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[
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[
2
]
.
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ter
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itab
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ito
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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:
2
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8
8
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8
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8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
2
,
Ap
r
il
20
25
:
1
8
1
7
-
1
8
2
5
1818
p
r
o
m
is
in
g
tech
n
o
l
o
g
ical
a
n
d
s
cien
tific
b
r
ea
k
th
r
o
u
g
h
s
.
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a
r
esu
lt,
we
p
r
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id
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wa
y
s
to
co
n
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o
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th
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cu
r
r
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ch
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ig
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s
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e
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d
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g
to
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en
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d
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ac
h
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lear
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in
g
alg
o
r
ith
m
s
[
3
]
.
On
e
o
f
th
e
m
o
s
t
p
o
p
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lar
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tr
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f
o
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s
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in
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tio
u
s
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s
es
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q
u
ar
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tin
e.
So
cial
is
o
latio
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m
ea
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wh
ile
f
o
cu
s
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wer
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n
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ter
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tio
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in
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lace
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at
th
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ay
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f
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te
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y
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ir
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.
I
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Ma
r
ch
2
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m
o
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MCO)
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Ac
co
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4
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o
latio
n
o
p
er
atio
n
s
an
d
th
e
u
s
ag
e
o
f
ap
p
licatio
n
s
f
o
r
s
p
r
e
ad
in
g
awa
r
en
ess
in
p
u
b
lic.
Dep
en
d
i
n
g
o
n
th
e
ch
o
s
en
s
ce
n
ar
io
,
g
eo
f
e
n
cin
g
allo
ws
an
in
d
iv
id
u
al
o
r
o
r
g
an
iz
atio
n
to
in
au
g
u
r
ate
n
o
tice
as
well
as
war
n
in
g
s
f
o
r
eith
er
tr
a
n
s
g
r
ess
in
g
o
r
en
t
er
in
g
with
in
th
at
s
p
ec
if
ie
d
ar
ea
o
f
g
e
o
f
en
ce
.
A
g
eo
g
r
a
p
h
ic
p
o
s
itio
n
in
g
s
y
s
tem
(
GPS
)
en
ab
le
d
d
e
v
ice,
a
p
p
licatio
n
,
an
d
p
er
s
o
n
ar
e
r
e
q
u
i
r
ed
f
o
r
g
eo
f
e
n
cin
g
.
Static
g
eo
f
en
ce
s
ar
e
f
ix
e
d
f
o
r
an
y
p
ar
ticu
la
r
d
e
v
ice
th
at
is
p
lace
d
at
a
s
tatio
n
ar
y
lo
ca
tio
n
[
5
]
.
An
ad
d
itio
n
al
d
eg
r
ee
o
f
v
er
if
icatio
n
is
ad
d
e
d
to
th
e
ex
is
tin
g
e
n
cr
y
p
tio
n
p
r
o
ce
s
s
th
at
p
r
o
v
i
d
es
a
lo
ca
tio
n
-
b
ased
cr
y
p
to
g
r
ap
h
y
ap
p
r
o
ac
h
en
a
b
lin
g
m
o
r
e
in
tr
i
ca
te
u
s
es
o
f
lo
ca
tio
n
d
ata
[
6
]
.
I
n
an
o
th
er
s
tu
d
y
,
a
n
etwo
r
k
o
f
I
o
T
s
y
s
tem
s
f
o
r
f
r
eq
u
e
n
cy
m
o
d
u
latio
n
(
FM)
au
d
io
r
ec
eiv
e
r
s
is
m
an
ag
ed
u
s
in
g
a
g
eo
-
f
e
n
ce
b
ased
o
n
lo
ca
tio
n
f
o
r
s
o
n
g
d
etec
tio
n
[
7
]
.
T
h
e
cu
r
r
e
n
t
tech
n
iq
u
es
u
n
d
er
g
o
v
ar
io
u
s
ch
allen
g
es
in
lo
ca
tin
g
th
e
p
atien
t
s
u
f
f
er
in
g
f
r
o
m
C
OVI
D.
I
n
itially
,
co
m
p
ass
es
an
d
th
eo
d
o
lites
wer
e
u
s
ed
y
ea
r
s
ag
o
to
d
eter
m
in
e
th
e
p
atien
ts
'
lo
ca
tio
n
.
T
h
e
h
ea
lth
o
f
th
e
p
atien
t
n
ee
d
s
t
o
b
e
p
r
o
tecte
d
an
d
p
e
r
s
o
n
al
d
ata
s
h
ar
in
g
s
h
o
u
ld
b
e
co
n
s
tr
ain
e
d
,
at
t
h
e
s
am
e
tim
e.
T
h
e
m
eth
o
d
o
f
d
ata
co
llectio
n
f
r
o
m
v
ar
io
u
s
r
eg
io
n
s
v
ar
ies
f
u
r
th
er
ca
u
s
es
lack
o
f
in
ter
o
p
er
ab
ilit
y
b
etwe
en
th
e
d
atab
ase
an
d
th
e
r
ep
o
r
tin
g
s
y
s
tem
s
.
