Synthesis and characterization of o-carboranylthioether derivatives

Article abstract of DOI:10.1515/znb-2011-0913

The reactions of the dithiolato-o-carborane salt (Et₃NH) ₂(S₂C₂B₁₀H₁₀) (2) with alkyl halides BrCH₂CH=CH₂, BrCH₂CH ₂CH₂Cl and C₆H₅CH₂Cl, afford the o-carboranyl-bisthioether derivatives 3a - c, which have been characterized by IR and NMR (¹H, ¹³C, ¹¹B) spectroscopy, mass spectrometry, and single-crystal X-ray diffraction (3c). The photoluminescent properties of the known compound 3c has been investigated. It exhibits a violet (chloroform solution) or blue (solid state) emission when excited with UV light.

Full text of DOI:10.1515/znb-2011-0913

DOI: 10.1515/ata-2016-0001
Acta Technologica Agriculturae 1/2016
Dušan Páleš et al.
Acta Technologica Agriculturae 1
Nitra, Slovaca Universitas Agriculturae Nitriae, 2016, pp. 1–5
ApproximAtion of Vehicle trAjectory with B-Spline curVe
Dušan PÁLEŠ*, Veronika VÁLIKOVÁ, Ján ANTL, František TÓTH
Slovak University of Agriculture in Nitra, Slovak Republic
In this contribution, we present the description of a B-spline curve. We deal with creation of its basis function as well as with
creation of the curve itself from entered control points. Following the literature, we formed an algorithm for B-spline modelling
and we used it for the planar and spatial curve. The planar curve is made of chosen points. The spatial curve approximates the
trajectory of a real vehicle, which trajectory was obtained by the set of measured points. The modelled curve very exactly describes
the polygon created from the linked control points. With the lowering degree of the curve, this one is more clamping to the given
polygon and for the extreme case it is transformed to the polygon itself. The advantage of the B-spline curve use is, for example in
comparison with a Bézier curve, high adaptability, which is expressed in its parameters – besides entered control points, these are
knots generated on the curve and degree of the curve.
Keywords: B-spline curve; Cox-de Boor formula; curve fitting
The most effective way for determination of curves for
Let U = {u0, u1, u2, ... , um} be non-decreasing sequence
practical use is to use a set of control points. These control of (m + 1) real numbers, thus u0 ≤ u1 ≤ u2 ≤ ... ≤ ui ≤ ui + 1
≤
points can be accompanied by other restriction for the ... ≤ um. These numbers are called knots, the set U is called
curve, for example boundary conditions or conditions the knot vector, and the half-opened interval 〈ui, ui + 1) is
for curve continuity (Sederberg, 2012). When a smooth the i-th knot span. Seeing that knots ui may be equal, some
curve runs only through some control points, we refer to knot spans may not exist, thus they are zero. If the knot ui
curve approximation. The B-spline curve is one of such appears p times, hence ui = ui + 1 = ... = ui + p - 1, where p >1,
approximation curves and is addressed in this contribution. ui is a multiple knot of multiplicity p, written as ui(p). If ui is
A special case of the B-spline curve is the Bézier curve (Páleš only a solitary knot, it is also called a simple knot. If the knots
and Rédl, 2014; Rédl et al., 2014).
are equally spaced, i.e. (ui + 1 - ui) = constant, for every 0 ≤ i
The B-spline curve is applied to a set of control points ≤ (m - 1), the knot vector or knot sequence is said uniform,
in a space, which were obtained by measurement of real otherwise it is non-uniform.
vehicle movement on a slope (Rédl, 2007, 2008). Data were
Knots can be considered as division points that
processed into the resulting trajectory (Rédl, 2012; Rédl and subdivide the interval 〈u0, um〉 into knot spans. All B-spline
Kučera, 2008). Except for this, the movement of the vehicle basis functions are supposed to have their domain on 〈u0,
was simulated using motion equations (Rédl, 2003; Rédl and um〉. We will use u0 = 0 and um = 1.
Kročko, 2007).
To define B-spline basis functions, we need one more
parameter k, which gives the degree of these basis functions.
