Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
10,
No.
2,
April
2020,
pp.
2192
2199
ISSN:
2088-8708,
DOI:
10.11591/ijece.v10i2.pp2192-2199
r
2192
The
best
sextic
appr
oximation
of
h
yperbola
with
order
twelv
e
Abedallah
Rababah
1
,
Esra’a
Rababah
2
1
Department
of
Mathematical
Sciences,
United
Arab
Emirates
Uni
v
ersity
,
Al
Ain,
U
AE
1,2
Department
of
Mathematics,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Jordan
Article
Inf
o
Article
history:
Recei
v
ed
Apr
8,
2019
Re
vised
Oct
20,
2019
Accepted
No
v
2,
2019
K
eyw
ords:
approximation
order
equioscillation
high
accurac
y
h
yperbola
se
xtic
parametric
approximation
ABSTRA
CT
In
this
article,
the
best
uniform
approximation
for
the
h
yperbola
of
de
gree
6
that
has
approximation
order
12
is
found.
The
associated
error
function
v
anishes
12
times
and
equioscillates
13
times.
F
or
an
arc
of
the
h
yperbola,
the
error
is
bounded
by
2
:
4
10
4
.
W
e
e
xplain
the
details
of
the
deri
v
ation
and
sho
w
ho
w
to
apply
the
method.
The
method
is
simple
and
this
encourages
and
moti
v
ates
people
w
ork-
ing
in
CG
and
CAD
to
apply
it
in
their
w
orks.
Copyright
c
2020
Insitute
of
Advanced
Engineeering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Abedallah
Rababah,
Department
of
Mathematical
Sciences,
United
Arab
Emirates
Uni
v
ersity
,
Al
Ain,
U
AE.
Department
of
Mathematics,
22110
Irbid,
Jordan.
Email:
rababah@uaeu.ac.ae
and
esrababah15@sci.just.edu.jo
1.
INTR
ODUCTION
B
´
ezier
curv
es
were
in
v
ented
by
Pierre
B
´
ezier
and
P
aul
de
Casteljau
in
the
1960’
s.
The
primary
application
w
as
in
the
automobile
industry
,
b
ut
today
the
y
are
widely
used
in
man
y
computer
applications
and
codes
to
sk
etch
curv
es.
Approximation
using
polynomials
of
lo
w
de
gree
is
f
a
v
ourable,
and
in
man
y
cases
is
a
must
issue.
Man
y
CAD
systems
are
limited
to
using
only
parametric
polynomial
curv
es
of
lo
w
de
gree
with
lo
w
errors.
The
big
error
causes
tw
o
major
disadv
antages:
high
accumulated
error
,
and
slo
w
and
costly
softw
are.
So,
it
is
f
a
v
ourable
to
approximate
using
lo
w
de
gree
polynomi
als.
P
arabolas
are
represented
e
xactly
using
parametric
curv
es
of
de
gree
2.
In
[1]
methods
of
best
uniform
approximation
of
a
circular
arc
of
de
gree
6
with
order
12
are
accomplished.
These
approximations
are
optimal
and
ha
v
e
13
equioscillations.
In
[2],
a
method
of
best
uniform
approximation
of
a
h
yperbola
of
de
grees
2
with
order
4
is
accomplished.
These
results
were
moti
v
ated
by
the
results
of
approximating
curv
es
with
high
accurac
y
[3-6].
In
this
paper
,
we
find
approximation
for
the
h
yperbola
of
de
gree
6
and
we
get
the
order
12
.
B
´
ezier
curv
e
techniques
are
used
to
represent
the
approximations
of
the
h
yperbola
using
parametric
polynomials
of
de
gree
6
that
ha
v
e
the
least
uniform
error
.
The
method
is
simple
and
practical.
T
o
achie
v
e
this
approximation,
proper
arrangements
and
symmetries
of
the
h
yperbola
are
applied
to
determine
the
B
´
ezier
points
that
define
the
approximation
of
the
h
yperbola,
and
therefore,
mak
es
a
CAD
system
more
ef
ficient
and
minimizes
the
cost.
