Inter
national
J
our
nal
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
r
3779
The
best
quintic
chebyshe
v
appr
oximation
of
cir
cular
ar
cs
of
order
ten
Abedallah
Rababah
Department
of
Mathematics,
Jordan
Uni
v
ersity
of
Science
and
T
echnology
,
Jordan
Article
Inf
o
Article
history:
Recei
v
ed
Feb
19,
2019
Re
vised
Apr
4,
2019
Accepted
Apr
9,
2019
K
eyw
ords:
High
performance
computing
B
´
ezier
curv
es
Quintic
approximation
Circular
arc
High
accurac
y
Approximation
order
CAD
ABSTRA
CT
Mathematically
,
circles
are
represented
by
trigonometric
parametric
equations
and
im-
plicit
equations.
Both
forms
are
not
proper
for
computer
applications
and
CAD
sys-
tems.
In
this
paper
,
a
quintic
polynomial
approximation
for
a
circular
arc
is
presented.
This
approximation
is
set
so
that
the
error
function
is
of
de
gree
10
rather
than
6
;
the
Chebyshe
v
error
function
equioscillates
11
times
rather
than
7
;
the
approximation
order
is
10
rather
than
6
.
The
method
approximates
more
than
the
full
circle
with
Chebyshe
v
uniform
error
of
1
=
2
9
.
The
e
xamples
sho
w
t
he
competence
and
simplicity
of
the
proposed
approximation,
and
that
it
can
not
be
impro
v
ed.
Copyright
c
2019
Insitute
of
Advanced
Engineeering
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
1.
INTR
ODUCTION
Only
NURBS
(Non
Uniform
Rational
B-Splines)
are
capable
to
properly
symbolize
circles.
NURBS
ha
v
e
mathematical
frame
and
harnessing
them
depends
on
mathematical
kno
wledge
and,
in
significant
cases,
demands
the
potential
of
emplo
ying
geometric
and
analytic
concepts.
Ho
we
v
er
,
circles
are
directed
to
be
used
by
people
of
limited
and
e
xclusi
v
e
kno
wledge
of
mathem
atical
notions
and
processes
lik
e
computer
graphics
designers,
stylers,
animators,
computer
aided
designers,
and
engineers.
Dra
wing
a
circle
in
an
y
CAD
system
is
the
primary
alphabet
in
an
y
softw
are.
So,
it
is
v
ery
genuine
to
ha
v
e
the
circl
e
as
a
primiti
v
e
and
as
a
b
uilt-
in-function
in
the
softw
are.
A
circle
can
be
represented
using
rational
B
´
ezier
curv
es
and
can
be
approximated
by
polynomial
curv
es.
Therefore,
approximating
a
circular
arc
by
polynomial
curv
es
with
highest
possible
accurac
y
is
a
v
ery
important
matter
[1].
W
e
treat
the
circular
arc
c
:
t
7!
(cos(
t
)
;
sin(
t
))
;
t
,
see
Figure
1,
to
be
approximated
by
a
polynomial
curv
e
with
superior
uniform
approximation.
T
o
come
to
this
consequence,
the
geometric
symmetries
of
the
circle
are
used
to
f
airly
choose
the
B
´
ezier
points
in
order
to
symbolize
the
quintic
B
´
ezier
curv
e
that
has
highest
approximation
order
of
10
.
The
circle
c
is
approximated
in
this
paper
using
a
quintic
parametrically
defined
polynomial
curv
e
p
:
t
7!
(
x
(
t
)
;
y
(
t
))
;
0
t
1
,
where
x
(
t
)
;
y
(
t
)
are
polynomials
of
de
gree
5
,
that
approximates
c
with
least
de
viation.
Man
y
researchers
ha
v
e
tackled
this
issue
using
dif
ferent
de
grees,
norms,
and
methods,
see
for
e
xample
[2,
3,
4,
5,
6,
7,
8,
9]
and
the
references
therein.
In
[10],
methods
for
approximating
circular
arcs
using
quintic
polynomial
curv
es
with
dif
ferent
bound-
ary
conditions
are
conside
red.
The
results
of
our
method
in
this
paper
outperform
the
results
in
[10],
J
ournal
homepage:
http://iaescor
e
.com/journals/inde
x.php/IJECE
Evaluation Warning : The document was created with Spire.PDF for Python.
