Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
10,
No.
2,
April
2020,
pp.
1648
1654
ISSN:
2088-8708,
DOI:
10.11591/ijece.v10i2.pp1648-1654
r
1648
A
ppr
oximating
offset
cur
v
es
using
B
´
ezier
cur
v
es
with
high
accuracy
Abedallah
Rababah
1
,
Moath
J
aradat
2
1
Department
of
Mathematical
Sciences,
United
Arab
Emirates
Uni
v
ersity
,
United
Arab
Emirates
1,2
Department
of
Mathematics,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Jordan
Article
Inf
o
Article
history:
Recei
v
ed
Apr
14,
2019
Re
vised
Oct
20,
2019
Accepted
Oct
30,
2019
K
eyw
ords:
Approximation
order
B
´
ezier
curv
es
Circular
arc
Cubic
approximation
High
accurac
y
ABSTRA
CT
In
this
paper
,
a
ne
w
method
for
the
approximation
of
of
fset
curv
es
is
presented
using
the
idea
of
the
parallel
deri
v
ati
v
e
curv
es.
The
best
uniform
approximation
of
de
gree
3
with
order
6
is
used
to
construct
a
method
to
find
the
approximation
of
the
of
fset
curv
es
for
B
´
ezier
curv
es.
The
proposed
method
is
based
on
the
best
uniform
approximation,
and
therefore;
t
he
proposed
method
for
constructing
the
of
fset
curv
es
induces
better
outcomes
than
the
e
xisting
methods.
Copyright
c
2020
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Abedallah
Rababah,
Department
of
Mathematics,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
22110
Irbid,
Jordan.
Email:
rababah@just.edu.jo
and
rababah@uaeu.ac.ae
1.
INTR
ODUCTION
The
of
fset
curv
es
appeared
in
the
19th
century
and
are
widely
used
in
Computer
Aided
Design/Computer
Aided
Manuf
actoring
CAD/CAM
applications,
and
has
other
applications
in
man
y
computer
fields.
Man
y
studies
on
the
of
fset
approximation
are
carried
out
by
man
y
researchers.
Hoschek
[1]
approximated
the
of
fset
curv
es
using
splines.
Rational
of
fset
curv
es
are
approximatedby
F
arouki
and
Sakkalis
[2]
by
constructing
the
Pythagorean-hodograph
(PH)
curv
es.
In
[3],
rational
of
fset
curv
es
based
on
the
quadratic
approximation
of
the
circular
arc
are
approximated.
Recently
,
of
fset
approximation
curv
es
based
on
the
circular
arc
approxima-
tions
are
presented
[4-6]
yielding
rational
of
fset
approximation
which
are
the
con
v
olution
of
the
unit
normal
v
ector
and
the
gi
v
en
curv
e.
The
of
fset
approximation
in
this
paper
is
based
on
the
best
uniform
approxima-
tion
of
the
circular
arc
and
yields
a
polynomial
of
fset
approximation
curv
e.
The
best
uniform
approxima-
tion
of
the
circular
arc
of
de
gree
3
presented
in
[7]
where
the
error
function
is
the
Chebyshe
v
polynomial
of
de
gree
6,
see
also
[8-16].
.
This
of
fset
method
is
constructed
as
follo
ws:
gi
v
en
a
B
´
ezier
curv
e
b
(
t
)
and
its
unit
normal
v
ector
N
(
t
)
which
is
a
circular
arc.
Then
we
use
the
best
uniform
approximation
of
de
gree
3
to
approximate
the
unit
normal
v
ector
of
the
gi
v
en
curv
e.
Since
the
best
uniform
approximation
is
of
high
accurac
y
then
it
is
anticipated
that
the
approximation
of
the
normal
v
ector
is
as
of
high
accurac
y
.
Thereafter
,
a
special
reparametrization
of
the
approximation
to
unit
normal
v
ector
N
a
(
t
)
is
carried
out
to
ha
v
e
the
same
length
as
the
unit
no
r
mal
v
ector
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;
L
1
norm,
L
2
norm,
and
L
1
norm
which
is
the
best
uniform
approximation
that
we
are
using
in
my
paper
.
