Approximating offset curves using B ´ ezier curves with high accuracy

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 10, No. 2, April 2020, pp. 1648∼1654
ISSN: 2088-8708, DOI: 10.11591/ijece.v10i2.pp1648-1654 Ì 1648
Approximating offset curves using Bézier curves
with high accuracy
Abedallah Rababah1
, Moath Jaradat2
1
Department of Mathematical Sciences, United Arab Emirates University, United Arab Emirates
1,2
Department of Mathematics, Jordan University of Science and Technology, Jordan
Article Info
Article history:
Received Apr 14, 2019
Revised Oct 20, 2019
Accepted Oct 30, 2019
Keywords:
Approximation order
Bézier curves
Circular arc
Cubic approximation
High accuracy
ABSTRACT
In this paper, a new method for the approximation of offset curves is presented using
the idea of the parallel derivative curves. The best uniform approximation of degree 3
with order 6 is used to construct a method to find the approximation of the offset curves
for Bézier curves. The proposed method is based on the best uniform approximation,
and therefore; the proposed method for constructing the offset curves induces better
outcomes than the existing methods.
Copyright c 2020 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Abedallah Rababah,
Department of Mathematics,
Jordan University of Science and Technology,
22110 Irbid, Jordan.
Email: rababah@just.edu.jo and rababah@uaeu.ac.ae
1. INTRODUCTION
The offset curves appeared in the 19th century and are widely used in Computer Aided Design/Computer
Aided Manufactoring CAD/CAM applications, and has other applications in many computer fields. Many
studies on the offset approximation are carried out by many researchers. Hoschek [1] approximated the offset
curves using splines. Rational offset curves are approximatedby Farouki and Sakkalis [2] by constructing the
Pythagorean-hodograph (PH) curves. In [3], rational offset curves based on the quadratic approximation of
the circular arc are approximated. Recently, offset approximation curves based on the circular arc approxima-
tions are presented [4-6] yielding rational offset approximation which are the convolution of the unit normal
vector and the given curve. The offset approximation in this paper is based on the best uniform approxima-
tion of the circular arc and yields a polynomial offset approximation curve. The best uniform approxima-
tion of the circular arc of degree 3 presented in [7] where the error function is the Chebyshev polynomial of
degree 6, see also [8-16]. .
This offset method is constructed as follows: given a Bézier curve b(t) and its unit normal vector N(t)
which is a circular arc. Then we use the best uniform approximation of degree 3 to approximate the unit normal
vector of the given curve. Since the best uniform approximation is of high accuracy then it is anticipated that
the approximation of the normal vector is as of high accuracy. Thereafter, a special reparametrization of the
approximation to unit normal vector Na
(t) is carried out to have the same length as the unit normal vector
N(t). In this method one step approximation is used so the error will be less than other methods. There are
three types of approximation with respect to the norm; L1 norm, L2 norm, and L∞ norm which is the best
uniform approximation that we are using in my paper. Cubic Bézier curves are commonly used in almost all
industrial companies; it is used in computer graphics, animation, modeling, CAD, CAGD, design, and many
Journal homepage: http://ijece.iaescore.com/index.php/IJECE

Int J Elec & Comp Eng ISSN: 2088-8708 Ì 1649
other related fields. In these and other applications in CG and CAGD, conic sections are the most commonly
used curves in any CAD system.
The Bernstein polynomials are one of the most important polynomials in mathematics. They serve
essential tasks in numerical, approximation and Bézier curves, because they form basis which are numerically
stable. The Bernstein basis polynomials of degree n are defined as [17-19]:
Bn
i (t) =
n
i
ti
(1 − t)n−i
, t ∈ [0, 1], i = 0, 1, 2, ..., n, (1)
where the binomial coeffcients are given by
n
i
=
n!
i!(n − i)!
.
The Bernstein polynomials are used as basis for the approximation and representation of curves and
are generalized to triangular surfaces [20,21]. The Bernstein polynomials are, in particular, important for the
construction of the Bézier curves that are defined as follow.
A Bézier curve of degree n is defined by
b(t) =
n
i=0
biBn
i (t) =
x(t)
y(t)
, t ∈ [0, 1], (2)
where bi’s are the control points, and Bn
i (t) are the Bernstein polynomials of degree n.
For a given Bézier curve b(t) in (2), the offset curve br(t) with offset distance r ∈ R+
is given by
br(t) = b(t) + rN(t), (3)
where N(t) is the unit normal vector of b(t) given by
N(t) =
(y (t), −x (t))
(x (t))2 + (y (t))2
. (4)
The error function e(t) is used to measure the error between N(t) and Na
(t) and is given by
e(t) = (
y (t)
x 2(t) + y 2(t)
−
ya(t)
x 2
a (t) + y 2
a (t)
)2
+ (
x (t)
x 2(t) + y 2(t)
−
xa(t)
x 2
a (t) + y 2
a (t)
)2
2. RESEARCH METHOD
In this section, we present a new method of offset curve approximation of the n-th degree Bézier curve
by a curve of degree 3. The best uniform approximation of the circular arc of degree 3 of order 6 is presented
in [7], see also [22-24]. The cubic approximation of circular arc p(t) has a parametrically defined polynomial
curve given by
p(t) =
−0.515647 + 5.99959t − 5.99959t2
−0.874847 − 2.25031t + 12t2 − 8t3
, t ∈ [0, 1]. (5)
Let b(t) be a regular planar Bézier curve of degree n given in (2) and N(t) be its unit normal vector
given in (4). As shown in Figure 1., given any Bézier curve b(t) then by the definition of the convolutio, the
tangent line of b(t) is parallel to the tangent line of N(t) which is the unit normal vector for b(t), ∀ t ∈ [0, 1].
Approximating offset curves using Bézier curves... (Abedallah Rababah)

