 1), for which the scalar expression is to be considered. For certain applications such as the analysis of the convergence rate in [2, 5] or [15] ) the mapping φ(t) =t should be a reparameterization of I intoÎ (i.e.φ > 0). In other situations such as robot's and drone path planning the extra trajectory looping ofγ is sometimes needed (e.g. for traction line posts' inspection while making circles by drone). Of course, in many other applications robot navigation requires trajectory planning with no loops whatsoever. In that context (as well as for length estimation) one of the conditions to exclude the local looping ofγ • φ is to require φ to be an injective function (see e.g. [13] ). From now on it is assumed thatγ =γ L which represents a piecewise-Lagrange cubicγ L :Î = [0,T ] → E n (see e.g. [1] ). More precisely, the inter- , for j = 0, 1, 2, 3. As already pointed out the unavailable knots T are estimated withT governed by exponential parameterization (3) . For simplicity we suppose that m = 3k, where k ∈ {1, 2, 3, . . .}. In a similar fashion, one selects here φ = ψ L defined as a track- On the other hand if ψ L i is not injective we may also have ψ L i : I i →Î i ⊂ Rg(ψ L i ). In order to construct the compositionγ L i • ψ L i as a well-defined function, each domain ofγ L i is here understood as naturally extendable fromÎ i to R. Such adjusted Lagrange piecewise-cubics denoted asγ L i satisfyγ L i |Î i =γ L i . The following result holds (see e.g. [7, 9] or [19] ): ) be a regular curve in E n sampled admissibly (see (1) ). Forγ L and λ = 1 in (3) each mapping ψ L i is a C ∞ reparameterization of I i intoÎ i and we have (uniformly over [0, T ]): In the remaining cases of λ ∈ [0, 1) from (3) let γ be sampled more-or-less uniformly (see (2)). Then for each mapping ψ L i combined withγ L i the following holds (uniformly over [0, T ]): Both (4) and (5) are sharp within the class of γ ∈ C 4 ([0, T ]) and within a given family of admitted samplings, assumed here as either (1) or (2), respectively. By the latter we understand the existence of at least one γ 0 ∈ C 4 ([0, T ])) and some admissible (or more-or-less uniform) sampling T 0 for which α(1) = 4 in (4) (or α(λ) = 1 for λ ∈ [0, 1) in (5)) are sharp according to Definition 1.3 -see also [9] or [12] . Note that ψ L as a track-sum of {ψ L i=3k } defines a piecewise C ∞ mapping of I into R at least continuous at T . If ψ L is a reparameterization (e.g. always holding asymptotically for λ = 1) then ψ L : I →Î. In particular for λ = 1 we also have d(γ) − d(γ L ) = O(δ 4 m ) -see [19] . In contrast, the injectivity of ψ L i and length estimation for λ ∈ [0, 1) has not been so far examined. In this paper we introduce two sufficient conditions enforcing each ψ L i : I i → I i to be injective, for λ ∈ [0, 1) governing the exponential parameterization (3). These two conditions are represented by the inequalities (6) and (7) . In the next step, Theorem 2.1 is established (the main result of this paper) to formulate several sufficient conditions enforcing (6) and (7) to hold asymptotically. Noticeably all derived conditions stipulating asymptotically the injectivity of ψ L are independent from γ and apply to any fixed λ ∈ [0, 1) and to any preselected β 0 -more-or-less-uniform samplings (i.e. to any 0 < β 0 < 1 fixed a priori). Additionally, all re-transformed algebraic constraints established here are visualized with the aid of 3D plots in Mathematica (see [22] ). The conditions can also be exploited once the incomplete information about samplings is available such as a priori knowledge of the respective upper and lower bounds for each triples (M im , N im , P im ) characterizing T as specified in (8) In this section we establish and discuss the asymptotic character (i.e. applicable for m sufficiently large) of two sufficient conditions enforcing ψ L i to be a genuine reparameterization of I i intoÎ i based on multidimensional reduced data Q m . Evidently the positivity of the quadraticψ L i (t) = a i t 2 + b i t + c i over I i is e.g. guaranteed (for both sparse and dense data Q m ) provided if e.g. either (6) or (7) hold: Noticeably, any admissible sampling (1) can be characterized as follows: where 0 < M im , N im , P im ≤ 1. The main theoretical contribution of this paper reads as: be sampled β 0 -more-or-less uniformly (see Definition (1. 