Recursive formula is defined as follows:
material and methods
N
i, 0 (u) = 1, u1 ≤ u ≤ ui + 1
(1)
B-spline basis functions
N
i, 0 (u) = 0, for other u
Bézier basis functions known as Bernstein polynomials are
used in a formula as a weighting function for parametric
representation of the curve (Shene, 2014). B-spline basis
functions are applied similarly, although they are more
complicated. They have two different properties in
comparison with Bézier basis functions and these are: 1)
solitary curve is divided by knots, 2) basis functions are not
nonzero on the whole area. Every B-spline basis function
is nonzero only on several neighbouring subintervals and
thereby it is changed only locally, so the change of one
control point influences only the near region around it and
not the whole curve.
u − ui
ui +k − ui
ui +k +1 − u
Ni , k −1(u) + Ni +1, k −1(u)
(2)
N
i , k (u) =
ui +k +1 − ui +1
This definition is usually referred to as the Cox-de Boor
recursion formula. If the degree is zero, i.e. k = 0, these basis
functions are all step functions that follows from Eq. (1).
i, 0 (u) = 1 is only in the i-th knot span 〈ui, ui + 1). For example,
if we have four knots u0 = 0, u1 = 1, u2 = 2 and u3 = 3, knot
spans 0, 1 and 2 are 〈0, 1), 〈1, 2) and 〈2, 3), and the basis
functions of degree 0 are N0, 0 (u) = 1 on interval 〈0, 1)
N
Contact address: *Dušan Páleš, Slovak University of Agriculture in Nitra, Faculty of Engineering, Department of Machine Design,
Tr. Andreja Hlinku 2, 949 76 Nitra, Slovakia, e-mail: dusan.pales@uniag.sk
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Dušan Páleš et al.
Acta Technologica Agriculturae 1/2016
figure 1
Basis function of zero order
figure 3
Basis function of second order
for u ∈ 〈1, 2) and N1, 1 (u) = (3 - u) for u ∈ 〈2, 3). Figure 2
shows that N0, 1 (u) is nonzero on intervals 〈0, 1) and 〈1, 2),
and N1, 1 (u) is nonzero on interval 〈1, 2) and 〈2, 3).
In general, for k = 1 we obtain the basis function:
ui +2 −u
u −ui
ui +1 −ui
(5)
Ni , 1(u) =
Ni , 0 (u) +
Ni +1, 0 (u)
ui +2 −ui +1
As:
N
i, 0 (u) = 1, u1 ≤ u ≤ ui + 1 Ni + 1, 0 (u) = 1, ui + 1 ≤ u ≤ ui + 2 (6)
i, 0 (u) = 0, for other u Ni + 1, 0 (u) = 0, for other u
N
hence:
u −ui
ui +1 −ui
figure 2
Basis function of first order
(7)
Ni , 1(u) =
Ni , 1(u) =
,
,
ui ≤ u ≤ ui+1
ui+1 ≤ u ≤ ui+2
and N0, 0 (u) = 0 elsewhere, N1, 0 (u) = 1 on interval 〈1, 2)
and N1, 0 (u) = 0 elsewhere and N2, 0 (u) = 1 on interval 〈2, 3)
and N2, 0 (u) = 0 elsewhere, see Figure 1.
ui +2 −u
ui +2 −ui+1
To compute Ni, 1 (u), Ni, 0 and Ni + 1, 0 are required.Therefore,
we can compute N0, 1 (u), N1, 1 (u), N2, 1 (u), N3, 1 (u) and so on.
Next, we compute all Ni, 2 (u), and we continue like this till
the computation of final Ni, k (u).
