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where
H
mn
a
a
mn
(
u
,
v
)=
4
a
mn
i
η
(
u
,
v
)
(27)
H
mn
b
b
mn
(
u
,
v
)=
4
a
mn
i
η
(
u
,
v
)
(28)
H
mn
c
c
mn
(
,
)=
(
,
)
.
u
v
4
a
mn
i
η
u
v
(29)
To ease the numerical optimization, the condition (16) is transformed. That is,
Function
G
(
,
)
u
v
is rewritten as
−
i
G
(
u
,
v
)=
G
1
(
u
,
v
)
,
(30)
3
π
(
u
2
+
v
2
)
where
√
3
2
√
3
2
1
2
u
1
2
u
G
1
(
u
,
v
)
≡
(
uH
a
(
u
,
v
)+(
+
v
)
H
b
(
u
,
v
)+(
−
+
v
)
H
c
(
u
,
v
))
F
(
u
,
v
)
m
,
n
a
mn
σ
mn
(
u
,
v
)
F
(
u
,
v
)
=
4
i
(31)
and
√
3
2
√
3
2
1
2
u
1
2
u
mn
mn
mn
σ
mn
(
u
,
v
)
≡
u
η
(
u
,
v
)+(
+
v
)
η
(
u
,
v
)+(
−
+
v
)
η
(
u
,
v
)
.
(32)
The expression to be minimized, Equation (9), is then rewritten as
2
dudv
D
Ψ
(
u
,
v
)
|
F
(
u
,
v
)
|
(33)
where
k
,
l
m
,
n
a
kl
a
mn
(
τ
a
kl
a
mn
b
kl
b
mn
c
kl
c
mn
Ψ
(
u
,
v
)=
16
τ
+
τ
τ
+
τ
τ
)
,
(34)
and
2
u
a
mn
a
mn
τ
(
u
,
v
)
≡
)
σ
mn
(
u
,
v
)
−
η
(
u
,
v
)
(35)
u
2
v
2
3
(
+
√
3
v
u
+
mn
mn
τ
(
u
,
v
)
≡
)
σ
mn
(
u
,
v
)
−
η
(
u
,
v
)
(36)
3
(
u
2
+
v
2
√
3
v
)
≡
−
+
u
mn
mn
τ
(
u
,
v
)
σ
mn
(
u
,
v
)
−
η
(
u
,
v
)
.
(37)
3
(
u
2
+
v
2
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