T
h
e
e
x
is
tin
g
s
y
s
tem
s
ca
u
s
e
d
elay
i
n
n
o
tif
y
in
g
th
e
p
atien
t
’
s
in
f
o
r
m
atio
n
th
at
f
u
r
t
h
er
lead
s
to
d
elay
ed
r
esp
o
n
s
e
an
d
lack
o
f
r
eso
u
r
ce
allo
ca
tio
n
.
Fu
r
th
er
m
o
r
e,
in
ac
c
u
r
ate
d
ata
also
lead
s
to
f
al
s
e
h
ea
lth
ac
tio
n
s
th
at
ar
e
to
b
e
t
ak
en
f
o
r
p
atien
ts
f
o
r
f
u
r
th
er
a
ctio
n
.
Priv
ate
d
ata
r
e
g
ar
d
in
g
t
h
e
lo
ca
tio
n
o
f
th
e
p
atien
t
ca
n
b
e
m
is
u
s
ed
.
An
o
th
er
r
esear
ch
g
ap
lies
in
d
etec
tin
g
th
e
lo
ca
tio
n
in
r
u
r
al
ar
ea
s
wh
er
e
th
e
h
ea
lth
ca
r
e
ac
ce
s
s
is
lim
ited
.
T
h
er
ef
o
r
e
,
th
e
o
b
jectiv
e
o
f
o
u
r
r
esear
ch
is
to
u
s
e
a
p
r
o
p
o
s
ed
m
eth
o
d
i.e
.
,
co
n
v
ex
h
u
ll
m
ec
h
an
is
m
in
o
r
d
er
to
o
v
er
co
m
e
th
e
ex
is
tin
g
ch
allen
g
es
lik
e
r
e
m
o
v
i
n
g
t
h
e
o
v
er
h
ea
d
s
,
an
d
to
b
u
ild
an
ef
f
ec
tiv
e
s
y
s
tem
wh
ich
is
r
o
b
u
s
t
to
d
etec
t
th
e
lo
ca
tio
n
o
f
th
e
p
atien
ts
th
at
en
h
an
ce
s
t
h
e
h
ea
lth
r
esp
o
n
s
e,
p
r
o
v
id
es th
e
o
p
tim
al
r
eso
u
r
ce
allo
ca
tio
n
an
d
ac
c
u
r
ate
r
esu
lts
.
Du
r
in
g
th
e
C
OVI
D
tim
e,
g
eo
f
en
ce
s
p
lay
e
d
a
v
e
r
y
im
p
o
r
ta
n
t
r
o
le
in
th
e
lo
ca
tio
n
d
etec
ti
o
n
o
f
th
e
af
f
ec
ted
p
atien
ts
.
T
h
e
p
atien
ts
ar
e
r
eq
u
ir
ed
to
s
tay
q
u
ar
a
n
tin
ed
if
th
e
y
g
et
af
f
ec
ted
f
r
o
m
C
OVI
D.
T
h
e
m
o
v
em
en
t
o
f
th
e
p
atien
ts
is
al
s
o
b
ein
g
tr
ac
ed
.
Als
o
,
th
e
cr
o
wd
ed
p
lace
s
a
r
e
m
o
n
ito
r
e
d
an
d
in
ca
s
e
th
e
c
r
o
wd
o
r
g
ath
er
in
g
is
f
o
u
n
d
,
it
is
in
s
tr
u
cted
to
b
e
d
is
p
er
s
ed
to
r
e
d
u
ce
th
e
s
p
r
ea
d
o
f
C
OVI
D.
I
t
also
co
n
tr
ib
u
ted
its
u
s
e
in
th
e
h
ea
lth
au
th
o
r
ities
b
y
s
en
d
in
g
th
e
n
o
tific
atio
n
s
to
ce
r
tain
in
d
iv
id
u
als
ab
o
u
t
th
e
b
r
ea
ch
o
f
th
e
d
is
tan
ce
in
ca
s
es th
ey
ar
e
af
f
e
cted
b
y
C
OVI
D
an
d
to
tak
e
ca
r
e
b
y
m
ai
n
tain
in
g
d
is
tan
ce
with
o
th
er
s
.
2.
T
H
E
CO
M
P
RE
H
E
NS
I
VE
T
H
E
O
RE
T
I
CA
L
B
ASI
S
Geo
f
en
cin
g
is
an
im
p
o
r
tan
t
te
ch
n
iq
u
e
u
s
ed
wid
ely
i
n
th
e
ar
ea
o
f
lo
ca
tio
n
d
etec
tio
n
alg
o
r
i
th
m
s
.
An
im
p
o
r
tan
t
s
tu
d
y
is
ca
r
r
ied
in
[
8
]
,
wh
er
e
n
ew
ap
p
r
o
ac
h
es
s
u
g
g
ested
to
ad
v
er
tis
in
g
ap
p
s
th
at
u
s
e
g
eo
f
e
n
ce
s
er
v
ices.
T
h
is
s
tu
d
y
en
h
an
ce
s
th
e
lo
ca
tio
n
-
b
ased
s
ea
r
ch
es
v
ia
an
ap
p
licatio
n
th
r
o
u
g
h
lo
ca
tio
n
s
p
ec
if
ic
s
er
v
ices
to
th
e
p
r
ac
tice
o
f
g
en
er
ally
s
ea
r
ch
in
g
ap
p
licatio
n
s
u
s
in
g
k
ey
wo
r
d
s
.