2)) with knots T represented by (8) . For data Q m combined with exponential parameterization (3) (with any fixed λ ∈ [0, 1)) the condition (6) yielding each ψ L i : I →Î i as a reparameterization holds asymptotically, if the following three inequalities are satisfied for sufficiently large m: with fixed ρ < 0, ρ 1 > 0 and ρ 2 > 0 but arbitrary small. Similarly, the condition (7) enforcingψ L i > 0 holds asymptotically if the following two inequalities are met for sufficiently large m: where constants ρ 3 > 0 and ρ 4 > 0 are fixed and small. Proof. Newton interpolation formula (see [1] ) based on divided differences of ψ L i yields over I i : We recall now the proof of (18) (see [9] or [12] ) since it is vital for further arguments. As γ is regular it can be assumed to be parameterized by arc-length rendering γ(t) = 1, for t ∈ [0, T ] (see [2] ). The latter due to The orthogonality ofγ andγ nullifies certain terms in the expression (for j = i+k with k = 0, 1, 2 and any λ ∈ [0, 1]): once Taylor expansion for γ ∈ C 3 is used: Indeed, upon substituting (16) into (15) and exploiting γ(t)|γ(t) = 0 one obtains:t For any admissible samplings the constants in the term (17) which is asymptotically small (for m large) due to the admissibility condition (1) and thus separated from −1. Hence the second-divided differences of ψ L i satisfy (with k = 0, 1, 2): (18) Thus, by (8) and (18) one obtains for each λ ∈ [0, 1] and k = 0, 1, 2 the following formula for the second divided differences of ψ L i (needed also in (15)): A similar argument leads to: Hence by (20) and (21) (for l = 0, 1) the third divided differences of ψ L i (needed in (15)) read as: Coupling again (20) and (21) The proof of (23) relies on . The second step resorts to more-or-less uniformity (3) of admitted samplings T for any λ ∈ [0, 1) (as λ − 1 < 0). However, to keep all constants in O(δ λ−1 m ) from (23) as independent from each representative of (3) from now on we admit only β 0 -more-or-less uniform samplings for some fixed 0 < β 0 ≤ 1 (see Definition 1.3). The latter permits to exploit the inequality to justify (23) with constants in O(δ λ−1 m ) depending on γ and λ (but not on samplings T ). Recalling now thatψ L i (t) = a i t 2 + b i t + c i over I i , by (15) we have: In the next steps both conditions (6) and (7) enforcingψ L i > 0 (for arbitrary m) are transformed into their asymptotic analogues applicable for sufficiently large m (i.e. for Q m sufficiently dense). This will ultimately complete the proof of Theorem 2.1. In doing so, both conditions (6) and (7) are reformulated into asymptotic counterparts expressed in terms of (M im , N im , P im ) (see Theorem 2.1). To save space only the first inequality from (6) i.e. a i < 0 is fully addressed here (which automatically covers both (i) and (iv) -see (9) and (12)). The remaining more complicated cases (ii), (iii) and (v) (listed below) are supplemented with the final asymptotic formulas (10), (11) and (13) . The proof of the latter shall be given in the full journal version of this paper. (i) By (24) the first inequality a i < 0 from (6) amounts to ψ L i [t i , t i+1 , t i+2 , t i+3 ] < 0 which in turn by (23) holds subject to: Asymptotically, for fixed λ ∈ [0, 1) the slowest term determining the sign of (25) accompanies δ λ−3 m and reads as (for all β 0more-or-less uniform samplings): provided θ 1 is not of any order Θ(δ 2+ε m ) with ε ≥ 0. A possible sufficient condition guaranteeing the latter is to require: to hold for any fixed ρ < 0. Evidently (26) amounts to the first inequality (9) assumed to hold in Theorem 2.1 in order to enforce in turn asymptotically the first inequality in (6) (for any fixed λ ∈ [0, 1)). (ii) A similar but longer argument shows that (upon combining (8), (15), (19) , (22) and (23)) the asymptotic fulfillment of the second inequality from (6) i.e.