Abstract
This paper discusses the issue of interpolating data points in arbitrary Euclidean space with the aid of Lagrange cubics 
and exponential parameterization. The latter is commonly used to either fit the so-called reduced data 
for which the associated exact interpolation knots remain unknown or to model the trajectory of the curve 
passing through 
. The exponential parameterization governed by a single parameter 
replaces such discrete set of unavailable knots 
(
- an internal clock) with some new values 
(
- an external clock). In order to compare 
with 
the selection of some 
should be predetermined. For some applications and theoretical considerations the function 
needs to form an injective mapping (e.g. in length estimation of 
with any 
fitting 
). We formulate and prove two sufficient conditions yielding 
as injective for given 
and analyze their asymptotic character which forms an important question for 
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 
by 
(once 
). The reparameterization has potential ramifications in computer graphics and robot navigation for trajectory planning e.g. to construct a new curve 
controlled by the appropriate choice of interpolation knots and of mapping 
(and/or possibly 
).
Introduction
Assume that 
represents a smooth regular curve (i.e. 
) of class 
(usually with 
) defined over a compact interval 
(with 
). Suppose that 
interpolation points 
(forming the so-called reduced data
) belong to an arbitrary Euclidean space 
. Here 
is not given (here 
). We introduce now (see e.g. [1, 7, 12] or [19]) some preliminary notions (applicable for 
).
Definition 1.1
The interpolation knots 
are admissible if:
 |
1 |
Definition 1.2
The interpolation knots 
are more-or-less uniform if there exist constants 
such that:
 |
2 |
for all 
and any 
. Alternatively, more-or-less uniformity amounts to the existence of some constant 
such that 
for all 
and arbitrary 
. Lastly, the subfamily 
of more-or-less uniform samplings represents a set of 
samplings if each of its representatives satisfies 
, for some 
fixed.
Having selected the fitting scheme 
of 
the unknown knots 
for the interpolant 
must somehow be replaced by estimates 
subject to 
. We use here the so-called exponential parameterization (see e.g. [17]) which depends on a single parameter 
according to:
 |
3 |
for 
. It is also assumed here that 
so that the extra condition 
is preserved as stipulated generically while fitting reduced data 
. The case of 
in (3) gives uniform knots
. Evidently the latter does not reflect the geometry of 
. On the other hand, 
yields the so-called cumulative chord parameterization which coincides with Euclidean distances between consecutive points 
and 
and as such it refers to the spread of 
. More information on the above topic and related issues can be found e.g. in [3, 5, 16, 17] or [18].
The selection of the specific interpolant 
(with 
) together with some knots’ estimates 
raises an important question concerning the convergence rate (if any) in approximating 
with 
(or its length) once 
. Recall first (see [1, 12] or [19]):
Definition 1.3
Consider a family 
of functions 
. We say that 
is of order 
(denoted as 
), if there is a constant 
such that, for some 
the inequality 
holds for all 
, uniformly over I.
For a given 
fitting dense data 
based on 
(and some a priori selected mapping 
) the natural question arises about the distance measurement 
tending to 0 (uniformly over I), while 
. Of course, by (1) proving 
not only guarantees the latter but also establishes lower bound on convergence speed (if 
). The coefficient 
appearing in Definition 1.3 is called the convergence rate in approximating 
by 
uniformly over [0, T]. If additionally such 
cannot be improved (once 
and 
are given) then 
is sharp. The latter analogously extends to the length estimation (with 
), for which the scalar expression 
is to be considered.
For certain applications such as the analysis of the convergence rate in 
(see e.g. [2, 5] or [15]) the mapping 
should be a reparameterization of I into 
(i.e. 
). In other situations such as robot’s and drone path planning the extra trajectory looping of 
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 of 
is to require 
to be an injective function (see e.g. [13]).