I
n
o
r
d
er
to
d
eliv
er
ap
p
licatio
n
s
u
g
g
esti
o
n
s
with
in
th
e
g
e
o
f
en
ce
ar
ea
an
d
lear
n
m
o
r
e
a
b
o
u
t
t
h
e
u
s
er
'
s
lo
ca
tio
n
i
n
f
o
r
m
atio
n
,
g
eo
f
en
ce
ap
p
r
o
ac
h
es a
r
e
u
s
ed
.
I
n
an
o
th
er
s
tu
d
y
,
a
s
y
s
tem
war
n
s
th
e
u
s
er
s
wh
en
ev
er
th
ey
e
n
ter
a
g
e
o
f
en
ce
ar
ea
[
9
]
.
2
.
1
.
F
r
a
m
ewo
r
k
A
f
r
am
ewo
r
k
is
v
er
y
im
p
o
r
t
an
t
in
th
e
cr
ea
tio
n
as
well
as
r
eq
u
ir
em
en
t
o
f
its
f
u
n
ctio
n
ality
f
o
r
g
o
v
er
n
m
en
t
f
u
n
ctio
n
s
in
wate
r
s
u
p
p
ly
,
p
o
wer
,
r
ec
y
clin
g
o
f
waste,
f
r
eig
h
t,
an
d
p
r
o
v
i
d
in
g
s
er
v
ice
in
an
ec
o
n
o
m
ic
way
an
d
h
en
ce
ac
h
i
ev
in
g
s
o
cial
g
o
als
[
1
0
]
.
A
p
r
e
r
eq
u
is
ite
f
o
r
im
p
lem
en
tin
g
r
e
g
io
n
al
d
ev
elo
p
m
en
t
is
th
e
p
r
esen
ce
o
f
s
u
itab
le
in
f
r
astru
ctu
r
e
[
1
1
]
.
I
n
f
r
astru
ctu
r
e
g
ap
s
h
av
e,
h
o
wev
e
r
,
b
ee
n
b
r
o
u
g
h
t
o
n
b
y
th
e
cu
r
r
en
t
u
n
ev
e
n
d
is
tr
ib
u
tio
n
o
f
in
f
r
astru
ctu
r
e.
Ma
n
y
co
n
s
tr
u
c
tio
n
p
r
o
jects
s
im
p
ly
p
a
y
atten
tio
n
to
t
h
e
city
'
s
o
r
a
r
eg
io
n
'
s
ec
o
n
o
m
ic
h
u
b
.
2
.
2
.
G
eo
f
ence
s
A
v
ir
tu
al
f
en
cin
g
th
at
is
co
n
s
tr
u
cted
to
r
ep
r
esen
t
a
r
eg
i
o
n
is
ca
lled
g
eo
f
e
n
cin
g
w
h
ich
is
cr
e
ated
f
r
o
m
n
u
m
er
o
u
s
g
eo
g
r
ap
h
ic
b
o
u
n
d
a
r
ies.
Ma
n
y
p
eo
p
le
u
s
e
g
eo
f
e
n
cin
g
in
d
aily
life
.
Fo
r
a
v
a
r
iety
o
f
p
u
r
p
o
s
es,
n
u
m
er
o
u
s
g
eo
f
e
n
cin
g
a
p
p
r
o
ac
h
es h
av
e
b
ee
n
cr
ea
ted
[
1
2
]
,
in
clu
d
in
g
:
a.
C
lo
s
en
ess
to
th
e
o
b
ject
o
f
in
ter
est:
a
g
eo
f
en
ce
is
cr
ea
ted
to
f
in
d
th
e
q
u
er
y
p
o
in
t
b
y
th
e
cr
ea
tio
n
o
f
a
cir
cu
lar
zo
n
e.
T
h
is
m
eth
o
d
s
ee
k
s
to
d
eter
m
in
e
h
o
w
clo
s
e
a
n
o
b
ject
is
to
th
e
o
b
ject
o
f
in
t
er
est.
On
ly
two
p
ar
am
eter
s
ar
e
u
s
ed
,
o
n
e
b
ein
g
th
e
r
ad
iu
s
v
alu
es
a
n
d
o
th
er
b
ein
g
th
e
ce
n
ter
co
o
r
d
in
ates.
As
a
r
esu
lt,
th
is
ap
p
r
o
ac
h
is
r
eg
ar
d
ed
as th
e
ea
s
iest
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
A
co
n
ve
x
h
u
ll b
a
s
ed
g
eo
fen
ci
n
g
s
ystem
to
era
d
ica
te
C
OV
I
D
(
P
a
r
u
l A
r
o
r
a
)
1819
b.
Ad
h
er
en
ce
t
o
th
e
r
o
u
te
an
d
t
im
etab
le
:
th
e
p
r
i
o
r
p
r
o
ce
d
u
r
e
an
d
t
h
is
o
n
e
a
r
e
n
ea
r
ly
id
e
n
tical.
Ho
wev
er
,
th
er
e
ar
e
o
th
er
s
ch
ed
u
lin
g
tech
n
iq
u
es
with
th
is
ap
p
r
o
ac
h
.