3780
r
ISSN:
2088-8708
are
optimal,
and
can
not
be
impro
v
ed.
The
error
function
to
measure
the
distance
between
p
and
c
is
the
follo
wing
radial
basis
distance:
E
(
t
)
:=
p
x
2
(
t
)
+
y
2
(
t
)
1
:
(1)
The
radial
distance
formula
contains
radicals
and
thus
will
be
replaced
by
the
follo
wing
de
viation
measure:
e
(
t
)
:=
x
2
(
t
)
+
y
2
(
t
)
1
:
(2)
Since
the
error
function
e
(
t
)
=
0
represents
the
implicit
equation
of
the
circle;
thus
it
is
the
suitable
measure
to
test
if
x
(
t
)
and
y
(
t
)
satisfy
the
equation
of
the
circle
and
to
measure
the
error
.
q
-
q
1
Figure
1.
A
circular
arc
Among
all
possible
kinds
of
approximations,
the
best
uniform
approximation
yields
the
most
accurate
approximation.
Unfortunately
,
in
the
li
terature,
there
is
no
method
to
find
the
best
uniform
approximation
of
de
gree
fi
v
e
that
approximates
with
order
6
[11].
In
this
paper
,
the
quintic
best
uniform
approximation
is
found
that
approximates
with
order
10
rather
that
6
.
This
is
a
significant
impro
v
ement
o
v
er
the
order
of
the
Chebyshe
v
and
Borel
theorems
[11].
In
particular
,
the
approximation
problem
that
is
considered
in
this
paper
is
to
locate
a
polynomial
curv
e
p
:
t
7!
(
x
(
t
)
;
y
(
t
))
;
0
t
1
,
where
x
(
t
)
;
y
(
t
)
are
quintic
polynomials,
that
”mimics”
c
and
minimizes
max
t
2
[0
;
1]
j
e
(
t
)
j
.
Moreo
v
er
,
the
follo
wing
conditions
are
fulfilled:
(a)
e
(
t
)
equioscillates
11
times
o
v
er
[0
;
1]
rather
than
7
,
(b)
p
approximates
c
with
order
10
rather
than
6
.
These
conditions
are
used
to
locate
the
B
´
ezier
points
and
to
get
the
v
alues
of
the
parameters
that
are
utilized
to
satisfy
the
geometric
conditions
of
the
circular
arc.
F
or
more
on
these
topics,
see
[1].
T
o
achie
v
e
the
conditions
of
the
issue
of
the
approximation,
the
follo
wing
feature
of
the
Chebyshe
v
polynomials
is
used.
Namely
,
the
monic
Chebyshe
v
polynomial
~
T
10
(
u
)
;
u
2
[
1
;
1]
,
gi
v
en
by
~
T
10
(
u
)
=
1
512
+
25
256
u
2
25
32
u
4
+
35
16
u
6
5
2
u
8
+
u
10
;
u
2
[
1
;
1]
(3)
is
the
unique
polynomial
of
de
gree
10
that
equioscillates
11
times
between
1
2
9
for
all
u
2
[
1
;
1]
and
has
the
least
de
viation
from
the
x
-axis
[11].
Since
the
uniform
error
(for
e
(
t
)
)
equals
2
9
,
so,
we
allo
w
the
angle
to
be
as
lar
ge
as
possible
in
order
to
approximate
the
lar
gest
circular
arc
with
this
specified
error
.
Ther
eafter
,
this
angle
has
to
be
scaled
by
a
f
actor
that
also
combined
with
a
reduction
in
the
uniform
error
,
see
the
last
conclusions
and
open
problems’
section.
Int
J
Elec
&
Comp
Eng,
V
ol.
9,
No.
5,
October
2019
:
3779
–
3785
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
3781
2.
RESEARCH
METHOD
The
curv
e
p
(
t
)
is
gi
v
en
in
B
´
ezier
form
in
order
to
use
the
B
´
ezier
technique
[13].
See
Figure
2
for
possible
B
´
ezier
points
of
a
quintic
B
´
ezier
curv
e.