From now on it is assumed that 
which represents a piecewise-Lagrange cubic 
(see e.g. [1]). More precisely, the interpolant 
is defined as a track-sum of Lagrange cubics 
with each 
satisfying 
, for 
. As already pointed out the unavailable knots 
are estimated with 
governed by exponential parameterization (3). For simplicity we suppose that 
, where 
. In a similar fashion, one selects here 
defined as a track-sum of Lagrange cubics 
mapping 
and fulfilling 
, for 
. Evidently if 
(as 
) then 
(here 
denotes the range of 
). On the other hand if 
is not injective we may also have 
. In order to construct the composition 
as a well-defined function, each domain of 
is here understood as naturally extendable from 
to 
. Such adjusted Lagrange piecewise-cubics denoted as 
satisfy 
. The following result holds (see e.g. [7, 9] or [19]):
Theorem 1.4
Assume 
be a regular curve in 
sampled admissibly (see (1)). For 
and 
in (3) each mapping 
is a 
reparameterization of 
into 
and we have (uniformly over [0, T]):
 |
4 |
In the remaining cases of 
from (3) let 
be sampled more-or-less uniformly (see (2)). Then for each mapping 
combined with 
the following holds (uniformly over [0, T]):
 |
5 |
Both (4) and (5) are sharp within the class of 
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 
) and some admissible (or more-or-less uniform) sampling 
for which 
in (4) (or 
for 
in (5)) are sharp according to Definition 1.3 - see also [9] or [12]. Note that 
as a track-sum of 
defines a piecewise 
mapping of I into 
at least continuous at 
. If 
is a reparameterization (e.g. always holding asymptotically for 
) then 
. In particular for 
we also have 
- see [19]. In contrast, the injectivity of 
and length estimation for 
has not been so far examined.
In this paper we introduce two sufficient conditions enforcing each 
to be injective, for 
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 
are independent from 
and apply to any fixed 
and to any preselected 
-more-or-less-uniform samplings (i.e. to any 
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 
characterizing 
as specified in (8) - see also Remark 3.1. The examples illustrate Theorem 2.1 and the relevance of this work (see Example 1). The conjecture concerning the sharp convergence rate 
in length estimation 
(combined with (3) for all 
yielding 
) is tested numerically (see Example 2 and Remark 3.2).
Sufficient Conditions for Injectivity of 
In this section we establish and discuss the asymptotic character (i.e. applicable for m sufficiently large) of two sufficient conditions enforcing 
to be a genuine reparameterization of 
into 
based on multidimensional reduced data 
.
Evidently the positivity of the quadratic
over 
is e.g. guaranteed (for both sparse and dense data 
) provided if e.g. either (6) or (7) hold:
 |
6 |
 |
7 |
Noticeably, any admissible sampling (1) can be characterized as follows:
 |
8 |
where 
. The main theoretical contribution of this paper reads as:
Theorem 2.1
Let 
be sampled 
-more-or-less uniformly (see Definition (1.2)) with knots 
represented by (8). For data 
combined with exponential parameterization (3) (with any fixed 
) the condition (6) yielding each 
as a reparameterization holds asymptotically, if the following three inequalities are satisfied for sufficiently large m:
 |
9 |
 |
10 |
 |
11 |
with fixed 
, 
and 
but arbitrary small. Similarly, the condition (7) enforcing 
holds asymptotically if the following two inequalities are met for sufficiently large m:
 |
12 |
 |
13 |
where constants 
and 
are fixed and small.