Cubic
B
´
ezier
curv
es
are
commonly
used
in
almost
all
industrial
companies;
it
is
used
in
computer
graphics,
animation,
modeling,
CAD,
CA
GD,
design,
and
man
y
J
ournal
homepage:
http://ijece
.iaescor
e
.com/inde
x.php/IJECE
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1649
other
related
fields.
In
these
and
other
applications
in
CG
and
CA
GD,
conic
sections
are
the
most
commonly
used
curv
es
in
an
y
CAD
system.
The
Bernstein
po
l
ynomials
are
one
of
the
most
important
polynomials
in
mathematic
s.
The
y
serv
e
essential
tasks
in
numerical,
approximation
and
B
´
ezier
curv
es,
because
the
y
form
basis
which
are
numerically
stable.
The
Bernstein
basis
polynomials
of
de
gree
n
are
defined
as
[17-19]:
B
n
i
(
t
)
=
n
i
t
i
(1
t
)
n
i
;
t
2
[0
;
1]
;
i
=
0
;
1
;
2
;
:::;
n;
(1)
where
the
binomial
coef
fcients
are
gi
v
en
by
n
i
=
n
!
i
!(
n
i
)!
:
The
Bernst
ein
polynomials
are
used
as
basis
for
the
approximation
and
representation
of
curv
es
and
are
generalized
to
triangular
surf
aces
[20,21].
The
Bernstein
polynomials
are,
in
particular
,
important
for
the
construction
of
the
B
´
ezier
curv
es
that
are
defined
as
follo
w
.
A
B
´
ezier
curv
e
of
de
gree
n
is
defined
by
b
(
t
)
=
n
X
i
=0
b
i
B
n
i
(
t
)
=
x
(
t
)
y
(
t
)
;
t
2
[0
;
1]
;
(2)
where
b
i
’
s
are
the
control
points,
and
B
n
i
(
t
)
are
the
Bernstein
polynomials
of
de
gree
n
.
F
or
a
gi
v
en
B
´
ezier
curv
e
b
(
t
)
in
(2),
the
of
fset
curv
e
b
r
(
t
)
with
of
fset
distance
r
2
R
+
is
gi
v
en
by
b
r
(
t
)
=
b
(
t
)
+
r
N
(
t
)
;
(3)
where
N
(
t
)
is
the
unit
normal
v
ector
of
b
(
t
)
gi
v
en
by
N
(
t
)
=
(
y
0
(
t
)
;
x
0
(
t
))
p
(
x
0
(
t
))
2
+
(
y
0
(
t
))
2
:
(4)
The
error
function
e
(
t
)
is
used
to
measure
the
error
between
N
(
t
)
and
N
a
(
t
)
and
is
gi
v
en
by
e
(
t
)
=
(
y
0
(
t
)
p
x
0
2
(
t
)
+
y
0
2
(
t
)
y
0
a
(
t
)
p
x
0
2
a
(
t
)
+
y
0
2
a
(
t
)
)
2
+
(
x
0
(
t
)
p
x
0
2
(
t
)
+
y
0
2
(
t
)
x
0
a
(
t
)
p
x
0
2
a
(
t
)
+
y
0
2
a
(
t
)
)
2
2.
RESEARCH
METHOD
In
this
section,
we
present
a
ne
w
method
of
of
fset
curv
e
approximation
of
the
n
-th
de
gree
B
´
ezier
curv
e
by
a
curv
e
of
de
gree
3
.
The
best
uniform
approximation
of
the
circular
arc
of
de
gree
3
of
order
6
is
presented
in
[7],
see
also
[22-24].