A
B
´
ezier
curv
e
p
(
t
)
of
de
gree
5
is
gi
v
en
by
p
(
t
)
=
5
X
i
=0
p
i
B
5
i
(
t
)
=:
x
(
t
)
y
(
t
)
;
0
t
1
:
(4)
The
points
p
0
;
p
1
;
p
2
;
p
3
;
p
4
and
p
5
are
called
the
B
´
ezier
points,
and
the
polynomials
B
5
0
(
t
)
=
(1
t
)
5
;
B
5
1
(
t
)
=
5
t
(1
t
)
4
;
B
5
2
(
t
)
=
10
t
2
(1
t
)
3
;
B
5
3
(
t
)
=
10
t
3
(1
t
)
2
;
B
5
4
(
t
)
=
5
t
4
(1
t
)
and
B
5
5
(
t
)
=
t
5
are
the
well-
kno
wn
quintic
Bernstein
polynomials.
Since
our
purpose
is
to
represent
the
arc
with
a
polynomial
curv
e
with
the
least
possible
error
,
it
is
not
substantial
for
the
errors
to
tak
e
place
at
the
endpoints
or
else
where;
it
is
significant
to
ensure
that
this
anno
yance
is
as
lo
w
as
concei
v
able
no
matter
where
the
error
occurs.
Our
approach
considers
lessen
the
wrongdoing
o
v
er
all
of
the
se
gment
[0
;
1]
.
T
o
e
xplore
the
B
´
ezier
form
approximation
of
a
circular
a
rc,
a
careful
selection
of
locations
of
the
B
´
ezier
points
should
be
well-done.
These
locations
are
substantial
to
earn
the
con
v
enient
curv
e
that
redeems
the
approximation
conditions.
Based
on
the
symm
etry
property
of
the
circle,
the
right
choice
for
the
be
ginning
control
point
p
0
should
obe
y
the
follo
wing
form:
p
0
=
(
0
cos(
)
;
0
sin(
))
,
where
v
alues
of
0
and
0
could
b
ut
should
not
be
the
same.
Similarly
,
for
symmetry
reasoning,
the
v
alid
option
for
the
end
control
point
p
5
is
p
5
=
(
0
cos(
)
;
0
sin(
))
.
Set
p
1
=
(
a
1
;
b
1
)
,
then
the
point
p
4
has
to
be
selected
to
satisfy
the
form
p
4
=
(
a
1
;
b
1
)
.
Set
the
point
p
2
=
(
a
2
;
b
2
)
,
then
the
point
p
3
has
to
be
selected
to
satisfy
the
form
p
3
=
(
a
2
;
b
2
)
.
Using
the
substitutions
a
0
=
0
cos(
)
;
b
0
=
0
sin(
)
,
then
the
con
v
enient
choices
for
the
B
´
ezier
points
ha
v
e
to
be,
see
Figure
2,
p
0
=
a
0
b
0
;
p
1
=
a
1
b
1
;
p
2
=
a
2
b
2
;
p
3
=
a
2
b
2
;
p
4
=
a
1
b
1
;
p
5
=
a
0
b
0
:
(5)
It
will
be
apparent
that
there
are
more
than
one
solution;
the
consonant
solution
of
best
approximation
be
gins
in
the
second
quadrant
and
ends
in
the
fourth
quadrant
counter
clockwise.
Therefore,
in
order
to
ha
v
e
the
B
´
ezier
curv
e
p
be
gin
in
the
second
quadrant,
go
counter
clockwise
through
fourth,
third,
first,
second,
and
ends
in
the
fourth
quadrant
as
the
circular
arc
c
,
the
follo
wing
stipulations
should
be
satisfied:
a
0
;
a
1
;
b
1
;
b
2
<
0
;
a
2
;
b
0
>
0
:
(6)
Emplo
y
the
B
´
ezier
points
in
(5)
in
the
B
´
ezier
curv
e
p
(
t
)
in
(4)
to
obtain:
p
(
t
)
=
x
(
t
)
y
(
t
)
=
a
0
B
5
0
(
t
)
+
B
5
5
(
t
)
+
a
1
B
5
1
(
t
)
+
B
5
4
(
t
)
+
a
2
B
5
2
(
t
)
+
B
5
3
b
0
B
5
0
(
t
)
B
5
5
(
t
)
+
b
1
B
5
1
(
t
)
B
5
4
(
t
)
+
b
2
B
5
2
(
t
)
B
5
3
(
t
)
;
0
t
1
:
(7)
The
B
´
ezier
curv
e
is
settled
by
the
6
parameters
a
0
;
a
1
;
a
2
;
b
0
;
b
1
;
b
2
[12].