ψ L i (t i ) > 0 is met subject to (10) satisfied for any fixed, but arbitrary small ρ 1 > 0 and sufficiently large m. (iii) The third inequalityψ L i (t i+3 ) > 0 determining (6) maps analogously into its asymptotic counterpart (11) assumed to be fulfilled for an arbitrary but fixed ρ 2 > 0 and m sufficiently large. (iv) Clearly the proof of (9) yields a symmetric sufficient condition for a i > 0 (representing the first inequality in (7)) to hold asymptotically. The latter coincides with (12) stipulated to be satisfied by any fixed ρ 3 > 0, subject to m getting large. (v) The reformulation of κ im =ψ L i ( −bi 2ai ) > 0 from (7) into (13) (assumed to hold for any fixed ρ 4 > 0 and sufficiently large m) involves a more intricate treatment (it is omitted here). The asymptotic conditions established in Theorem 2.1 in the form of specific inequalities depend (for each i) exclusively on triples (M im , N im , M im ) ∈ [β 0 , 1] 3 and fixed λ ∈ [0, 1) (not on curve γ). Consequently, they can all be also visualized geometrically in 3D for each i = 3k and λ ∈ [0, 1) as well as for any regular curve γ. Several examples with 3D plots are presented in Sect. 3 with the aid of Mathematica Package [22] . We note that all asymptotic conditions from Theorem 2.1 can be extended to their 2D analogues (with extra argument used establishing in fact a new theorem) which in turn can be visualized in more appealing 2D plots. Again it is omitted here as exceeding the scope of this paper. Recall that uniform sampling, for which M im = N im = P im = 1 (i.e. where β 0 = 1) combined with λ ∈ [0, 1) or λ = 1 with (1) both yieldψ L i = 1 + O(δ 2 m ) > 0 (see [9] and [19] )). Noticeably, conditions (10), (11) and (13) are met for either λ = 1 or T uniform and λ ∈ [0, 1). In contrast none of (9) or (12) (participating in either (6) or (7)) holds for the above two eventualities. A possible remedy to incorporate these two special cases in adjusted asymptotic representations of either a i > 0 or a i < 0 is to apply the fourth-order Taylor expansion for γ ∈ C 4 -see (16) . The analysis (left out here) yields a modified condition for a i > 0 (and thus for a i < 0), this time hinging not only on triples (M im , N im , P im ) ∈ [β 0 , 1] 3 , λ ∈ [0, 1) but also on γ curvature γ(t i ) 2 along T (see [9] and [19] ) -here γ(t) = 1 as γ is a regular curve and as such can be assumed to be parameterized by arc-length (see [2] ). The latter may not always be given in advance. Alternatively, one could rely on a priori imposed restrictions on curvatures of γ belonging to the prescribed family of admissible curves. In doing so, in a preliminary step, for a given fixed β 0 two families of β 0more-or-less uniform samplings (27) and (29) are introduced. Next the fulfillment of the asymptotic sufficient conditions enforcing the injectivity ofψ L > 0 (see Theorem 2.1) is examined for various λ ∈ [0, 1) and both samplings (27) and (29). In particular, the inequalities (9), (10), (11) , (denoted in this section by (6) * ) and (12), (13) (marked here with (7) * ) representing asymptotically in 3D both (6) and (7) are tested for different sets of triples (M im , N im , P im ) ∈ [β 0 , 1] 3 characterizing either (27) or (29). The algebraic calculations performed herein (assuming m is sufficiently large) are supplemented by geometrical visualizations with 3D plots in Mathematica. At this point, we re-emphasize that the asymptotic conditions from Theorem 2.1 can be extended further into respective 2D counterparts upon some laborious calculations. In return, the latter gives some advantage in visualizing more appealing 2D (versus 3D) plots. To save the space the relevant theory and testing concerning this extra 2D case are left out here. The second example reports on tests designed to numerically evaluate α(λ) in length estimation d(γ) − d(γ L ) = O(δ α(λ) ), for any λ ∈ [0, 1] yielding each ψ L i as an injective function. The conjecture concerning α(λ) is proposed in Remark 3.2 based on our numerical results. The tests reported here are performed for 2D and 3D curves γ sp , γ S introduced in Example 2 (i.e. for n = 2, 3). However all established results with the accompanied experimentation are equally applicable to arbitrary multidimen- Example 1. Consider first the following family T 1 of more-or-less uniform sampling (for geometrical distribution of {γ(t i )} 15 i=0 with sampling (27) see also Fig. 3(a) and Fig. 4(a) ): for which K l = 1 2 , K u = 