Using
suitable
translation
and
scaling,
the
h
yperbola
can
be
written
in
the
basic
forms:
y
2
x
2
=
1
and
x
2
y
2
=
1
.
Ev
ery
form
has
tw
o
branches.
So,
there
are
four
branches.
Geometrically
,
all
of
these
branches
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
2193
are
identical.
Therefore,
it
is
suf
ficient
to
represent
one
branch
and
the
other
three
branches
can
be
represented
using
rotation
of
this
branch.
So,
we
consider
the
upper
branch
of
the
h
yperbola
y
2
x
2
=
1
,
see
Figure
1.
It
can
be
written
in
parametric
form
c
:
t
7!
(sinh(
t
)
;
cosh(
t
))
;
t
2
<
.
W
e
are
interested
in
finding
out
the
longest
arc
of
the
h
yperbola
that
can
be
approximate
d
and
that
the
error
function
is
the
Chebyshe
v
polynomial,
see
[2].
It
is
impossible
to
e
xactly
repres
ent
a
h
yperbola
with
a
polynomial
curv
e
[7-11].
It
can
be
represented
e
xactly
using
rational
B
´
ezier
curv
es,
a
polynomial
parametric
form
is
preferred
in
man
y
applications.
The
ability
to
represent
a
primiti
v
e
h
yperbola
is
a
must
issue
especia
lly
in
computer
graphics
and
data
and
image
processing.
Thus,
there
is
a
demand
to
find
a
parametrically
defined
polynomial
curv
e
p
6
:
t
7!
(
x
6
(
t
)
;
y
6
(
t
))
;
0
t
1
,
where
x
6
(
t
)
;
y
6
(
t
)
are
polynomials
of
de
gree
6
.
The
p
6
has
to
approximate
c
within
tolerable
error
.
In
this
paper
,
de
gree
6
parametric
curv
es
are
considered,
and
it
is
sho
wn
that
the
error
is
v
ery
small
and
the
results
are
competiti
v
e.
-
20
-
10
10
20
-
20
-
10
10
20
Figure
1.
The
Hyperbola
y
2
x
2
=
1
The
Euclidean
error
function
is
used
to
measure
the
error
between
p
6
and
c
.
The
error
is
defined
by
:
E
(
t
)
:=
q
y
2
6
(
t
)
x
2
6
(
t
)
1
:
(1)
E
(
t
)
is
replaced
by
the
follo
wing
error
function:
e
(
t
)
:=
y
2
6
(
t
)
x
2
6
(
t
)
1
:
(2)
The
approximation
problem
is
formulated
as
follo
ws.
The
appr
oximation
pr
oblem
is
to
find
p
6
:
t
7!
(
x
6
(
t
)
;
y
6
(
t
))
;
0
t
1
,
where
x
6
(
t
)
;
y
6
(
t
)
are
polynomials
of
de
gree
6
,
that
approximates
c
by
satisfying
the
follo
wing
three
conditions:
(a)
p
6
minimizes
max
t
2
[0
;
1]
j
e
(
t
)
j
,
(b)
p
6
approximates
c
with
order
twelv
e
[12],
(c)
e
(
t
)
equioscillates
thirteen
times
o
v
er
[0
;
1]
.
The
solution
to
this
problem
is
sho
wn
in
section
3
.
This
sol
ution
is
presented
in
Figure
2
and
Figure
3;
the
corresponding
error
is
sho
wn
in
Figure
4.
More
related
results
can
be
found
in
[13-15]
and
the
references
therein.
The
best
se
xtic
appr
oximation
of
...
(Abedallah
Rababah)
Evaluation Warning : The document was created with Spire.PDF for Python.
2194
r
ISSN:
2088-8708
21
-
21
21
Figure
2.
The
Hyperbola
and
the
se
xtic
approximating
B
´
ezier
curv
e
Figure
3.
Zoom
in
the
b
usy
part
of
Figure
2
0.2
0.4
0.6
0.8
1.0
t
-
0.0004
-
0.0002
0.0002
0.0004
e
H
t
L
Figure
4.
The
error
of
the
se
xtic
approximating
B
´
ezier
curv
e
2.