Proof
Newton interpolation formula (see [1]) based on divided differences of 
yields over 
:
 |
which for each 
renders 
 |
14 |
We recall now the proof of (18) (see [9] or [12]) since it is vital for further arguments. As 
is regular it can be assumed to be parameterized by arc-length rendering 
, for 
(see [2]). The latter due to 
results in 
over 
. The orthogonality of 
and 
nullifies certain terms in the expression (for 
with 
and any 
):
 |
15 |
once Taylor expansion for 
is used:
 |
16 |
Indeed, upon substituting (16) into (15) and exploiting 
one obtains:
 |
17 |
For any admissible samplings the constants in the term 
depend on the third derivative of 
which is bounded over [0, T] as 
. Again Taylor Th. applied to the function 
at 
yields for all 
(with some fixed 
) the existence of some 
satisfying 
such that 
. For 
we exclude the singularity of 
at 
(with 
) which forces 
to be bounded over 
. Thus for 
we have 
- the constant standing along 
depends now on 
(which is fixed). Take now 
determined in (17) which is asymptotically small (for m large) due to the admissibility condition (1) and thus separated from 
. Hence the second-divided differences of 
satisfy (with 
):
 |
18 |
Thus, by (8) and (18) one obtains for each 
and 
the following formula for the second divided differences of 
(needed also in (15)):
 |
19 |
with 
, 
and 
. Furthermore still by (18) combined with 
(for 
) and telescoped 
the third-divided difference of 
is equal to 
 |
20 |
A similar argument leads to:
 |
21 |
Hence by (20) and (21) (for 
) the third divided differences of 
(needed in (15)) read as:
 |
22 |
Coupling again (20) and (21) with telescoped 
and 
reduces the fourth divided difference of 
into:
 |
which ultimately yields 
 |
23 |
The proof of (23) relies on 
. The second step resorts to more-or-less uniformity (3) of admitted samplings 
for any 
(as 
). However, to keep all constants in 
from (23) as independent from each representative of (3) from now on we admit only 
-more-or-less uniform samplings for some fixed 
(see Definition 1.3). The latter permits to exploit the inequality 
to justify (23) with constants in 
depending on 
and 
(but not on samplings 
).
Recalling now that 
over 
, by (15) we have:
 |
24 |
In the next steps both conditions (6) and (7) enforcing 
(for arbitrary m) are transformed into their asymptotic analogues applicable for sufficiently large m (i.e. for 
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 
(see Theorem 2.1). To save space only the first inequality from (6) i.e. 
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
from (6) amounts to 
which in turn by (23) holds subject to:
 |
25 |
for 
. Asymptotically, for fixed 
the slowest term determining the sign of (25) accompanies 
and reads as (for all 
-more-or-less uniform samplings):
 |
provided 
is not of any order 
with 
. A possible sufficient condition guaranteeing the latter is to require:
 |
26 |
to hold for any fixed 
. 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 
).
(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. 
is met subject to (10) satisfied for any fixed, but arbitrary small 
and sufficiently large m.
(iii) The third inequality 
determining (6) maps analogously into its asymptotic counterpart (11) assumed to be fulfilled for an arbitrary but fixed 
and m sufficiently large.
(iv) Clearly the proof of (9) yields a symmetric sufficient condition for 
(representing the first inequality in (7)) to hold asymptotically. The latter coincides with (12) stipulated to be satisfied by any fixed 
, subject to m getting large.
(v) The reformulation of 
from (7) into (13) (assumed to hold for any fixed 
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 
and fixed 
(not on curve 
). Consequently, they can all be also visualized geometrically in 3D for each 
and 
as well as for any regular curve 
. 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 
(i.e. where 
) combined with 
or 
with (1) both yield 
(see [9] and [19])). Noticeably, conditions (10), (11) and (13) are met for either 
or 
uniform and 
. 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 
or 
is to apply the fourth-order Taylor expansion for 
- see (16). The analysis (left out here) yields a modified condition for 
(and thus for 
), this time hinging not only on triples 
, 
but also on 
curvature 
along 
(see [9] and [19]) - here 
as 
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 
belonging to the prescribed family of admissible curves.
Experimentation and Testing
In this section first Theorem 2.1 is illustrated with some examples based on algebraic tests supported by 3D plots generated in Mathematica (see Subsect. 3.1). Next the convergence rate 
for 
is numerically investigated. A special attention is given to 
yielding 
as a piecewise 
reparameterization of [0, T] into 
(see Subsect. 3.2).
In doing so, in a preliminary step, for a given fixed 
two families of 
-more-or-less uniform samplings (27) and (29) are introduced. Next the fulfillment of the asymptotic sufficient conditions enforcing the injectivity of 
(see Theorem 2.1) is examined for various 
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 
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 
in length estimation 
, for any 
yielding each 
as an injective function. The conjecture concerning 
is proposed in Remark 3.2 based on our numerical results.