These
parameters
are
hired
to
get
the
best
uni
form
approximation.
W
e
w
ant
to
impose
the
conditions
on
the
polynomial
curv
e
p
;
the
polynomials
x
(
t
)
and
y
(
t
)
are
substituted
into
e
(
t
)
.
This
leads
to
a
polynomail
of
de
gree
10
that
is
solv
ed
using
a
computer
algebra
system.
These
proceedings
are
demonstrated
in
the
ne
xt
section.
3.
RESUL
TS
AND
AN
AL
YSIS
The
v
alues
of
a
0
;
a
1
;
a
2
;
b
0
;
b
1
;
b
2
that
satisfy
the
conditions
of
the
approximation
problem
are
speci-
fied
numerically
and
rounded
fittingly
in
the
follo
wing
theorem.
Theor
em
1:
The
B
´
ezier
curv
e
(7)
together
with
the
B
´
ezier
points
in
(5)
and
the
v
alues
of
the
parameters
a
0
;
a
1
;
a
2
;
b
0
;
b
1
,
and
b
2
gi
v
en
by
a
0
=
0
:
978179455549407
;
a
1
=
1
:
338058000836784
;
a
2
=
2
:
365283682287911
;
(8)
b
0
=
0
:
21241016393227463
;
b
1
=
1
:
6287837424896061
;
b
2
=
2
:
4356328876380307
(9)
achie
v
es
the
follo
wing
three
conditions:
p
minimizes
the
uniform
norm
of
the
error
function
max
t
2
[0
;
1]
j
e
(
t
)
j
and
approximates
c
with
order
10
,
and
the
error
function
e
(
t
)
equioscillates
11
times
in
[0
;
1]
.
The
best
quintic
c
hebyshe
v
appr
oximation
of
...
(Abedallah
Rababah)
Evaluation Warning : The document was created with Spire.PDF for Python.
3782
r
ISSN:
2088-8708
The
error
functions
fulfill:
1
2
9
e
(
t
)
1
2
9
;
1
2
9
(2
)
E
(
t
)
1
2
9
(2
+
)
;
where
=
max
0
t
1
j
E
(
t
)
j
2
10
;
8
t
2
[0
;
1]
:
(10)
Pr
oof:
W
e
be
gin
by
considering
the
polynomials
x
(
t
)
and
y
(
t
)
in
equation
(7)
and
substituting
them
into
the
error
function
e
(
t
)
in
(2).
Disposition
the
phrase
and
performing
se
v
eral
simplifications
gi
v
es
the
follo
wing
equation:
e
(
t
)
=
4
b
2
0
40
b
0
b
1
+
100
b
2
1
+
80
b
0
b
2
400
b
1
b
2
+
400
b
2
2
t
10
+
20
b
2
0
+
200
b
0
b
1
500
b
2
1
400
b
0
b
2
+
2000
b
1
b
2
2000
b
2
2
t
9
+
25
a
2
0
+
150
a
0
a
1
+
225
a
2
1
100
a
0
a
2
300
a
1
a
2
+
100
a
2
2
+
65
b
2
0
570
b
0
b
1
+
1225
b
2
1
+
1060
b
0
b
2
4500
b
1
b
2
+
4100
b
2
2
t
8
+
100
a
2
0
600
a
0
a
1
900
a
2
1
+
400
a
0
a
2
+
1200
a
1
a
2
400
a
2
2
140
b
2
0
+
1080
b
0
b
1
1900
b
2
1
1840
b
0
b
2
+
6000
b
1
b
2
4400
b
2
2
t
7
+
200
a
2
0
+
1100
a
0
a
1
+
1500
a
2
1
700
a
0
a
2
1900
a
1
a
2
+
600
a
2
2
+
220
b
2
0
1420
b
0
b
1
+
2000
b
2
1
+
2100
b
0
b
2
5100
b
1
b
2
+
2600
b
2
2
t
6
+
250
a
2
0
1200
a
0
a
1
1350
a
2
1
+
700
a
0
a
2
+
1500
a
1
a
2
400
a
2
2
254
b
2
0
+
1320
b
0
b
1
1450
b
2
1
1540
b
0
b
2
+
2700
b
1
b
2
800
b
2
2
t
5
+
210
a
2
0
+
830
a
0
a
1
+
700
a
2
1
420
a
0
a
2
600
a
1
a
2
+
100
a
2
2
+
210
b
2
0
850
b
0
b
1
+
700
b
2
1
+
700
b
0
b
2
800
b
1
b
2
+
100
b
2
2
t
4
+
120
a
2
0
360
a
0
a
1
200
a
2
1
+
140
a
0
a
2
+
100
a
1
a
2
120
b
2
0
+
360
b
0
b
1
200
b
2
1
180
b
0
b
2
+
100
b
1
b
2
)
t
3
+
45
a
2
0
+
90
a
0
a
1
+
25
a
2
1
20
a
0
a
2
+
45
b
2
0
90
b
0
b
1
+
25
b
2
1
+
20
b
0
b
2
t
2
+
10
a
2
0
10
a
0
a
1
10
b
2
0
+
10
b
0
b
1
t
+
(
a
2
0
+
b
2
0
1)
:
The
approximation
conditions
are
satisfied
if
the
error
function
is
equalized
with
the
polynomial
of
least
de
viation
among
all
monic
polynomials
of
de
gree
10
.