T
h
e
tim
e
at
wh
ich
th
e
au
to
m
o
b
ile
m
u
s
t
p
ass
ea
ch
d
eter
m
in
e
d
co
o
r
d
in
ates a
lo
n
g
th
e
p
ath
ar
e
p
r
o
v
id
ed
.
c.
Geo
f
en
ce
d
ter
r
ito
r
y
:
u
s
in
g
t
h
is
tech
n
iq
u
e
,
th
e
m
o
v
in
g
o
b
jects
in
a
g
eo
f
e
n
ce
a
r
ea
ar
e
au
to
m
atica
lly
tr
ac
k
ed
.
T
h
is
v
ir
tu
al
b
o
u
n
d
ar
y
is
s
h
ap
ed
eith
er
in
th
e
f
o
r
m
o
f
a
cir
cle
o
r
a
ce
r
tain
co
m
p
l
ex
f
ig
u
r
e,
lik
e
a
p
o
ly
g
o
n
.
B
u
ild
in
g
s
,
a
g
r
icu
ltu
r
al
f
ield
s
,
an
d
o
f
f
ice
s
p
ac
es a
r
e
all
r
ep
r
esen
ted
b
y
p
o
ly
g
o
n
s
[
1
3
]
.
2
.
3
.
E
x
is
t
ing
wo
r
k
T
h
e
b
elo
w
g
iv
en
m
eth
o
d
s
ar
e
u
s
ed
to
d
etec
t
th
e
lo
ca
tio
n
o
f
th
e
n
o
d
es
in
s
id
e
an
d
o
u
ts
id
e
th
e
b
o
u
n
d
ar
y
.
T
h
ese
tech
n
iq
u
es
h
elp
in
d
eter
m
in
in
g
th
e
C
OVI
D
p
atien
ts
wh
eth
er
th
ey
lie
with
in
th
e
is
o
latio
n
r
an
g
e
o
r
o
u
t
o
f
th
e
r
an
g
e
o
f
is
o
latio
n
.
Alth
o
u
g
h
th
ese
ar
e
m
ath
em
atica
l
alg
o
r
it
h
m
s
,
h
o
wev
er
,
we
h
av
e
u
s
ed
th
ese
in
d
eter
m
in
in
g
th
e
lo
ca
ti
o
n
o
f
th
e
p
atien
ts
.
2
.
3
.
1
.
Dela
un
a
y
t
ria
ng
ula
t
io
n m
et
ho
d
A
tech
n
iq
u
e
f
o
r
f
ig
u
r
i
n
g
o
u
t
a
co
o
r
d
in
ate'
s
lo
ca
tio
n
is
ca
ll
ed
Dela
u
n
a
y
tr
ian
g
u
latio
n
m
e
th
o
d
.
T
h
is
tech
n
iq
u
e
is
also
k
n
o
wn
as
‘
i
n
s
id
e
p
o
ly
g
o
n
’
m
eth
o
d
.
B
y
d
r
awin
g
im
ag
in
ar
y
h
o
r
izo
n
tal
lin
es,
it
is
p
o
s
s
ib
le
to
f
in
d
o
u
t
th
e
lo
ca
tio
n
o
f
d
etaile
d
p
o
in
ts
o
v
e
r
a
p
lan
e.
T
h
is
lin
e
is
m
ad
e
u
p
o
f
ac
q
u
i
r
ed
s
ev
e
r
al
co
o
r
d
in
ates.
T
h
e
to
tal
n
u
m
b
er
o
f
co
o
r
d
in
ates
th
at
cr
o
s
s
o
v
er
th
e
ed
g
e
o
f
g
eo
f
e
n
ce
will
th
en
b
e
u
s
e
d
to
ca
lcu
late
th
e
co
o
r
d
in
ates.
T
h
e
c
o
o
r
d
in
ates
ar
e
in
th
e
p
o
ly
g
o
n
r
e
g
io
n
wh
e
n
it
i
s
an
o
d
d
n
u
m
b
er
.
T
h
e
c
o
o
r
d
in
ates,
h
o
wev
e
r
,
ar
e
o
u
ts
id
e
th
e
g
e
o
f
en
ce
wh
e
n
th
e
in
teg
er
is
ev
en
[
1
4
]
.
I
n
o
th
er
wo
r
d
s
,
r
ay
ca
s
tin
g
is
a
tech
n
iq
u
e
th
at
is
em
p
lo
y
ed
in
co
m
p
u
tatio
n
al
g
e
o
m
etr
y
.
I
t
is
ca
p
ab
le
o
f
co
n
v
e
r
tin
g
a
2
-
D
s
ce
n
e
in
to
a
3
-
D
v
iewp
o
in
t.
A
r
ay
is
d
ir
ec
ted
at
th
e
p
o
ly
g
o
n
u
n
d
er
o
b
s
er
v
atio
n
f
r
o
m
a
ce
r
tain
p
lace
.
W
h
en
an
en
d
less
r
ay
is
p
r
ed
icted
f
r
o
m
an
y
p
o
in
t
'
'
to
war
d
s
a
s
p
ec
if
ied
p
o
ly
g
o
n
al
f
en
ce
‘
’
,
th
is
m
eth
o
d
d
eter
m
in
es
wh
en
th
e
q
u
er
y
p
o
in
t
‘
’
is
with
i
n
th
e
p
o
l
y
g
o
n
o
r
n
o
t,
as
s
h
o
wn
in
Fig
u
r
e
1
.