3 2 and β 1 = 1 3 (see Definition 1.2). Here 0 ≤ i ≤ m = 3k, where k ∈ {1, 2, . . . }, so that t 0 = 0 and t m = T = 1. Upon resorting to (8) the following 3D compact asymptotic representation T 3D 1 of T 1 reads as (for m = 3k): The last two points in (28) are generated for m = 3k as t m = 1. We set β 0 = 0.16 and hence as β 0 ≤ β 1 the sampling (27) is also β 0 -more-or-less uniform. We also admit another β 0 -more-or-less uniform sampling T 2 defined according to (for geometrical spread of {γ(t i )} 15 i=0 with sampling (29) see also Fig. 3 (b) and Fig. 4(b) ): with K l = 1 3 , K u = 5 3 and β 2 = 1 5 ≥ β 0 (see Definition 1.2). Again we set t 0 = 0 and t m = T = 1 with 0 ≤ i ≤ m = 3k, for k ∈ {1, 2, . . . }. By (8) the 3D asymptotic form T 3D 2 of (29) reads as: The last two points in (30) come for m = 3k as t m = 1 and the first point is due to t 0 = 0. The inequalities (9), (10), (11) marked as (6) * (or (12) and (13) denoted by (7) * ) enforcing asymptotically (6) (or (7)) to hold are tested over [β 0 , 1] 3 for both samplings (27) and (29). The fixed parameter λ is set either to λ = 0.3 or to λ = 0.9 with ρ = −0.001, ρ 1 = ρ 2 = 0.05, ρ 3 = 0.001 and ρ 4 = 0.005 -see Table 1 and Table 2 . The corresponding sets of triples (M im , N im , P im ) ∈ [β 0 , 1] 3 (6) and (7) (implied asymptotically by (6) * and (7) * ) for sampling (27) (represented by (28)) and for λ = 0.3 and λ = 0.9 with ρ = −0.001, ρ1 = 0.05, ρ2 = 0.05, ρ3 = 0.001 and ρ4 = 0.005. Here T stands for true and F for false, respectively. Table 2 . Testing conditions (6) and (7) (implied asymptotically by (6) * and (7) * ) for sampling (29) (represented by (30)) and for λ = 0.3 and λ = 0.9 with ρ = −0.001, ρ1 = 0.05, ρ2 = 0.05, ρ3 = 0.001 and ρ4 = 0.005. Here T stands for true and F for false, respectively. satisfying either (6) * or (7) * represent the respective solids D λ β0 ⊂ [β 0 , 1] 3 plotted in 3D by Mathematica as shown in Fig. 1 and Fig. 2 . Noticeably different points from T 3D k , for k = 1, 2 may interchangeably satisfy one of the sufficient conditions enforcing either (6) or (7) to hold asymptotically. The latter is demonstrated in Table 1 and Table 2 . Indeed for λ = 0.3 all conditions from (6) * are not satisfied by both T 3D k (for k = 1, 2) as we have F in the respective columns of both Table 1 and Table 2 . Moreover, the conditions from (7) * are only fulfilled by some points (not all) from T 3D k . Consequently the injectivity of ψ L i for either T 3D 1 or T 3D 2 is not guaranteed. Geometrically both T 3D k (for k = 1, 2) are not contained in the respective injectivity zones D λ=0.3 β0 (for either (6) * or (7) * ). In contrast for λ = 0.9, a simple inspection of Table 1 and Table 2 reveals that all points from T 3D k (for k = 1, 2) can be split into two subsets each contained in the injectivity zones D λ=0.9 β0 determined by either (6) * or by (7) * , respectively. Algebraically the latter yields at least one T in the last two columns of all rows for both Table 1 and Table 2 . Remark 3.1. Note that if for a given family of β 0 -more-or-less uniform samplings T β0 the subfamily T ν β0 ⊂ T β0 with extra constraints ν 1 ≤ M im ≤ ν 2 , ν 3 ≤ N im ≤ ν 4 and ν 5 ≤ P im ≤ ν 6 (here ν = (ν 1 , ν 2 , ν 3 , ν 4 , ν 5 , ν 6 )) is chosen one can also examine (for a fixed λ ∈ [0, 1)) whether I 3D ν 6 ). By Theorem 2.1, should the latter holds the entire subfamily of T ν β0 yields asymptotically ψ L i as injective functions. The incomplete information on input samplings T carried by T ν β0 can in certain situations accompany Q m . We Both curves γ sp , γ S (from (31) and (32)) sampled according to either (27) or (29) are plotted in Fig. 3 and Fig. 4 , respectively. The numerical results assess- m )) are presented in Table 3 . Recall that here, a linear regression to computeα(λ) is applied to the collections of points {(log(m), − log(E m )} mmax=201 mmin=120 , with E m = |d(γ) − d(γ L )| and for various λ ∈ {0.3, 0.7, 0.9}. The results from Table 3 suggest that for all λ ∈ {0.3, 0.7, 0.9} renderingψ L > 0 (e.g. the latter is guaranteed if Theorem 2.1 holds) one may expect lim m→∞ E m = 0 with the quadratic convergence rate α(λ) = 2 ≈α(λ). In fact the numerical results from Example 2 