The
cubic
approximation
of
circular
arc
p
(
t
)
has
a
parametrically
defined
polynom
ial
curv
e
gi
v
en
by
p
(
t
)
=
0
:
515647
+
5
:
99959
t
5
:
99959
t
2
0
:
874847
2
:
25031
t
+
12
t
2
8
t
3
;
t
2
[0
;
1]
:
(5)
Let
b
(
t
)
be
a
re
gular
planar
B
´
ezier
curv
e
of
de
gree
n
gi
v
en
in
(2)
and
N
(
t
)
be
its
unit
normal
v
ector
gi
v
en
in
(4)
.
As
sho
wn
in
Figure
1.,
gi
v
en
an
y
B
´
ezier
curv
e
b
(
t
)
then
by
the
definition
of
the
con
v
olutio,
the
tangent
line
of
b
(
t
)
is
parallel
to
the
tangent
line
of
N
(
t
)
which
is
the
unit
normal
v
ector
for
b
(
t
)
,
8
t
2
[0
;
1]
.
Appr
oximating
of
fset
curves
using
B
´
ezier
curves...
(Abedallah
Rababah)
Evaluation Warning : The document was created with Spire.PDF for Python.
1650
r
ISSN:
2088-8708
Figure
1.
T
angent
of
b
(
t
)
(thick)
parallel
to
the
tangent
of
N
(
t
)
(dashed)
Thus
b
r
N
(
t
)
=
b
(
t
)
+
r
N
(
t
)
=
b
r
(
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,
N
a
(
s
(
t
))
and
b
(
t
)
ha
v
e
the
same
unit
normal
v
ector
.
So,
the
of
fset
approximation
is
gi
v
en
by
b
a
r
(
t
)
=
b
r
N
a
(
s
(
t
))
=
b
(
t
)
+
r
N
a
(
s
(
t
))
;
t
2
[0
;
1]
:
(6)
The
construction
of
the
approximation
N
a
(
s
(
t
))
,
t
2
[0
;
1]
for
the
cubic
B
´
ezeir
curv
e
is
considered.
N
a
(
s
)
=
(
2
:
25031
+
24
s
24
s
2
)
;
(5
:
99959
11
:
9992
s
)
p
(5
:
99959
11
:
9992
s
)
2
+
(
2
:
25031
+
24
s
24
s
2
)
2
;
(7)
where
s
=
s
(
t
)
,
t
2
[0
;
1]
is
re
gular
reparametrization
to
mak
e
both
curv
es
be
gin
and
end
at
the
same
points.
The
curv
e
defined
by
b
a
r
(
t
)
=
b
N
a
(
s
(
t
))
=
b
(
t
)
+
r
N
a
(
s
(
t
))
;
t
2
[0
;
1]
;
is
the
approximation
of
the
of
fset
curv
e
by
cubic
B
´
ezier
curv
e
where
N
a
(
s
(
t
))
is
as
in
(7)
.
The
computation
of
the
reparametrization,
s
=
s
(
t
)
,
where
t
2
[0
;
1]
is
considered.
N
a
(
t
)
and
N
(
t
)
ha
v
e
dif
ferent
parameters,
both
of
them
are
circular
arcs
,
b
ut
the
y
do
not
ha
v
e
the
same
start
and
end
points.
Figure
2.
sho
ws
N
a
(
t
)
and
N
(
t
)
for
a
B
´
ezier
curv
e
b
(
t
)
.
A
reparametrization
s
=
s
(
t
)
is
pres
ented,
so
that
the
curv
e
and
its
approximation
be
gin
and
end
at
the
same
points.
Figure
2.
N
(
t
)
(thick
)
for
B
´
ezier
curv
e
and
the
approximation
N
a
(
t
)
(dashed)
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
1648
–
1654
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1651
Let
b
(
t
)
be
the
curv
e
in
(2)
and
N
(
t
)
be
its
unit
normal
v
ector
.
The
reparametrization
is
s
(
t
)
=
t
b
+
1
t
a
;
t
2
[0
;
1]
;
where
a;
b
are
gi
v
en
by
the
foll
wing:
the
first
step
we
find
N
(0)
and
N
(1)
for
the
gi
v
en
curv
e,
b
(
t
)
.