T
h
e
q
u
e
r
y
p
o
in
t
‘
’
is
in
s
id
e
o
f
‘
’
if
th
e
lim
itles
s
r
ay
cr
o
s
s
es
th
e
o
d
d
n
u
m
b
e
r
o
f
n
o
d
es; o
t
h
er
wis
e,
it is
o
u
ts
id
e
o
f
.
Fig
u
r
e
1
.
I
n
s
id
e
p
o
l
y
g
o
n
m
o
d
el
A
p
o
ly
g
o
n
in
Fig
u
r
e
1
h
as
an
in
f
in
ite
n
u
m
b
er
o
f
r
ay
s
ca
s
t
o
n
to
it.
T
h
e
co
n
n
ec
tio
n
p
o
in
ts
ar
e
0
,
1
,
2
,
3
,
o
r
4
,
as
s
h
o
wn
.
W
h
en
ev
e
r
t
h
e
in
ter
s
ec
tio
n
s
a
r
e
o
d
d
,
th
is
p
o
in
t
is
s
h
o
wn
to
b
e
in
s
id
e
t
h
e
g
eo
f
en
ce
;
h
o
we
v
er
,
wh
en
th
er
e
a
r
e
an
e
v
en
n
u
m
b
e
r
o
f
in
ter
s
ec
tio
n
s
,
it is
s
h
o
wn
t
o
b
e
o
u
ts
id
e
th
e
p
o
l
y
g
o
n
.
2
.
3
.
2
.
T
ria
ng
le
weig
ht
m
et
ho
d
T
h
e
tr
ian
g
le
weig
h
t
b
o
r
d
er
b
r
ea
ch
d
etec
tio
n
m
eth
o
d
co
m
p
r
is
es
two
s
tag
es:
in
itial
izatio
n
an
d
r
u
n
-
tim
e.
W
h
en
th
e
s
y
s
tem
f
ir
s
t
co
m
es
o
n
,
b
o
th
k
ee
p
-
in
an
d
k
ee
p
-
o
u
t
g
eo
f
en
ce
s
m
u
s
t
g
o
th
r
o
u
g
h
th
e
s
tar
tu
p
p
r
o
ce
d
u
r
e.
An
y
ex
clu
s
io
n
o
r
in
clu
s
io
n
g
eo
f
en
ce
th
at
is
am
en
d
ed
s
h
o
u
ld
b
e
estab
lis
h
ed
f
r
o
m
s
cr
atch
if
a
n
y
m
o
d
if
icatio
n
s
ar
e
m
a
d
e
to
a
n
y
o
f
t
h
e
ex
is
tin
g
b
o
r
d
er
s
.
Du
r
in
g
th
e
in
itializatio
n
p
r
o
ce
s
s
,
all
th
e
b
asic
b
o
u
n
d
ar
ies
ar
e
d
iv
i
d
ed
in
to
a
s
tr
aig
h
tf
o
r
war
d
p
o
ly
g
o
n
al
b
o
u
n
d
ar
y
with
y
-
m
o
n
o
t
o
n
o
u
s
p
o
ly
g
o
n
al
lim
its
,
an
d
is
f
in
ally
d
iv
id
ed
in
t
o
tr
ian
g
le
s
.
T
h
e
r
u
n
-
tim
e
m
eth
o
d
d
eter
m
in
es
wh
eth
er
ea
ch
tr
ian
g
le
h
as
th
e
in
ter
est
p
o
in
t.
T
h
e
p
lace
is
in
s
id
e
t
h
e
p
o
ly
g
o
n
if
a
n
y
o
f
t
h
e
p
o
ly
g
o
n
s
ar
o
u
n
d
th
e
p
o
in
t
o
f
in
ter
est
c
o
n
tain
it;
if
n
o
t,
it
is
o
u
ts
id
e.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
2
,
Ap
r
il
20
25
:
1
8
1
7
-
1
8
2
5
1820
A
g
eo
f
en
c
e
v
io
latio
n
o
cc
u
r
s
wh
en
ev
er
th
e
s
u
b
ject
is
eith
e
r
with
in
th
e
i
n
clu
s
io
n
g
eo
f
e
n
c
e
o
r
o
u
t
o
f
th
e
ex
clu
s
io
n
g
eo
f
en
ce
as
s
h
o
wn
in
Fig
u
r
e
2
.
An
y
g
eo
f
en
c
e
p
er
im
eter
ca
n
b
e
d
i
v
id
ed
i
n
to
n
o
n
-
in
ter
s
ec
tin
g
tr
ian
g
les
u
s
in
g
tr
ian
g
u
latio
n
ap
p
r
o
ac
h
,
wh
ich
is
b
ased
o
n
th
e
r
eg
u
lar
izatio
n
m
eth
o
d
.
T
r
ian
g
le
weig
h
t
ch
ar
ac
ter
izatio
n
(
T
W
C
)
m
eth
o
d
is
o
p
er
ated
o
n
th
e
t
r
ian
g
le,
in
o
r
d
er
to
s
ee
h
o
w
a
r
a
n
d
o
m
p
o
ly
g
o
n
is
d
iv
id
ed
,
as
s
h
o
wn
in
Fig
u
r
e
3
.