RESEARCH
METHOD
Let
p
6
(
t
)
=
(
x
6
(
t
)
;
y
6
(
t
))
be
a
se
xtic
polynomial
parametric
representat
ion
of
the
curv
e
c
.
In
CA
GD,
curv
es
are
presented
using
the
B
´
ezier
form,
see
Figure
5
for
possible
choice
of
the
B
´
ezier
points.
P
0
P
1
P
2
P
3
P
4
P
5
P
6
21
-
21
21
Figure
5.
Possible
B
´
ezier
points
of
the
h
yperbola
The
B
´
ezier
curv
e
p
6
(
t
)
has
the
follo
wing
form:
p
6
(
t
)
=
6
X
i
=0
p
i
B
6
i
(
t
)
=:
x
6
(
t
)
y
6
(
t
)
;
0
t
1
;
(3)
where
p
0
;
p
1
;
p
2
;
p
3
;
p
4
;
p
5
;
p
6
are
the
B
´
ezier
points,
and
the
Bernstein
polynomial
basis
of
de
gree
6
is
defined
by:
B
6
i
(
t
)
=
6
i
(1
t
)
6
i
t
i
;
i
=
0
;
:
:
:
;
6
;
t
2
[0
;
1]
:
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
2192
–
2199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2195
The
simplicity
of
this
method
should
encourage
people
w
orking
in
the
fields
of
Computer
Graphics,
Image
Processing,
CAD,
and
Data
Processing
to
adopt
it
in
their
design
and
applications.
The
symmetry
in
the
h
yperbola
is
used
to
better
locate
the
B
´
ezier
points.
W
e
be
gin
by
letting
p
0
=
(
;
)
.
The
h
yperbola
is
symmetric
around
the
y
-axis,
so,
to
obe
y
this
symmetry
,
the
point
p
6
should
ha
v
e
the
form
p
6
=
(
;
)
.
The
point
p
1
=
(
;
)
,
so,
to
obe
y
the
symmetry
,
the
point
p
5
should
ha
v
e
the
form
p
5
=
(
;
)
.
The
point
p
2
=
(
;
)
,
so,
to
obe
y
the
symmetry
,
the
point
p
4
should
ha
v
e
the
form
p
4
=
(
;
)
.
There
is
one
remaining
point;
if
this
point
lies
in
either
halv
es
of
the
plane
around
the
y
-axis,
then
the
symmetry
of
the
h
yperbola
i
s
kick
ed.
Thus,
the
point
p
3
must
lie
on
the
y
-axis
and
has
the
form
p
3
=
(0
;
!
)
.
Therefore,
the
proper
choice
for
the
B
´
ezier
points
is
p
0
=
;
p
1
=
;
p
2
=
;
p
3
=
0
!
;
p
4
=
;
p
5
=
;
p
6
=
:
(4)
The
B
´
ezier
polynomial
curv
e
p
6
(
t
)
in
(3)
is
gi
v
en
in
the
form
0
@
x
6
(
t
)
y
6
(
t
)
1
A
=
0
@
B
6
0
(
t
)
B
6
1
(
t
)
B
6
2
(
t
)
+
B
6
4
(
t
)
+
B
6
5
(
t
)
+
B
6
6
(
t
)
B
6
0
(
t
)
+
B
6
1
(
t
)
+
B
6
2
(
t
)
+
!
B
6
3
(
t
)
+
B
6
4
(
t
)
+
B
6
5
(
t
)
+
B
6
6
(
t
)
1
A
;
(5)
where
0
t
1
.
The
se
v
en
parameters
;
;
;
;
;
;
!
are
used
to
ha
v
e
the
polynomial
approximation
p
6
comply
with
the
conditions
of
the
approximation
problem;
this
is
done
in
the
follo
wing
section.
3.
RESUL
TS
AND
AN
AL
YSIS
The
v
alues
of
;
;
;
;
;
;
!
that
minimize
the
uniform
error
and
satisfy
the
conditions
of
the
approximation
problem
are
gi
v
en
in
the
follo
wing
theorem.
A
symbolic
programming
language
is
used
to
get
the
v
alues
of
the
parameters
in
(4)
and
are
rounded
in
decimal
form.