combined with (5) in conjunction with the argument used to prove d(γ)−d(γ L ) = O(δ 4 m ) for λ = 1 (see [7] ) or [19] ) lead to expect α(λ) = 2 in d(γ) − d(γ L ) = O(δ α(λ) m ), for all λ ∈ [0, 1) yielding ψ L as a piecewise C ∞ reparametrization. The latter forms an open problem which can be stated as: Remark 3.2. Assume γ ∈ C 4 ([0, T ]) be a regular curve in E n sampled more-orless uniformly (see Definition 1.2). For the interpolantγ L and any λ ∈ [0, 1) in (3) yielding each ψ L i : I →Î as a C ∞ genuine reparameterization Example 2 suggests a sharp quadratic convergence rate in: In particular if Theorem 2.1 holds (and β 0 -more-or-less uniform samplings are used) the mapping ψ L is asymptotically a reparameterization which in turn hints to expect (33). Recall that by sharpness of (33) we understand the existence of at least one regular curve of class C 4 and of at least one samplings from T β0 such that in (33) the convergence rate α(λ) has exactly order 2 (i.e. is not faster than quadratic). Fitting reduced data (see e.g. [3] or [16] ) constitutes an important task in computer vision and graphics, engineering, microbiology, physics and other applications like medical image processing (e.g. for area, length and boundary estimation or trajectory planning) -see e.g. [4, 6, 8, 11, 14, 15, 17, 20] or [21] . Two sufficient conditions (6) and (7) are first formulated to ensure that the Lagrange piecewise-cubic ψ L : [0, T ] → [0,T ] (introduced in Sect. 1) is a genuine reparameterization. The latter applies to both sparse and dense reduced data Q m . Here the unknown interpolation knots T are replaced byT which in turn is determined by exponential parameterization (3) controlled by a single parameter λ ∈ [0, 1] and Q m . The main contribution established in Theorem 2.1 (see Sect. 2) reformulates (6) and (7) into respective asymptotic representatives valid for sufficiently large m (i.e. for Q m getting denser). These new transformed conditions (specified in Theorem 2.1) depend exclusively on λ ∈ [0, 1) and T characterized by (8) within the admitted class of β 0 -more-or-less uniform samplings (see Definition 1.2) and apply to any regular curve γ ∈ C 3 ([0, T ]) (with 0 < T < ∞). Lastly, in Sect. 3 two illustrative examples are presented. The attached 3D plots generated in Mathematica [22] illustrate the algebraic character of the asymptotic conditions justified in Theorem 2.1 (see Example 1). In addition, the numerical examination of the convergence rate in length estimation of interpolated γ for λ ∈ {0.3, 0.7, 0.9} are performed. Consequently, based on the latter the conjecture suggesting the quadratic convergence rate for d(γ) − d(γ L ) = O(δ 2 m ) is posed (see Example 2 and Remark 3.2), subject to the injectivity of ψ L . At this point we remark that all asymptotic formulas from Theorem 2.1 are extendable to the corresponding inequalities expressed in (x, y)-variables. This can be achieved by converting first (with the aid of special homogeneous mapping) each triple (M im , N im , P im ) from (8) into a pair (x(M im , N im , P im ), y(M im , N im , P im )) and then by reformulating all conditions from Theorem 2.1, accordingly in terms of (x, y). The satisfaction of such new conditions enforces (9), (10) and (11) or (12) and (13) asymptotically (and thus of (6) or (7)). It is a big advantage to reduce the illustrations from 3D to more appealing 2D analogues. We omit here the theoretical discussion and the geometrical insight of this 2D extension of Theorem 2.1. Similarly, recall that only items (i) and (iv) (see Sect. 2) are given here a full proof. In contrast, the final steps of proving (ii), (iii) and (v) are left out as treated later exhaustively in a journal version of this work (together with the mentioned above 2D extension of Theorem 2.1). Future work may include various interpolation schemesγ or φ based on Q m combined with either (3) or with otherT compensating the unknown knots T (see e.g. [3, 10, 13] or [16] ). Searching for alternative sufficient conditions enforcing ψ L i to be injective forms an interesting topic. Lastly the theoretical justification of (33) poses another open problem. 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