1650 Ì ISSN: 2088-8708
Figure 1. Tangent of b(t) (thick) parallel to the tangent of N(t) (dashed)
Thus
b ∗ rN(t) = b(t) + rN(t) = br(t).
Since N(t) is circular arc, the tangent line of N(t) is parallel to the tangent line of b(t), then the
approximation of N(t) is also circular arc and parallel to b(t). Note that, Na
(s(t)) and b(t) have the same unit
normal vector. So, the offset approximation is given by
ba
r (t) = b ∗ rNa
(s(t)) = b(t) + rNa
(s(t)), t ∈ [0, 1]. (6)
The construction of the approximation Na
(s(t)), t ∈ [0, 1] for the cubic Bézeir curve is considered.
Na
(s) =
(−2.25031 + 24s − 24s2
), −(5.99959 − 11.9992s)
(5.99959 − 11.9992s)2 + (−2.25031 + 24s − 24s2)2
, (7)
where s = s(t), t ∈ [0, 1] is regular reparametrization to make both curves begin and end at the same points.
The curve defined by
ba
r (t) = b ∗ Na
(s(t)) = b(t) + rNa
(s(t)), t ∈ [0, 1],
is the approximation of the offset curve by cubic Bézier curve where Na
(s(t)) is as in (7).
The computation of the reparametrization, s = s(t), where t ∈ [0, 1] is considered. Na
(t) and N(t)
have different parameters, both of them are circular arcs, but they do not have the same start and end points.
Figure 2. shows Na
(t) and N(t) for a Bézier curve b(t). A reparametrization s = s(t) is presented, so that the
curve and its approximation begin and end at the same points.
Figure 2. N(t) (thick ) for Bézier curve and the approximation Na
(t) (dashed)
Int J Elec & Comp Eng, Vol. 10, No. 2, April 2020 : 1648 – 1654