T
h
e
f
i
r
s
t
s
tep
in
th
is
in
itializatio
n
p
r
o
ce
d
u
r
e
is
to
s
p
lit
th
e
p
o
ly
g
o
n
in
t
o
m
o
n
o
to
n
e
p
o
ly
g
o
n
al
b
o
u
n
d
ar
ies,
an
d
t
h
e
s
ec
o
n
d
is
to
d
i
v
id
e
ea
ch
m
o
n
o
to
n
e
p
o
l
y
g
o
n
al
p
er
im
eter
in
to
tr
ian
g
les
[
1
5
]
.
T
h
ese
o
p
er
atio
n
s
ca
n
all
b
e
p
e
r
f
o
r
m
e
d
o
n
th
e
x
-
ax
is
as we
ll,
b
u
t th
ey
a
r
e
all
d
o
n
e
with
th
e
y
-
ax
is
in
m
in
d
.
Fig
u
r
e
2
.
T
r
ian
g
le
weig
h
t c
h
ar
ac
ter
izatio
n
f
lo
wch
ar
t
a.
Po
ly
g
o
n
s
in
to
s
u
b
-
p
o
ly
g
o
n
s
I
f
all
p
lan
es
p
ar
allel
t
o
th
e
x
-
ax
is
d
o
n
o
t
c
r
o
s
s
m
o
r
e
th
an
t
wo
ed
g
es
o
f
a
p
o
ly
g
o
n
,
it
is
s
aid
to
b
e
y
-
m
o
n
o
to
n
e.
T
h
e
m
o
n
o
to
n
e
p
o
ly
g
o
n
s
ar
e
cr
ea
ted
b
y
cr
ea
tin
g
th
e
lin
k
ag
es
b
etwe
en
th
e
v
e
r
tices
af
ter
iter
atin
g
th
e
p
o
ly
g
o
n
al
p
o
i
n
ts
f
r
o
m
lar
g
est
to
s
m
alles
t
y
-
lo
ca
tio
n
,
an
d
th
en
f
r
o
m
lo
west
to
h
ig
h
est
v
a
lu
es
o
f
y
-
lo
ca
tio
n
.
Geo
f
en
cin
g
ig
n
o
r
es
n
ewly
ad
d
ed
ed
g
es
th
at
ex
ten
d
b
ey
o
n
d
th
e
in
itial
p
o
ly
g
o
n
s
b
ec
a
u
s
e
it
o
n
ly
tak
es
in
to
ac
co
u
n
t
t
h
e
ar
ea
co
v
e
r
ed
b
y
th
e
o
r
i
g
in
al
p
o
ly
g
o
n
[
1
6
]
.
I
t
i
s
ass
er
ted
th
at
th
e
e
d
g
e
s
its
o
u
ts
id
e
th
e
o
r
ig
in
al
p
o
ly
g
o
n
wh
en
th
e
n
ewly
g
en
e
r
ated
p
o
ly
g
o
n
'
s
ed
g
e
v
e
r
tices a
r
e
o
r
d
e
r
ed
clo
c
k
wis
e
r
ath
er
t
h
an
an
ticlo
ck
wis
e.
b.
C
r
ea
tio
n
o
f
tr
ian
g
les
Fro
m
lar
g
est
to
lo
west
y
-
v
alu
e,
th
e
v
er
tices
o
f
ea
ch
m
o
n
o
to
n
e
p
o
ly
g
o
n
ar
e
iter
ativ
ely
r
e
p
e
ated
wh
ile
jo
in
in
g
ed
g
es
to
f
o
r
m
t
r
ian
g
l
es
at
ea
ch
s
tep
.
Du
e
to
th
e
f
ac
t
th
at
th
ey
ar
e
co
n
tain
ed
in
s
id
e
y
-
m
o
n
o
to
n
e
p
o
ly
g
o
n
s
,
all
ed
g
es
th
at
ar
e
cr
ea
ted
ar
e
p
r
eser
v
ed
.
T
h
e
ex
a
m
p
le
g
eo
f
en
ce
is
s
u
b
jecte
d
to
th
is
p
r
o
ce
s
s
f
o
r
f
i
v
e
tim
es,
wh
er
e
iter
atio
n
is
to
b
e
p
er
f
o
r
m
ed
f
o
r
ea
ch
m
o
n
o
to
n
e
p
o
ly
g
o
n
.
3.
P
RO
P
O
SE
D
AP
P
RO
ACH
AND
M
E
T
H
O
DO
L
O
G
Y
T
h
e
s
im
u
latio
n
was
ca
r
r
ied
o
u
t
in
MA
T
L
AB
an
d
im
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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C
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p
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g
I
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N:
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8
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s
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th
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T
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m
eth
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s
d
r
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ac
k
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a.
Setu
p
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d
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tim
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s
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es
ar
e
p
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esen
t
in
T
W
C
A.
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h
e
k
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in
an
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k
ee
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t
g
e
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ce
s
m
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t
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in
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ag
ain
if
th
e
g
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f
e
n
ce
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n
d
a
r
y
ch
an
g
es a
f
ter
in
itializatio
n
is
co
m
p
lete.
b.