When we compute N0, 1 (u) and N1, 1 (u), we can express
0, 2 (u) as:
N
u −u0
u2 −u0
u3 −u
u3 −u1
(8)
N
0, 2 (u) =
N
0, 1(u) +
N1, 1(u)
In the above, we have obtained N0, (u), N1, (u)
0
and N2, 0 (u) for the knot vector U = {0, 1, 2, 3}. Let us com0pute
N
0, 1 (u), i = 0, k = 1 from the definition:
Similar substitution as in the formula for N0, 1 (u) for knot
values gives us:
u −u0
u1 −u0
u2 −u
u2 −u1
(3)
(4)
Ni , 0 (u) =
N
0, 0 (u) +
N1, 0 (u)
N0, 2 (u) = 0.5 ⋅ u ⋅ N0, 1(u) + 0.5(3 - u) ⋅ N1, 1 (u)
(9)
Since u0 = 0, u1 = 1 and u2 = 2, Eq. (3) simplifies to:
0, 1 (u) = u ⋅ N0, 0 (u) + (2 - u) ⋅ N1, 0 (u)
N
0, 1 (u) is nonzero on intervals 〈0, 1) and 〈1, 2), and N1, 1 (u)
is nonzero on intervals 〈1, 2) and 〈2, 3); therefore, the basis
function has to be divided into three parts, Figure 3:
First: u ∈ 〈0, 1), then only N0, 1 (u) contributes to the
function N0, 2 (u) and hence:
N
Because N0, 0 (u) is nonzero on interval 〈0, 1) and N1,0 (u)
on interval 〈1, 2), if u ∈ 〈0, 1), only N0, 0 (u) contributes
N
0, 2 = 0.5 ⋅ u2
(10)
to N0, (u) and analogically, only N1, (u) contributes
to N0, 11(u) if u ∈ 〈1, 2). Therefore, if u ∈ 〈0, 1), then N0, 1
(u) = u ∙ N0, 0 (u) = u, and if u ∈ 〈1, 2), then N0, 1 (u) = (2 - u) ∙
N1, 0 (u) = (2 - u). Similar computation gives N1, 1 (u) = (u - 1)
0
Second: u ∈ 〈1, 2), N0, 1 (u) = (2 - u) and N1, 1 (u) = (u - 1)
contribute to the function N0, 2 (u)
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Dušan Páleš et al.
N
0, 2 = 0.5 ⋅ u(2 - u) + 0.5(3 - u) ⋅ (u - 1) = 0.5(-2u2 + 6u - 3) (11)
Third: u ∈ 〈2, 3), only N1, 1 (u) = (3 - u) contributes to the
function N0, 2 (u)
0, 2 (u) = 0.5(3 - u) ⋅ (3 - u) = 0.5(3 - u)2
B-spline curve
N
(12)
If there is (n + 1) given control points P0, P1, ... Pn and (m + 1)
knots u0, u1,... um, then the B-spline curve of order k is defined
with the formula:
n
(13)
C(u) =
Ni , k (u)⋅P
i
∑
i=0
where Ni, (u) are the B-spline basis functions of order
figure 4
B-spline curve of the degree k = 5
k
k explained above. The form of the B-spline curve notation
is very similar to the Bézier curve notation. Unlike the
Bézier curve, the B-spline curve contains more information,
besides the set of (n + 1) control points also the knot vector
of (m + 1) knots and degree of the curve k. For n, m and
k must hold: m = n + k + 1. More precisely, if we want to
define the B-spline curve of order k by help of (n + 1) control
points, we need to create (n + k + 2) knots u0, u1,... un + k + 1
.
If we have (n + 1) given control points and (m + 1) knots,
then the degree of the B-spline curve is k = m - n - 1. The
knot points divide the B-spline curve on the parts that are
the Bézier curves of order k. While the degree of the Bézier
curve depends on the number of control points and holds
not to be influenced, the degree of the B-spline curve is the
entering parameter for it.
If the knots do not have any ordered structure and
they are only regularly placed, an opened B-spline curve is
created and it does not run through the first and last point
and it also does not have the first and the last flowline of
control points as tangent lines. If we want to obtain the
so-called clamped B-spline curve, the first and the last knot
has to be (k + 1) multiple. At that time, it resembles to the
Bézier curve, and we have modelled this type of the clamped
B-spline curve. The closed B-spline curve is the last time and
it enables creating loops, the first control point of which is
equal with the last one.
figure 5
5 B-spline curve of the degree k = 3
results and discussion
We have formed an algorithm for computation of knot
points u0, u1,... un + k + 1 (14) according to (Meung, 2014):
UP(n, k): = for i ∈ 0 ... n + k + 1
upi ← 0 if i < k + 1
otherwise
upi ← n - k + 1 if i > n
upi ← i - k otherwise
(14)
figure 6
B-spline curve of the degree k = 2
The algorithms were debugged with various parameters
on a test example, the results of which are shown in Figs
4–7. The control points in the example were defined by the
coordinates P0[1, 1], P1[2, 4], P2[5, 5], P3[4, 2], P4[5, 1] and
P5[7, 3]. As can be seen in the figures, with the lowering
degree k, the curve is more clamping to the polygon created
up
Their illustration is shown in Figs 4–7. Another algorithm
(15) is used for computation of the basis function of B-spline
curves, which are defined by Eqs (1), (2) (Meung, 2014).