The tests reported here are performed for 2D and 3D curves 
, 
introduced in Example 2 (i.e. for 
). However all established results with the accompanied experimentation are equally applicable to arbitrary multidimensional reduced data 
with 
.
Testing Injectivity of 
Example 1
Consider first the following family 
of more-or-less uniform sampling (for geometrical distribution of 
with sampling (27) see also Fig. 3(a) and Fig. 4(a)):
 |
27 |
for which 
, 
and 
(see Definition 1.2). Here 
, where 
, so that 
and 
. Upon resorting to (8) the following 3D compact asymptotic representation 
of 
reads as (for 
):
 |
28 |
The last two points in (28) are generated for 
as 
. We set 
and hence as 
the sampling (27) is also 
.
Fig. 3.
A spiral curve
from (31) sampled according to: a) (27) or b) (29), for 
.
Fig. 4.
A Steinmetz curve
from (32) sampled according to: a) (27) or b) (29), for 
(with dotted gray point 
).
We also admit another 
defined according to (for geometrical spread of 
with sampling (29) see also Fig. 3(b) and Fig. 4(b)):
 |
29 |
with 
, 
and 
(see Definition 1.2). Again we set 
and 
with 
, for 
. By (8) the 3D asymptotic form 
of (29) reads as:
 |
30 |
The last two points in (30) come for 
as 
and the first point is due to 
.
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 
for both samplings (27) and (29). The fixed parameter 
is set either to 
or to 
with 
, 
, 
and 
- see Table 1 and Table 2. The corresponding sets of triples 
satisfying either (6)
or (7)
represent the respective solids 
plotted in 3D by Mathematica as shown in Fig. 1 and Fig. 2.
Table 1.
Table 2.
Fig. 1.
Condition (6) enforced asymptotically by (6)
visualized in 3D plots as two solids 
, for 
or 
, respectively. Here 
with dotted points representing samplings: a) (27) mapped into (28) or b) (29) mapped into (30) both for 
and samplings: c) (27) mapped into (28) or d) (29)mapped into (30) both for 
.
Fig. 2.
Condition (7) enforced asymptotically by (7)
visualized in 3D plots as two solids 
, for 
or 
, respectively. Here 
with dotted points representing samplings: a) (27) mapped into (28) or b) (29) mapped into (30) both for 
and samplings: c) (27) mapped into (28) or d) (29) mapped into (30) both for 
.
Noticeably different points from 
, for 
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 
all conditions from (6)
are not satisfied by both 
(for 
) as we have 
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 
. Consequently the injectivity of 
for either 
or 
is not guaranteed. Geometrically both 
(for 
) are not contained in the respective injectivity zones 
(for either (6)
or (7)
). In contrast for 
, a simple inspection of Table 1 and Table 2 reveals that all points from 
(for 
) can be split into two subsets each contained in the injectivity zones 
determined by either (6)
or by (7)
, respectively. Algebraically the latter yields at least one 
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 
-more-or-less uniform samplings 
the subfamily 
with extra constraints 
, 
and 
(here 
) is chosen one can also examine (for a fixed 
) whether 
, where 
. By Theorem 2.1, should the latter holds the entire subfamily of 
yields asymptotically 
as injective functions. The incomplete information on input samplings 
carried by 
can in certain situations accompany 
.
Numerical Testing for Length Estimation
We pass now to the experiments designed to investigate convergence rate 
in length approximation by examining 
- see Definition 1.3. The coefficient 
is estimated numerically by 
which in turn is computed using a linear regression on the pairs 
, where 
, for a given m. The slope a of the regression line 
found in Mathematica with the aid of Normal[LinearModelFit[data]] yields 
forming a numerical estimate of 
.
Example 2
Consider a 2D
spiral
(a regular curve with 
and 
):
 |
31 |
and the so-called 3D
Steinmetz curve
(a regular closed curve with 
- see a dotted gray point in Fig. 4):
 |
32 |
Both curves 
, 
(from (31) and (32)) sampled according to either (27) or (29) are plotted in Fig. 3 and Fig. 4, respectively. The numerical results assessing the estimate 
of 
(for 
) are presented in Table 3. Recall that here, a linear regression to compute 
is applied to the collections of points 
, with 
and for various 
. The results from Table 3 suggest that for all 
rendering 
(e.g. the latter is guaranteed if Theorem 2.1 holds) one may expect 
with the quadratic convergence rate
. 