T
W
C
A
p
r
o
d
u
ce
s
in
ac
cu
r
ate
f
in
d
in
g
s
wh
e
n
n
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d
es
ar
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d
y
n
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m
ic
s
in
ce
it
is
u
n
ab
le
t
o
tak
e
u
p
d
ates
in
to
ac
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n
t w
ith
r
e
g
ar
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t
o
d
y
n
am
i
c
g
eo
f
en
ce
s
d
u
e
to
laten
c
y
.
T
h
e
f
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llo
win
g
ar
e
s
o
m
e
d
r
awb
ac
k
s
o
f
th
e
Dela
u
n
ay
tr
ian
g
u
la
tio
n
m
eth
o
d
:
a.
An
o
v
er
h
ea
d
is
m
ad
e
wh
ile
ta
k
in
g
th
e
c
o
m
p
lex
p
o
ly
g
o
n
s
in
t
o
ac
co
u
n
t.
b.
Fo
r
in
tr
icate
p
o
ly
g
o
n
s
with
a
g
r
ea
ter
n
u
m
b
er
o
f
e
d
g
es,
p
r
e
p
r
o
ce
s
s
in
g
is
n
ec
ess
ar
y
.
C
o
n
v
ex
h
u
ll
o
f
p
o
in
ts
ca
n
b
e
u
s
ed
to
ex
ten
d
th
is
r
esear
ch
b
ec
au
s
e
th
e
s
u
r
f
ac
e
ca
n
b
e
s
p
li
t
in
to
two
h
alf
b
ased
o
n
co
o
r
d
i
n
ates.
A
d
iv
id
e
an
d
co
n
q
u
e
r
ap
p
r
o
ac
h
ar
e
u
s
ed
to
c
alcu
late
th
e
an
ti
clo
ck
wis
e
p
o
in
ts
.
T
h
e
p
lan
e'
s
ce
n
tr
al
ax
is
i
s
d
eter
m
in
ed
b
y
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ig
h
est
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alu
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T
h
e
ch
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en
p
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t
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jo
in
ed
to
th
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ev
elo
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ar
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itio
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if
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ati
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(
,
,
)
=
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e
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o
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y
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t
h
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in
t
[
1
7
]
.
As
a
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lt,
ea
ch
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tin
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ally
cir
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ll
,
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wn
in
Fig
u
r
e
3
.
T
h
e
Dela
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ter
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o
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A
(
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tech
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eter
m
in
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if
th
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p
o
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t
is
in
s
id
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ter
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esp
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ely
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ith
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W
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A
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d
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ay
tr
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latio
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s
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th
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llo
win
g
f
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to
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s
:
a.
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o
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v
ex
h
u
ll
d
o
es n
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t
h
av
e
d
if
f
er
en
t in
itializatio
n
an
d
r
u
n
tim
e
s
tep
s
h
en
ce
m
ak
in
g
it less
co
m
p
licated
.
b.
C
o
n
v
ex
h
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ll
alg
o
r
ith
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g
iv
es
a
cc
u
r
ate
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esu
lts
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o
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th
e
d
y
n
am
i
c
n
o
d
es.
c.
Pre
p
r
o
ce
s
s
in
g
is
n
o
t n
ec
ess
ar
y
f
o
r
ch
ec
k
in
g
th
e
b
o
u
n
d
ar
y
p
o
in
ts
th
u
s
av
o
id
in
g
th
e
o
v
er
h
ea
d
s
.
Alg
o
r
ith
m
1
.
C
o
n
v
ex
h
u
ll
Input: Determine a vector of pair ‘v’ to take query points as the input.
Output: Arrange vector v.
Initialize first element of ‘v’ as
pair ‘x’.
Repeat the above steps while one reaches point ‘x’ again.
Define a pair ‘y’that is set to next element of ‘v’.
For ith pair of v[i]if [x,v[i],y] have anti clockwise orientation, update value of ‘y’ as
v[i]. This loop works till the pair with most anti clockwise orientation is found.
end if
end for
Save the output value of ‘y’ in the vector pair
set ‘x’= ‘y’ for next step
Output is vector of pairs.
Fig
u
r
e
3
.
C
o
n
v
ex
h
u
ll
4.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
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N
T
h
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o
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jectiv
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o
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th
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less
tim
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n
tim
e
an
d
h
av
e
least
s
tan
d
ar
d
d
ev
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b
y
u
s
in
g
c
o
n
v
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x
h
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m
eth
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d
.
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e
ex
p
er
im
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0
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o
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r
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o
r
1
0
v
er
tices.
T
h
e
ex
ec
u
tio
n
tim
e,
m
ea
n
tim
e
an
d
s
tan
d
ar
d
d
ev
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ar
e
co
n
s
id
er
ed
f
o
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t
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e
p
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ith
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im
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T
h
is
test
is
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
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&
C
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g
,
Vo
l.
15
,
No
.
2
,
Ap
r
il
20
25
:
1
8
1
7
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1
8
2
5
1822
r
u
n
f
o
r
th
e
alg
o
r
ith
m
s
:
T
W
C
A,
Dela
u
n
ay
tr
ian
g
u
latio
n
an
d
co
n
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e
x
h
u
ll.