So,
the
last
equation
which
e
x
emplifies
the
error
function
has
to
be
equalized
with
the
Chebys
he
v
polynomial
of
first
kind
of
de
gree
10
,
~
T
10
(2
t
1)
=
512
.
W
e
kno
w
that
~
T
10
(
u
)
=
cos(10
arccos(
u
))
;
u
2
[
1
;
1]
is
the
unique
monic
polynomial
of
de
gree
10
that
has
the
least
de
viation.
It
is
gi
v
en
by
equation
(3),
see
[11].
Comparing
the
coef
ficients
of
equal
po
wers
of
both
sides
and
using
the
utilities
of
the
computer
algebra
system
in
Mathematica,
the
solution
that
fulfills
the
conditions
in
(6)
is
established.
Unfortunately
,
the
solution
is
a
collection
of
length
y
fractions
and
radicals
that
is
impractical
to
write
do
wn
the
v
alues
of
the
parameters
in
this
paper
,
so,
we
write
them
in
decimal
forms
in
equat
ions
(8)
and
(9).
This
sho
ws
that
p
fulfills
the
three
conditions
of
the
approximation
problem.
T
o
pro
v
e
the
error
formula
for
E
(
t
)
,
the
relation
to
e
(
t
)
is
established.
The
error
function
e
(
t
)
minimized
is
link
ed
to
the
radial
error
E
(
t
)
by
the
formulation:
e
(
t
)
=
x
2
(
t
)
+
y
2
(
t
)
1
=
(
p
x
2
(
t
)
+
y
2
(
t
)
+
1)
(
p
x
2
(
t
)
+
y
2
(
t
)
1)
=
(2
+
E
(
t
))
E
(
t
)
:
Thus
E
(
t
)
=
e
(
t
)
2
+
E
(
t
)
:
Substituting
the
bounds
for
e
(
t
)
gi
v
es
1
2
9
(2
)
E
(
t
)
1
2
9
(2
+
)
;
where
=
max
0
t
1
j
E
(
t
)
j
2
10
;
t
2
[0
;
1]
:
This
completes
the
proof
of
Theorem
1.
Int
J
Elec
&
Comp
Eng,
V
ol.
9,
No.
5,
October
2019
:
3779
–
3785
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
3783
The
circular
arc
and
the
approximating
B
´
ezier
curv
e
are
plotted
in
Figure
2
.
The
resulting
error
between
the
curv
e
and
the
approximation
is
not
identified
by
the
human
e
ye
which
is
clear
from
figure
of
the
corresponding
error
plotted
in
Figure
3
.
-
q
P
0
P
5
P
1
P
2
P
3
P
4
1
Figure
2.
Circular
arc
and
its
quintic
B
´
ezier
curv
e
in
Theorem
1
.
0.2
0.4
0.6
0.8
1.0
-
0.0010
-
0.0005
0.0005
0.0010
Figure
3.
Radial
Error
of
the
quintic
B
´
ezier
curv
e
in
Theorem
1
.
The
resulting
B
´
ezier
curv
e
re
v
eals
a
brilliant
positioning
of
the
B
´
ezier
points
to
embrace
more
than
a
whole
circle
whilst
possessing
the
Chebyshe
v
error
.
