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where
k
i
=
g
(γ
,
ζ ) /(λ
i
−
2
γζ)
(
k
i
<
0 only for
i
>
j
).
$
λ
j
−
2
γζ
(λ
i
−
2
γζ)
+
k
q
j
q
k
(λ
k
−
2
γζ)(λ
i
−
2
γζ)
=
j
b
T
b
λ
i
−
2
γ(
1
−
ζ)
λ
i
−
+
γζ
−
for
i
=
j
γζ
2
2
k
i
=
%
λ
j
−
2
γζ
2
+
q
j
q
k
(λ
k
−
2
γζ)
λ
j
−
2
γζ
k
=
j
b
T
b
λ
j
−
2
γ(
1
−
ζ)
λ
j
−
+
γζ
−
for
i
=
j
γζ
2
2
(5.96)
For
γ
→
λ
j
/
2
ζ
−
,
o
k
j
$
for
i
=
j
q
j
k
i
→
(5.97)
%
for
i
=
j
λ
γζ
2
−
2
j
Hence, the asymptotic form of the hyperboloid is
z
j
k
j
≈
1
(5.98)
Recalling eq. (5.93), the barrier locus is given by
k
j
v
j
+
x
c
(γ
q
j
λ
j
−
2
γζ
v
j
+
x
c
(γ
x
bar
≈
±
,
ζ)
=±
,
ζ)
(5.99)
where
v
j
is the
j
th column of
V
. Consider the expression (5.45) for
x
c
(γ
,
ζ)
in
the limit
proj
v
j
A
T
b
ζ
−
1
1
λ
j
−
2
γζ
γ
→
λ
j
/
2
λ
j
−
2
γζ
v
j
v
j
A
T
b
=
x
c
(γ
,
ζ)
−→
(5.100)
v
j
v
T
j
is the
n
-dimensional projection matrix defining the vector parallel
to
v
j
with norm given by the orthogonal projection onto the line of orientation
v
j
passing through the origin. Recalling Theorem 102, we have
where
q
j
v
j
λ
j
−
2
γζ
ζ
−
γ
→
λ
j
/
2
x
c
(γ
ζ)
−→
,
(5.101)
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