Let b(t) be the curve in (2) and N(t) be its unit normal vector. The reparametrization is
s(t) =
t
b
+
1 − t
a
, t ∈ [0, 1],
where a, b are given by the follwing: the ﬁrst step we ﬁnd N(0) and N(1) for the given curve, b(t). We get to
solve the equations
Na
(
1
a
) =
xa
(1
a )
ya(1
a )
= N(0) =
x(0)
y(0)
(8)
Na
(
1
b
) =
xa
(1
b )
ya(1
b )
= N(1) =
x(1)
y(1)
. (9)
By the symmetry of the approximation of the circular arc, in equation (8), xa
(1
a ) and ya
(1
a ) equal to
zero at the same parameters, and (9), xa
(1
b ) and ya
(1
b ) equal to one at the same parameters, then a equals the
parameter in (8) and b equals the parameter in (9).
By solving the following equations
(−2.25031 + 24(1
a ) − 24(1
a )2
)
(5.99959 − 11.9992(1
a ))2 + (−2.25031 + 24(1
a ) − 24(1
a )2)2
= x(0)
and
−(5.99959 − 11.9992(1
a ))
(5.99959 − 11.9992(1
a ))2 + (−2.25031 + 24(1
a ) − 24(1
a )2)2
= y(0)
we get the value of the parameter a.
And by solving
(−2.25031 + 24(1
b ) − 24(1
b )2
)
(5.99959 − 11.9992(1
b ))2 + (−2.25031 + 24(1
b ) − 24(1
b )2)2
= x(1)
and
−(5.99959 − 11.9992(1
b ))
(5.99959 − 11.9992(1
b ))2 + (−2.25031 + 24(1
b ) − 24(1
b )2)2
= y(1)
we get the value of the parameter b.
Then the approximation of Na
(t) for the cubic case is
Na
(t) =
(−2.25031 + 24(t
b + 1−t
a ) − 24(t
b + 1−t
a )2
), −(5.99959 − 11.9992(t
b + 1−t
a ))
(5.99959 − 11.9992(t
b + 1−t
a ))2 + (−2.25031 + 24(t
b + 1−t
a ) − 24(t
b + 1−t
a )2)2
.
3. RESULTS AND ANALYSIS
The method is applied for the following cubic parametric curve:
x(t)
y(t)
=
27.2688t3
+ 341.56752t2
(1 − t) + 351.1t(1 − t)2
+ 51.1(1 − t)3
47.1461t3 + 338.8523975t2(1 − t) + 333.324t(1 − t)2 + 21.4(1 − t)3
, (10)
where t ∈ [0, 1] and the unit normal vector is given by:
N(t) =
(35.772(1−t)2
+33.1704(1−t)t+24.8811t2
)
√
(35.772(1−t)2+33.1704(1−t)t+24.8811t2)2+(0.(1−t)2−57.1949(1−t)t−42.8962t2)2
(−(0.(1−t)2−57.1949(1−t)t−42.8962t2))
√
(35.772(1−t)2+33.1704(1−t)t+24.8811t2)2+(0.(1−t)2−57.1949(1−t)t−42.8962t2)2
.

1652 Ì ISSN: 2088-8708
Figure 3. represents the graph of the cubic parametric curve and Figure 4. is the cubic parametric
curve with its offset curve computed by the formula. Figure 5. illustrates the parametric cubic curve with
the cubic approximation of the offset curve and the original offset curve. And Figure 6. illustrates the error
between the offset curve and the approximation of the offset curve.
Figure 3. The cubic parametric curve Figure 4. Cubic parameteric curve (thick)
and its offset curve (dashed)
By solving eqautions (8) and (9), we get
a = 2, b = 1.28862.
Figure 5. Cubic curve (thick) and its
offset curve (dashed) and the cubic
approximation of the offset curve
(dotted)
Figure 6. Error between offset curve
and the approximation offset curve
4. CONCLUSION
In this article, cubic approximation of offset curve is established. The method is based on the best
uniform approximation of the circular arc of degree 3 with order 6. The numerical examples reveal how
efﬁcient this method is. The maximum error is 5 × 10−16
, thus the proposed method induced better outcomes
than the existing methods. The results in this paper can be used to improve the results obtained in [25], see also
the results in [26].
ACKNOWLEDGEMENTS
The authors owe thanks for the reviewers for invaluable suggestions to improve an earlier version of
this paper. This research was funded by Jordan University of Science and Technology.

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BIOGRAPHIES OF AUTHORS
Abedallah Rababah is a professor of mathematics at United Arab Emirates University and is on
leave from Jordan University of Science and Technology. He is working in the field of Computer
Aided Geometric Design, abbreviated CAGD. In particular, his research is on degree raising and
reduction of Bézier curves and surfaces with geometric boundary conditions, Bernstein polynomials,
and their duality. He is known for his research in describing approximation methods that significantly
improve the standard rates obtained by classical (local Taylor, Hermite) methods. He proved the
following conjecture for a particular set of curves of nonzero measure: Conjecture: A smooth regular
planar curve can, in general, be approximated by a polynomial curve of degree n with order 2n. The
method exploited the freedom in the choice of the parametrization and achieved the order 4n/3, rather
than n + 1. Generalizations were also proved for space curves. Professor Rababah is also doing
research in the fields of classical approximation theory, orthogonal polynomials, Jacobi-weighted
orthogonal polynomials on triangular domains, and best uniform approximations. Since 1992, He
has been teaching at German, Jordanian, American, Canadian, and Emirates’ universities. He is
active in the editorial boards of many journals in mathematics and computer science. Further info
can be found on his homepage at ResearchGate:
https://www.researchgate.net/profile/Abedallah−Rababah
or at http://www.just.edu.jo/eportfolio/Pages/Default.aspx?email=rababah
Moath Jaradat graduated from Jordan University of Science and Technology. He obtained both
Bachelor and Master Degrees in Mathematics with research interests in the field of approximating
offset curves using Bézier curves with high accuracy and numerical and approximations. Further
info can be found on his homepage at ResearchGate:
https://www.researchgate.net/profile/Moad−Jaradat

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