W
e
could
not
foresee
a
quintic
polynomial
to
approximate
more
than
the
full
circle
further
accurately
than
this
approximation.
4.
CONCLUSION
In
this
article,
quintic
approximation
of
the
circle
is
established.
The
approximation
fulfills
e
xtremely
conclusi
v
e
circumstances.
Unlik
e
the
classical
approximation
that
a
w
ards
order
of
approximation
of
6
,
this
approximation
has
order
of
approximation
of
10
;
this
is
an
superb
acquisition.
Moreo
v
er
,
in
the
significance
of
the
Chebyshe
v
norm,
this
approximation
is
the
best
and
can
not
be
impro
v
ed.
The
error
function
equioscillates
11
times
rather
than
7
.
The
numerical
e
xamples
re
v
eal
ho
w
ef
ficient
this
method
is.
The
approximation
intersects
the
circular
arc
10
times
with
maximum
error
2
9
and
thus
outperforming
an
y
other
approximation.
Further
research
and
in
v
estig
ations
can
be
accomplished
by
considering
the
follo
wing
issues:
(a)
Finding
the
quintic
best
polynomial
approximation
for
other
curv
es
that
are
widely
used
in
the
computer’
s
applications
lik
e
the
h
yperbolas
and
the
spirals.
(b)
Using
the
qui
ntic
best
polynomial
approximation
in
this
paper
to
find
the
of
fset
curv
e
approximations
of
the
B
´
ezier
curv
es.
(c)
Finding
the
quintic
best
polynomial
approximation
with
geometric
conditions,
G
1
-
and
G
2
-conditions
at
the
end
points.
The
best
quintic
c
hebyshe
v
appr
oximation
of
...
(Abedallah
Rababah)
Evaluation Warning : The document was created with Spire.PDF for Python.
3784
r
ISSN:
2088-8708
(d)
Applying
the
method
in
this
paper
to
approximate
the
bit
error
rate
(BER)
e
xpressions
of
the
dif
ferential
M-PSK
modulation
[14].
(e)
Finding
the
quintic
best
polynomial
approximation
to
shorter
arcs
of
the
circle
that
has
an
error
reduced
according
to
the
length
of
the
arc
relati
v
e
to
the
gi
v
en
approximation
to
more
than
the
whole
circle.
(f)
Using
the
res
ults
in
this
paper
to
obtain
the
optimal
delays
between
consecuti
v
e
requests
(DCR)
in
LoW
-
P
AN
border
routers
to
limit
request
send
rates
[15].
(g)
Making
a
v
alid
conjecture
similar
to
the
conjecture
in
[16,
17]
that
impro
v
es
the
results
of
Chebyshe
v’
s
and
Borel’
s
theorems
to
approximate
a
curv
e
with
2
n
+
1
equioscillations
rather
than
for
functions
with
n
+
2
equioscillations.
This
is
a
no
v
el
break
through
and
treasure
tro
v
e
in
the
field
of
approximation
theory;
ho
we
v
er
,
this
is
a
challenging
issue
and
our
today’
s
kno
wledge
is
not
rich
enough
to
tackle
it.
A
CKNO
WLEDGEMENT
The
author
o
wes
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
.
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ollig,
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y
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urv
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und
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art
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ersit
¨
at
,
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est
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,
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[17]
A.
Rababah,
”T
aylor
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for
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curv
es,
”
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of
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Mathematical
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pp.
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Int
J
Elec
&
Comp
Eng,
V
ol.
9,
No.
5,
October
2019
:
3779
–
3785
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
3785
BIOGRAPHY
OF
A
UTHORS
Abedallah
Rababah
is
a
professor
of
Mathematics
at
Jordan
Uni
v
ersity
of
Science
and
T
ech-
nology
.
He
is
w
orking
in
the
field
of
Computer
Aided
Geometric
Design,
abbre
viated
CA
GD.
In
particular
,
his
research
is
on
de
gree
raising
and
reduction
of
B
´
ezier
curv
es
and
s
urf
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
follo
wing
conjec-
ture
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
also
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
uniform
approxi-
mations.
Since
1992,
He
has
been
teaching
at
German,
Jordania
n,
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
The
best
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appr
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of
...
(Abedallah
Rababah)
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