First
a
g
eo
f
en
c
e
is
cr
ea
ted
with
1
0
v
er
tices
o
n
a
m
ap
an
d
th
en
th
ese
alg
o
r
ith
m
s
ar
e
u
s
ed
to
lo
c
ate
th
e
n
o
d
es.
T
h
e
g
eo
f
en
ce
b
o
u
n
d
a
r
y
is
u
s
ed
to
d
eter
m
in
e
th
e
n
o
d
es
wh
eth
e
r
th
ey
lie
with
in
th
e
g
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f
e
n
c
e
ar
ea
o
r
o
u
ts
id
e
th
e
g
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f
en
ce
ar
ea
.
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ea
l
-
tim
e
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tr
ac
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e
with
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em
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s
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er
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y
,
ev
en
wh
ile
it
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ield
s
p
r
ec
is
e
f
in
d
in
g
s
,
[
1
8
]
.
T
h
e
GPS
is
u
s
ed
to
an
aly
ze
t
h
is
in
o
r
d
er
to
p
r
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v
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d
e
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Dep
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in
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o
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n
ee
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v
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in
d
icatio
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(
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at
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led
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y
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e
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ig
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h
if
ts
f
r
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m
W
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-
Fi
to
m
o
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ile
d
ata
[
1
9
]
.
Ho
wev
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it
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u
s
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a
ce
r
tain
am
o
u
n
t
o
f
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elay
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ile
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tio
n
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etec
tio
n
.
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n
th
is
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esear
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,
r
ea
l
-
tim
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o
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ito
r
in
g
is
n
o
t
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;
in
s
tead
,
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o
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n
d
ar
y
o
r
g
e
o
f
en
ce
n
o
tific
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s
ar
e
u
s
ed
t
o
aler
t
u
s
wh
en
th
e
b
o
u
n
d
ar
y
is
cr
o
s
s
ed
[
2
0
]
.
T
h
e
co
n
s
tr
u
ctio
n
o
f
g
eo
f
e
n
ce
s
a
llo
ws
f
o
r
m
o
r
e
r
esear
ch
to
b
e
d
o
n
e
b
ec
a
u
s
e
th
e
cu
r
r
en
t m
et
h
o
d
r
eq
u
ir
es m
o
r
e
tim
e
an
d
en
er
g
y
[
2
1
]
.
Op
en
Stre
etM
ap
s
ar
e
u
s
ed
,
wh
er
e
th
e
g
eo
f
e
n
ce
s
ar
e
d
esig
n
ed
as p
o
ly
g
o
n
s
,
wh
e
r
e
th
e
b
o
u
n
d
ar
y
ex
it is
r
ep
o
r
ted
t
o
th
e
au
th
e
n
ticated
u
s
er
wh
ile
tak
in
g
th
e
b
o
u
n
d
ar
y
v
i
o
latio
n
in
to
c
o
n
s
id
er
a
tio
n
[
2
2
]
.
T
en
m
ap
v
er
tices
ar
e
u
s
ed
to
d
r
aw
th
e
g
eo
f
e
n
ce
b
o
u
n
d
ar
y
,
as
s
h
o
w
n
in
Fig
u
r
e
4
.
E
v
er
y
tim
e
th
e
T
r
ian
g
le
W
eig
h
t
alg
o
r
ith
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r
u
n
s
,
th
e
n
o
d
es
th
at
is
in
s
id
e
th
e
f
en
ce
as
well
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y
tim
e
th
e
y
cr
o
s
s
it
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e
r
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o
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ized
a
n
d
d
is
p
lay
ed
[
2
3
]
.
T
en
v
er
tices
ar
e
u
tili
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to
estab
lis
h
th
e
b
o
r
d
er
in
th
e
s
h
ap
e
o
f
a
p
o
l
y
g
o
n
,
as
s
h
o
wn
in
Fig
u
r
e
4
,
a
n
d
th
e
y
ar
e
th
e
n
u
s
ed
to
ass
ess
if
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Fig
u
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4
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u
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4
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u
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ates
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I
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I
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15
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2
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Ap
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20
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e
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RE
F
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NC
E
S
[
1
]
G
.
Li
,
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.
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[
2
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[
3
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.
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.
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n
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[
4
]
B
.
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.
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[
5
]
E.
H
a
ss
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n
,
M
.
Y
.
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h
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ms,
N
.
A
.
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3
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p
p
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0
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–
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.
[
6
]
P
.
S
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[
7
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.
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[
8
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.
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m,
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s:
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2
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.
[
9
]
B
.
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d
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[
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0
]
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.
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l
.
,
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1
1
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.
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.
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s:
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[
1
2
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.
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n
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l
.
,
“
P
r
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t
Tr
a
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s:
To
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[
1
3
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A
.
A
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r
a
,
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.
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b
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.
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h
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ssessm
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ma
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.
[
1
4
]
A
.
S
.
K
w
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a
-
R
a
sh
i
d
,
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.
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.
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se
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se
(
C
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D
-
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)
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ses
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it
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re
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s.
S
h
e
c
a
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tac
ted
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t
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m
a
il
:
su
m
a
n
d
e
sw
a
l.
c
se
@d
c
ru
stm
.
o
rg
.
”
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