Table 3.
The numerical estimates of 
for a spiral
from (31) and a Steinmetz curve
from (32) computed for 
and 
. Here 
stands for true and 
for false, respectively.
| Curve | Sampling |
|
|
|
(6) or (7)
|
|---|---|---|---|---|---|
| (31) | (27) | 0.3 | 0.0735200 | 0.044 | F |
| 0.7 | 0.0000083 | 1.945 | T | ||
| 0.9 | 0.0000016 | 1.885 | T | ||
| (29) | 0.3 | 2.4619100 | -0.012 | F | |
| 0.7 | 0.0050445 | 0.007 | F | ||
| 0.9 | 0.0000319 | 1.989 | T | ||
| (32) | (27) | 0.3 | 0.2036000 | 0.033 | F |
| 0.7 | 0.0000897 | 2.015 | T | ||
| 0.9 | 0.0000181 | 2.092 | T | ||
| (29) | 0.3 | 6.7392400 | -0.009 | F | |
| 0.7 | 0.0132964 | -0.080 | F | ||
| 0.9 | 0.0003419 | 1.985 | T |
In fact the numerical results from Example 2 combined with (5) in conjunction with the argument used to prove 
for 
(see [7]) or [19]) lead to expect 
in 
, for all 
yielding 
as a piecewise 
reparametrization. The latter forms an open problem which can be stated as:
Remark 3.2
Assume 
be a regular curve in 
sampled more-or-less uniformly (see Definition 1.2). For the interpolant 
and any 
in (3) yielding each 
as a 
genuine reparameterization Example 2 suggests a sharp quadratic convergence rate in:
 |
33 |
In particular if Theorem 2.1 holds (and 
-more-or-less uniform samplings are used) the mapping 
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 
and of at least one samplings from 
such that in (33) the convergence rate 
has exactly order 2 (i.e. is not faster than quadratic). 
Conclusions
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 
(introduced in Sect. 1) is a genuine reparameterization. The latter applies to both sparse and dense reduced data 
. Here the unknown interpolation knots 
are replaced by 
which in turn is determined by exponential parameterization (3) controlled by a single parameter 
and 
. 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 
getting denser). These new transformed conditions (specified in Theorem 2.1) depend exclusively on 
and 
characterized by (8) within the admitted class of 
-more-or-less uniform samplings (see Definition 1.2) and apply to any regular curve 
(with 
). 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 
for 
are performed. Consequently, based on the latter the conjecture suggesting the quadratic convergence rate for 
is posed (see Example 2 and Remark 3.2), subject to the injectivity of 
. 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 
from (8) into a pair 
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 schemes 
or 
based on 
combined with either (3) or with other 
compensating the unknown knots 
(see e.g. [3, 10, 13] or [16]). Searching for alternative sufficient conditions enforcing 
to be injective forms an interesting topic. Lastly the theoretical justification of (33) poses another open problem.
Contributor Information
Valeria V. Krzhizhanovskaya, Email: V.Krzhizhanovskaya@uva.nl
Gábor Závodszky, Email: G.Zavodszky@uva.nl.
Michael H. Lees, Email: m.h.lees@uva.nl
Jack J. Dongarra, Email: dongarra@icl.utk.edu
Peter M. A. Sloot, Email: p.m.a.sloot@uva.nl
Sérgio Brissos, Email: sergio.brissos@intellegibilis.com.
João Teixeira, Email: joao.teixeira@intellegibilis.com.
Ryszard Kozera, Email: ryszard.kozera@sggw.edu.pl, Email: ryszard.kozera@gmail.com.
Lyle Noakes, Email: lyle.noakes@uwa.edu.au.
Magdalena Wilkołazka, Email: magda.wilkolazka@gmail.com, Email: magda8310@kul.lublin.pl.
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