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Dušan Páleš et al.
Acta Technologica Agriculturae 1/2016
x in m
figure 8
Comparison of measured points of vehicle
trajectory with the B-spline curve in the plain xy
figure 7
B-spline curve of the degree k = 1
from the control points. From this follows the essential benefit of the B-spline curve in comparison with the Bézier curve,
with the lowering k we can more precisely describe the given polygon, as can be seen when comparing all four figures. An
extreme case occurs for the degree of the curve k = 1, Figure 7. In this case, the curve correctly copies the polygon of control
points and exactly control points are its knots:
N(i, k, n, u, up): = if k = 0
if i + 1 = n + k + 1
if u ≤ upi + 1
sum ← 1 if u ≥ upi
sum ← 0 otherwise
sum ← 0 otherwise
(15)
otherwise
if u < upi + 1
sum ← 1 if u ≥ upi
sum ← 0 otherwise
sum ← 0 otherwise
otherwise
(u −u pi )
sum ←
∙ B(i, k - 1, n, u, up) if upi + k > upi
(u pi +k ) −u pi )
(u pi+k+1 −u)
sum ← sum +
∙ B(i + 1, k - 1, n, u, up) if upi + k + 1 > upi + 1
(u pi +k ) −u pi+1
)
sum
Another issue to note in Figs 4–7 is that this is the clamped B-spline curve, whereby the first and the last knot are (k + 1)
multiple. For Figure 4, the degree is k = 5 and the first and the last knot are six-fold. For Figure 5, the degree is k = 3 and the
ending knots are four-fold. Similarly for Figure 6, k = 2 and for Figure 7, k = 1.
The tested procedure in a plain was applied to a spatial curve (Rusnák et al., 2008). We solved the real vehicle movement
on a slope, given by the set of points, each one with three coordinates. The number of control points is n = 138, whereby we
chose the degree of the curve k = 5. The basis functions of the B-spline curve (15) were projected into all three directions x, y
and z. In Figs 8–10, we show the projections of control points and of the B-spline curve into all three plains of the space xy, yz,
zx. The curve very exactly copies measured control points, illustrated by empty circles, which is obvious from all three figures.
The last Figure 11, represents the spatial B-spline curve created by the trajectory of the vehicle moving on the slope.
Seeing that the curve completely approximates the measured coordinates of the control points, these were not individually
depicted in the figure.
Conclusion
We applied the algorithm for computation of knot points and basis B-spline functions for creation of the planar and
spatial curve. The planar curve was only a test example; the spatial curve was used for the coordinates obtained in real
measurement during the vehicle movement on the slope. Following the obtained results, we can state that the B-spline
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Acta Technologica Agriculturae 1/2016
Dušan Páleš et al.
follows the higher accuracy of approximation with the
B-spline curve in comparison with the Bézier curve. Its other
advantage is high flexibility because due to its modification
we can move the control points, we can change the knots
on the curve and last but not least, we can change the
degree of the curve. The curve here creates one unit and we
do not have to worry about its continuity. The only minus
occurring may be longer and more complicated algorithm
for formation of basis functions (15) such as the necessity of
the next algorithm (14) for knot creation.
Acknowledgment
The research performed at the Department of Machine
Design, Faculty of Engineering, Slovak University of
Agriculture in Nitra was supported by the Scientific Grant
Agency, grant VEGA No 1/0575/14 “Minimising the risks of
environmental factors in animal production buildings‘.
y in m
figure 9
Comparison of measured points of vehicle
trajectory with the B-spline curve in the plain yz
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Trajectory of the vehicle in the space modelled
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curve very exactly describes the continuance of given
control points. With the lowering of the curve degree, the
curve is more clamping to the polygon created from linked
control points. The segments between the knots always
create the Bézier curve of corresponding order. From this
nnn
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