Theor
em
1:
The
B
´
ezier
curv
e
in
(5)
with
the
B
´
ezier
points
in
(4),
where
=
21
:
396696163346007
;
=
21
:
420062908119476
;
=
0
:
34812943434024657
;
=
0
:
3015887594987887
;
=
0
:
5937616113806532
;
=
0
:
705589524343983
;
!
=
0
:
1813438330271954
(6)
satisfies
the
three
conditions
of
the
Approximation
Problem.
More
precisely
,
the
error
functions
satisfy:
1
2
11
e
(
t
)
1
2
11
;
1
2
11
(2
)
E
(
t
)
1
2
11
(2
+
)
;
where
=
max
0
t
1
j
E
(
t
)
j
:
(7)
Pr
oof:
From
equation
(5),
we
get
x
6
(
t
)
=
B
6
6
(
t
)
B
6
0
(
t
)
+
B
6
5
(
t
)
B
6
1
(
t
)
+
B
6
4
(
t
)
B
6
2
(
t
)
;
y
6
(
t
)
=
B
6
6
(
t
)
+
B
6
0
(
t
)
+
B
6
5
(
t
)
+
B
6
1
(
t
)
+
B
6
4
(
t
)
+
B
6
2
(
t
)
+
!
B
6
3
(
t
)
:
Substituting
x
6
(
t
)
and
y
6
(
t
)
into
the
error
function
e
(
t
)
in
(2)
and
re
writing
the
result
in
terms
of
po
wers
of
t
we
get
the
follo
wing
equality:
e
(
t
)
=
t
12
4
2
48
+
144
2
+
120
720
+
900
2
80
!
+
480
!
1200
!
+
400
!
2
+
t
11
24
2
+
288
864
2
720
+
4320
5400
2
+
480
!
2880
!
+
7200
!
2400
!
2
+
t
10
36
2
+
288
576
2
360
+
1440
900
2
+
96
2
1032
+
2736
2
+2400
12600
+
14400
2
1560
!
+
8160
!
18600
!
+
6000
!
2
+
t
9
180
2
1440
+
2880
2
+
1800
7200
+
4500
2
260
2
+
2520
5760
2
The
best
se
xtic
appr
oximation
of
...
(Abedallah
Rababah)
Evaluation Warning : The document was created with Spire.PDF for Python.
2196
r
ISSN:
2088-8708
5400
+
23400
22500
2
+
3400
!
14400
!
+
27000
!
8000
!
2
+
t
8
465
2
+
3480
6480
2
4170
+
15480
9225
2
+
525
2
4440
+8640
2
+
8430
29880
+
22725
2
5040
!
+
17040
!
24000
!
+
6000
!
2
+
t
7
780
2
5280
+
8640
2
+
5880
18720
+
9900
2
804
2
+
5808
9504
2
9240
+
26640
15300
2
+
5040
!
13440
!
+
13200
!
2400
!
2
+
t
6
922
2
+
5460
7488
2
5460
+
13860
5850
2
+
926
2
5628
+7632
2
+
7140
16380
+
6750
2
3360
!
+
6720
!
4200
!
+
400
!
2
+
t
5
792
2
3948
+
4320
2
+
3360
6300
+
1800
2
792
2
+
3972
4320
2
3840
+
6660
1800
2
+
1440
!
1920
!
+
600
!
+
t
4
495
2
+
1980
1620
2
1320
+
1620
225
2
+
495
2
1980
+
1620
2
+1380
1620
+
225
2
360
!
+
240
!
+
t
3
220
2
660
+
360
2
+
300
180
220
2
+
660
360
2
300
+180
+
40
!
