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e
22
−
;
K
1
G
22
×
(1.115)
−
(
e
12
−
K
1
G
12
)
2nd left eigencolumns
,K
2
:
F
12
F
22
=
1
G
22
(
e
11
K
2
G
12
)
2
×
−
K
2
G
11
)
2
−
2
G
12
(
e
11
−
K
2
G
11
)(
e
12
−
K
2
G
12
)+
G
11
(
e
12
−
−
;
(
e
12
K
2
G
12
)
e
11
− K
2
G
11
−
×
(1.116)
1st right eigencolumns
,κ
1
:
f
11
f
21
=
1
g
11
(
E
22
−
κ
1
g
12
)
2
×
(1.117)
κ
1
g
22
)
2
−
2
g
12
(
E
12
−
κ
1
g
12
)(
E
22
−
κ
1
g
22
)+
g
22
(
E
12
−
E
22
−
;
κ
1
g
22
×
−
(
E
12
−
κ
1
g
12
)
2nd right eigencolumns,
κ
2
:
f
12
f
22
=
1
g
22
(
E
11
κ
2
g
12
)
2
×
(1.118)
−
κ
2
g
11
)
2
−
2
g
12
(
E
11
−
κ
2
g
11
)(
E
12
−
κ
2
g
12
)+
g
11
(
E
12
−
−
(
E
12
− κ
2
g
12
)
E
11
− κ
2
g
11
.
×
End of Lemma.
The proof of these relations follows the line of thought of the proof of Lemma
1.6
. Accordingly,
we skip any proof here.
The canonical forms of the
scale difference
(d
s
)
2
(d
S
)
2
and (d
S
)
2
(d
s
)
2
, respectively, have been
−
−
interpreted as
1
1
left Euler-Lagrange circle
S
right Euler-Lagrange circle
S
versus
versus
left Euler-Lagrange ellipse
right Euler-Lagrange ellipse
(1.119)
1
√
K
1
,
√
K
2
1
√
κ
1
,
√
κ
2
E
(
K
i
>
0
∀
i
=1
,
2)
,
and
E
(
κ
i
>
0
∀
i
=1
,
2)
,
left Euler-Lagrange hyperbola
right Euler-Lagrange hyperbola
1
√
K
1
,
√
K
2
1
√
K
1
,
√
K
2
H
(
K
1
>
0
,K
2
<
0)
,
H
(
κ
1
>
0
,κ
2
<
0)
,
r
, respectively. A deforma-
tion portrait with a positive eigenvalue
K
(E
l
,
G
r
)or
κ
(E
r
,
G
l
) is referred to as
extension
, with
a negative eigenvalue
K
(E
l
,
G
r
)or
κ
(E
r
,
G
l
)as
compression
. Obviously, Cauchy-Green deforma-
tion and Euler-Lagrange deformation are related as outlined in Corollary
1.9
. The four cases of
the eigenspace analysis of the left and the right Euler-Lagrange deformation are illustrated in
Figs.
1.12
,
1.13
,
1.14
,and
1.15
.
l
and the right tangent space
T
u
R
on the left tangent space
T
U
M
M
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