Geography Reference
In-Depth Information
Table 22.26
Simultaneous diagonalization of the matrix pair (
G
l
,
G
r
): distortion
“
characteristic equation
(22.151)
Λ
2
G
11
C
12
−
Λ
2
G
12
C
11
−
= 0
(22.152)
Λ
2
G
12
C
22
−
Λ
2
G
22
C
12
−
x
L
+
y
L
−
Λ
2
G
11
x
L
y
B
+
x
B
y
L
x
L
y
B
+
x
B
y
L
x
2
B
+
y
B
−
= 0
(22.153)
Λ
2
G
22
Λ
4
−
tr
C
l
G
−
l
Λ
2
+det
C
l
G
−
1
= 0
(22.154)
l
tr
C
l
G
−
l
2
Λ
1
,
2
=
Λ
2
±
=
1
1
2
2
tr
C
l
G
−
1
4det
C
l
G
−
1
±
−
(22.155)
l
l
Hilbert invariants
J
1
:=
tr
C
l
G
−
1
(22.156)
l
J
2
:= det
C
l
G
−
1
(22.157)
l
“
the Cayley matrixproduct
=
x
L
+
y
L
G
−
1
(
x
L
x
B
+
y
L
y
B
)
G
−
1
22
C
l
G
−
1
11
x
2
B
+
y
B
G
−
1
(22.158)
l
(
x
L
x
B
+
y
L
y
B
)
G
−
1
11
22
J
1
=
tr
C
l
G
−
1
l
=
G
−
1
11
x
L
+
G
−
1
x
2
B
+
G
−
1
y
L
+
G
−
1
y
B
(22.159)
22
11
22
x
L
+
y
L
x
2
B
+
y
B
−
(
x
L
x
B
+
y
L
y
B
)
2
J
2
=
G
−
1
G
−
1
22
(22.160)
11
Appendix 2
The Euler-Lagrange equation of minimal distortion energy in Gauss surface normal coordinates
Based upon the matrix pair
{
C
l
G
l
}
of type “left Cauchy-Green deformation tensor
C
l
”and
“left metric tensor
G
l
” given for an oblate ellipsoid-of-revolution
E
A
1
,A
2
by Eqs. (
22.18
)-(
22.20
),
and (
22.32
) we aim at deriving the stationary point set of distortion energy which is based upon
the
Hilbert invariant density
tr
C
l
G
−
l
of distortion energy which we already encountered in the
eigenspace analysis of the matrix pair
. Outlined in Table
22.30
,firstwederivetheleft
surface element
dS
l
in terms of Gauss surface normal
{
C
l
G
l
}
{
longitude
L
, latitude
B
}
by Eqs. (
22.191
)and(
22.192
).
Second
,Eqs.(
22.193
)and(
22.194
) offer a representation of distor-
tion energy density tr
C
l
G
−
l
as a
Hilbert invariant.
The distortion energy, the integral over the
distortion energy density with
fixed boundaries
,Eq.(
22.195
) is additively decomposed
•
2
,Eq.(
22.196
),
into an integral I proportional to
Grad x
2
,Eq.(
22.197
)!
Such a representation is proven by computing
Grad x
and
Grad y
, respectively, in terms of
the left Cartan 2-leg
•
into an integral II proportional to
Grad y
2
and
2
,
{
C
1
,
C
2
|
X
}
attached to the point
X
,aswellas
Grad x
Grad y
respectively, by Eqs.(
22.198
)-(
22.203
). The first variations
• δ|
=0
,
Eq.(
22.204
),
• δ||
=0
,
Eq.(
22.205
)
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