)
+
t
2
66
2
+
132
36
2
30
+
66
2
132
+
36
2
+
30
+
t
12
2
12
12
2
+
12
+
(
1
2
+
2
)
:
W
e
substitute
the
v
alues
of
;
;
;
;
;
and
!
from
(6)
into
the
last
equation
to
get
e
(
t
)
=
4096
t
12
24576
t
11
+
64512
t
10
97280
t
9
+
93024
t
8
58752
t
7
+
24752
t
6
6864
t
5
+
19305
16
t
4
1001
8
t
3
+
429
64
t
2
9
64
t
+
1
2048
;
t
2
[0
;
1]
:
Making
the
substitution
t
=
u
+1
2
reduces
the
error
function
to
the
follo
wing
form
e
(
u
)
=
1
2048
9
256
u
2
+
105
256
u
4
7
4
u
6
+
27
8
u
8
+
3
t
10
+
t
12
;
u
2
[
1
;
1]
:
The
last
form
of
e
(
u
)
coincides
with
the
monic
Chebyshe
v
polynomial
~
T
12
(
u
)
;
u
2
[
1
;
1]
,
which
is
the
unique
polynomial
of
de
gree
12
that
equioscillates
13
times
between
1
2
11
for
all
u
2
[
1
;
1]
and
has
the
least
de
viation
[12].
This
sho
ws
that
p
6
satisfies
the
conditions
of
the
approximation
problem.
The
error
formula
for
E
(
t
)
can
be
pro
v
ed
using
its
relation
to
the
error
function
e
(
t
)
.
This
pro
v
es
Theorem
1.
It
is
clear
that
the
approximation
is
the
best
uniform
approximation
from
Figure
4.
Figure
2
and
Figure
3
sho
w
the
h
yperbola
and
the
approximating
B
´
ezier
curv
e,
Figure
4
sho
ws
the
corresponding
error
,
and
Figure
6
sho
ws
the
Euclidean
error
.
4.
CONCLUSION
In
this
paper
,
the
best
uniform
approximation
of
the
h
yperbola
with
parametrically
defined
polyno-
mial
curv
es
of
de
gree
6
are
e
xplicitly
gi
v
en.
The
error
function
equioscillates
13
times;
the
approximation
order
is
12
.
The
method
of
construction
demonstrates
the
ef
ficienc
y
and
simplicity
of
the
approximation
method.
The
approximation
intersects
the
h
yperbola
12
times
with
maximum
error
2
:
4
10
4
.
Reflecting
the
upper
branch
of
the
h
yperbola
around
the
x
-axis
gi
v
es
the
l
o
wer
branch.
The
h
yperbola
is
sho
wn
in
Figure
7.
Note
t
hat
the
points
p
1
(
p
5
)
and
p
2
(
p
4
)
i
n
both
branches
are
v
ery
close
to
each
other
and
can
not
be
distin-
guished
from
each
other
.
The
results
in
this
paper
can
be
used
to
impro
v
e
the
results
obtained
in
[16-23]
see
also
the
results
in
[24,
25].
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
2192
–
2199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2197
0.2
0.4
0.6
0.8
1.0
t
-
0.0002
-
0.0001
0.0001
0.0002
E
H
t
L
Figure
6.
The
Euclidean
error
of
the
se
xtic
B
´
ezier
curv
e
21
-
21
21
-
21
-
21
21
Figure
7.
Both
branches
of
the
Hyperbola
using
tw
o
B
´
ezier
curv
es
A
CKNO
WLEDGMENTS
The
authors
o
we
thanks
for
the
re
vie
wers
for
in
v
aluable
sugges
tions
to
impro
v
e
an
e
arlier
v
ersion
of
this
paper
.
This
research
w
as
funded
by
Jordan
Uni
v
ersity
of
Science
and
T
echnology
.
REFERENCES
[1]
A.
Rababah,
”Best
se
xtic
approximation
of
circular
arcs
with
thirteen
equioscillations,
”
Pro-
ceedings
of
the
Jangjeon
Mathematical
Societ
y
,
v
ol.
21,
Issue
1,
pp.
111–123,
2018.
DOI:
http://doi.or
g/10.17777/pjms2018.21.1.111
[2]
A.
Rababah,
”The
Best
Quadratic
Approximation
of
Hyperbola
with
Order
F
our
,
”
Journal
of
Inequalities
and
Special
Functions
,
v
ol.
7,
Issue
3,
pp.
1–11,
2016.
[3]
M.
Floater
,
”
High
order
approximation
of
conic
sections
by
quadratic
splines,
”
Computer
Aided
Geomet-
ric
Design
,
v
ol.
12,
pp.