W
e
get
to
solv
e
the
equations
N
a
(
1
a
)
=
x
a
(
1
a
)
y
a
(
1
a
)
=
N
(0)
=
x
(0)
y
(0)
(8)
N
a
(
1
b
)
=
x
a
(
1
b
)
y
a
(
1
b
)
=
N
(1)
=
x
(1)
y
(1)
:
(9)
By
the
symmetry
of
the
approximation
of
the
circular
a
rc,
in
equation
(8)
,
x
a
(
1
a
)
and
y
a
(
1
a
)
equal
to
zero
at
the
same
parameters,
and
(9)
,
x
a
(
1
b
)
and
y
a
(
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
follo
wing
equations
(
2
:
25031
+
24(
1
a
)
24(
1
a
)
2
)
q
(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
))
q
(5
:
99959
11
:
9992(
1
a
))
2
+
(
2
:
25031
+
24(
1
a
)
24(
1
a
)
2
)
2
=
y
(0)
we
get
the
v
alue
of
the
parameter
a
.
And
by
solving
(
2
:
25031
+
24(
1
b
)
24(
1
b
)
2
)
q
(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
))
q
(5
:
99959
11
:
9992(
1
b
))
2
+
(
2
:
25031
+
24(
1
b
)
24(
1
b
)
2
)
2
=
y
(1)
we
get
the
v
alue
of
the
parameter
b
.
Then
the
approximation
of
N
a
(
t
)
for
the
cubic
case
is
N
a
(
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
))
q
(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.
RESUL
TS
AND
AN
AL
YSIS
The
method
is
applied
for
the
follo
wing
cubic
parametric
curv
e:
x
(
t
)
y
(
t
)
=
27
:
2688
t
3
+
341
:
56752
t
2
(1
t
)
+
351
:
1
t
(1
t
)
2
+
51
:
1(1
t
)
3
47
:
1461
t
3
+
338
:
8523975
t
2
(1
t
)
+
333
:
324
t
(1
t
)
2
+
21
:
4(1
t
)
3
;
(10)
where
t
2
[0
;
1]
and
the
unit
normal
v
ector
is
gi
v
en
by:
N
(
t
)
=
(35
:
772(1
t
)
2
+33
:
1704(1
t
)
t
+24
:
8811
t
2
)
p
(35
:
772(1
t
)
2
+33
:
1704(1
t
)
t
+24
:
8811
t
2
)
2
+(0
:
(1
t
)
2
57
:
1949(1
t
)
t
42
:
8962
t
2
)
2
(
(0
:
(1
t
)
2
57
:
1949(1
t
)
t
42
:
8962
t
2
))
p
(35
:
772(1
t
)
2
+33
:
1704(1
t
)
t
+24
:
8811
t
2
)
2
+(0
:
(1
t
)
2
57
:
1949(1
t
)
t
42
:
8962
t
2
)
2
:
Appr
oximating
of
fset
curves
using
B
´
ezier
curves...
(Abedallah
Rababah)
Evaluation Warning : The document was created with Spire.PDF for Python.
1652
r
ISSN:
2088-8708
Figure
3.
represents
the
graph
of
the
cubic
parametric
curv
e
and
Figure
4.
is
the
cubic
para
metric
curv
e
with
its
of
fset
curv
e
computed
by
the
formula.
Figure
5.
illustrates
the
parametric
cubic
curv
e
with
the
cubic
approximation
of
the
of
fset
curv
e
and
the
original
of
fset
curv
e.
And
Figure
6.
illustrates
the
error
between
the
of
fset
curv
e
and
the
approximation
of
the
of
fset
curv
e.
Figure
3.
The
cubic
parametric
curv
e
Figure
4.
Cubic
parameteric
curv
e
(thick)
and
its
of
fset
curv
e
(dashed)
By
solving
eqautions
(8)
and
(9)
,
we
get
a
=
2
;
b
=
1
:
28862
:
Figure
5.
Cubic
curv
e
(thick)
and
its
of
fset
curv
e
(dashed)
and
the
cubic
approximation
of
the
of
fset
curv
e
(dotted)
Figure
6.