617–637,
1995.
[4]
C.
de
Boor
.,
et
al.,
”High
accurac
y
geometric
Hermite
interpolation,
”
Computer
Aided
Geometric
Design
,
v
ol.
4,
pp.
269–278,
1988.
[5]
A.
Rababah,
”T
aylor
theorem
for
planar
curv
es,
”
Proceedings
of
the
American
Mathematical
Society
,
v
ol.
119,
Issue
3,
pp.
803–810,
1993.
[6]
A.
Rababah,
”High
accurac
y
Hermite
approximation
for
space
curv
es
in
<
d
,
”
Journal
of
Mathematical
Analysis
and
Applications
,
v
ol.
325,
Issue
2,
pp.
920–931,
2007.
[7]
K.
H
¨
ollig
and
J.
H
¨
orner
,
”Approximation
and
Modeling
with
B-Splines,
”
SIAM,
T
itles
i
n
Applied
Math-
ematics
,
v
ol.
132,
2013.
[8]
H.
Prautzsch.,
et
al.,
”B
´
ezier
and
B-Spline
T
echniques,
”
Springer
2002.
The
best
se
xtic
appr
oximation
of
...
(Abedallah
Rababah)
Evaluation Warning : The document was created with Spire.PDF for Python.
2198
r
ISSN:
2088-8708
[9]
P
.
B
´
ezier
,
”The
mathematical
basis
of
the
UNISURF
CAD
system,
”
Butterw
orth-Heinemann
Ne
wton
,
MA,
USA,
ISBN
0-408-22175-5,
1986.
[10]
Carl
de
Boor
,
A
Practical
Guide
to
Splines,
Series
T
itle:
Applied
Mathematical
Sciences,
V
ol
27,
Springer
-
V
erlag
Ne
w
Y
ork,
1978.
[11]
Gerald
F
arin,
Curv
es
and
surf
aces
for
computer
-aided
geometric
design:
a
practical
guide.
Else
vier
,
2014.
[12]
J.
Rice,
”The
approximation
of
functions,
”
linear
theory
,
Addison-W
esle
y
,
v
ol.
1,
1964.
[13]
Y
.
J.
Ahn
and
C.
Hof
fmann,
”Circle
approximation
using
LN
B
´
ezier
curv
es
of
e
v
en
de
gree
and
its
ap-
plication,
”
Journal
of
Mathematical
Analysis
and
Applications
,
v
ol.
410
Issue
1,
pp.
257–266,
February
2014.
[14]
S.
W
.
Kim
and
Y
.
J.
Ahn,
”Circle
approximation
by
quartic
G
2
spline
using
alternation
of
error
function,
”
J.
KSIAM
,
v
ol.
17,
Issue
5,
pp.
171–179,
2013.
[15]
A.
Rababah.,
et
al.,
”Multiple
De
gree
Reduction
and
Ele
v
ation
of
B
´
ezier
Curv
es
Using
Jacobi-Bernstein
Basis
T
ransformations,
”
Numerical
Functional
Analysis
and
Opt
imization
,
v
ol.
28,
Issue
9-10,
pp.
1179–
1196,
2007.
[16]
S.
Alabed,
”Performance
Analysis
of
Dif
ferential
Beamforming
in
Decentralized
Netw
orks,
”
Interna-
tional
Journal
of
Electrical
and
Computer
Engineering
(IJECE)
,
v
ol.
8
Issue
3,
pp.
1692-1700,
2018.
DOI:
http://doi.or
g/10.11591/ijece.v8i3.pp1692-1700
[17]
C.
Suw
annapong
and
C.
Khunboa,
”An
Approximation
Delay
betwee
n
Consecuti
v
e
Requests
for
Con-
gestion
Control
in
Unicast
CoAP-based
Group
Communication,
”
International
Journal
of
Electrical
and
Computer
Engineering
(IJECE)
,
v
ol.
9,
Issue
3,
June
2019.
DOI:
http://doi.or
g/10.11591/ijece.v9i3.
[18]
Bhar
g
a
v
Bhatkalkar
,
Abhishek
Joshi,
Srikanth
Prabhu,
Sulatha
Bhandary
,
Automated
fundus
image
qual-
ity
assessment
and
se
gmentation
of
optic
disc
using
con
v
olutional
neural
netw
orks.
International
Journal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.10,
No.1,
2020,
pp.
816
827,
ISSN:
2088-8708,
DOI:
10.11591/ijece.v10i1.pp816-827.
[19]
Rodiah,
Sarifuddin
Madenda,
Diana
T
ri
Susetianigtias,
De
wi
Agushinta
Rahayu,
EtySutanty
,
Optic
Disc
and
Macula
Localization
from
Retinal
Optical
Coherence
T
omograph
y
and
Fundus
Image,
International
Journal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.8,
No.6,
2018,
pp.
5050-5060,
ISSN:
2088-
8708,
DOI:
10.11591/ijece.v8i6.pp5050-5060.
[20]
G.
Da
w
ay
,
Hana
H.
kareem,
Ahmed
Rafid
Hashim,
Pupil
Detection
Based
on
Color
Dif
ference
and
Circu-
lar
Hough
T
ransformHazim,
International
Journal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.8,
No.5,
2018,
pp.
3278-3284,
ISSN:
2088-8708,
DOI:
10.11591/ijece.v8i5.pp3278-3284.
[21]
A.
R
ababah,
L-2
de
gree
reduction
of
triangular
B
´
ezier
surf
aces
with
common
tangent
planes
at
v
er
-
tices.
International
Journal
of
Computational
Geometry
&
Applications,
V
ol.
15,
No.
5
(2005)
477-490.
http://www
.w
orldscinet.com/ijcg
a/15/1505/S02181959051505.html
[22]
A.
Rababah,
Distances
with
rational
triangular
B
´
ezier
surf
aces.
Applied
Mathematics
and
Computation
160,
(
2005
)
379-386.
http://www
.else
vier
.com/locate/amc
[23]
A.
Rababah
and
Moath
Jaradat,
Approximating
of
fset
Curv
es
using
B
´
ezier
curv
es
with
high
accurac
y
,
International
Journal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.10,
No.2,
2020
(P
AR
T
I).
[24]
A.
Rababah,
”Approximation
v
on
K
urv
en
mit
Polynomen
und
Splines,
”
Ph.
Dissertation,
Stuttg
art
Uni-
v
ersit
¨
at
,
W
est
German
y
,
1992.
[25]
A.
Rababah,
The
best
quintic
Chebyshe
v
approximation
of
circular
arcs
of
order
ten,
International
Journal
of
Electrical
and
Computer
Engineering
(IJECE)V
ol.
9,
No.
5,
October
2019,
pp.
3779-3785,
ISSN:
2088-
8708,
DOI:
10.11591/ijece.v9i5.pp3779-3785.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
2192
–
2199
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2199
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
echnol
ogy
.
He
is
w
orking
in
the
field
of
Computer
Aided
Geometric
Design,
abbre
viated
CA
GD.
In
particular
,
his
rese
arch
is
on
de
gree
raising
and
reduction
of
B
´
ezier
curv
es
and
surf
aces
with
geometric
boundary
conditions,
Bernstein
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
classical
(local
T
aylor
,
Hermite)
methods.
He
pro
v
ed
the
fol-
lo
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
wi
th
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
also
pro
v
ed
for
space
curv
es.
Professor
Rababah
is
also
do-
ing
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’
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
Esra’a
Rababah
graduated
from
Jordan
Uni
v
ersity
of
Science
and
T
echnology
.
She
obtained
both
Bachelor
and
Master
De
grees
in
Mathematics
with
research
interests
in
the
field
of
numerical
analy-
sis,
a
pproximation
theory
,
and
B
´
ezier
techniques.
She
w
ork
ed
as
teacher
assistant
at
Jordan
Uni
v
er
-
sity
of
Science
and
T
echnology
.
Further
info
can
be
found
on
her
homepage
at
ResearchGate:
https://www
.researchg
ate.net/profile/Esraa
Rababah
The
best
se
xtic
appr
oximation
of
...
(Abedallah
Rababah)
Evaluation Warning : The document was created with Spire.PDF for Python.