Error
between
of
fset
curv
e
and
the
approximation
of
fset
curv
e
4.
CONCLUSION
In
this
article,
cubic
approximation
of
of
fset
curv
e
is
established.
The
method
is
based
on
the
best
uniform
approximation
of
the
circular
arc
of
de
gree
3
with
order
6
.
The
numerical
e
xamples
re
v
eal
ho
w
ef
ficient
this
method
is.
The
maximum
er
ror
is
5
10
16
,
thus
the
proposed
method
induced
better
outcomes
than
the
e
xisting
methods.
The
results
in
this
paper
can
be
used
to
impro
v
e
the
results
obtained
in
[25],
see
also
the
results
in
[26].
A
CKNO
WLEDGEMENTS
The
authors
o
we
thanks
for
the
re
vie
wers
for
in
v
aluable
suggestions
to
impro
v
e
an
earlier
v
ersion
of
this
paper
.
This
research
w
as
funded
by
Jordan
Uni
v
ersity
of
Science
and
T
echnology
.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
1648
–
1654
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1653
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Appr
oximating
of
fset
curves
using
B
´
ezier
curves...
(Abedallah
Rababah)
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1654
r
ISSN:
2088-8708
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BIOGRAPHIES
OF
A
UTHORS
Abedallah
Rababah
is
a
professor
of
mathematics
at
United
Arab
Emirates
Uni
v
ersity
and
is
on
lea
v
e
from
Jordan
Uni
v
ersity
of
Science
and
T
echnology
.
He
is
w
orking
in
the
field
of
Computer
Aided
Geome
tric
Design,
abbre
viated
CA
GD.
In
particular
,
his
research
is
on
de
gree
raising
and
reduction
of
B
´
ezier
curv
es
and
surf
aces
with
geometric
boundary
conditions,
Bernst
ein
polynomials,
and
their
duality
.
He
is
kno
wn
for
his
research
in
describing
approximation
methods
that
significantly
impro
v
e
the
standard
rates
obtained
by
classic
al
(local
T
aylor
,
Hermite)
methods.
He
pro
v
ed
the
follo
wing
conjecture
for
a
particular
set
of
curv
es
of
nonzero
measure:
Conjecture:
A
smooth
re
gular
planar
curv
e
can,
in
general,
be
approximated
by
a
polynomial
curv
e
of
de
gree
n
with
order
2n.
The
method
e
xploited
the
freedom
in
the
choice
of
the
parametrization
and
achie
v
ed
the
order
4n/3,
rather
than
n
+
1.
Generalizations
were
al
so
pro
v
ed
for
space
curv
es.
Professor
Rababah
is
also
doing
research
in
the
fields
of
classical
approximation
theory
,
orthogonal
polynomials,
Jacobi-weighted
orthogonal
polynomials
on
triangular
domains,
and
best
unifor
m
approximati
ons.
Since
1992,
He
has
been
teaching
at
German,
Jordanian,
American,
Canadian,
and
Emirates’
uni
v
ersities.
He
is
acti
v
e
in
the
editorial
boards
of
man
y
journals
in
mathematics
and
computer
science.
Further
info
can
be
found
on
his
homepage
at
ResearchGate:
https://www
.researchg
ate.net/profile/Abedallah
Rababah
or
at
http://www
.just.edu.jo/eportfolio/P
ages/Def
ault.aspx?email=rababah
Moath
J
aradat
graduated
from
J
ordan
Uni
v
ersity
of
Science
and
T
echnology
.
He
obtained
both
Bachelor
and
Ma
ster
De
grees
in
Mathematics
with
research
interests
in
the
field
of
approximating
of
fset
curv
es
using
B
´
ezier
curv
es
with
high
accurac
y
and
numerical
and
approximations.
Further
info
can
be
found
on
his
homepage
at
ResearchGate:
https://www
.researchg
ate.net/profile/Moad
Jaradat
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
1648
–
1654
Evaluation Warning : The document was created with Spire.PDF for Python.