Geography Reference
In-Depth Information
Corollary F.8 (Left principal stretches of the vertical-horizontal mean of mixed equiareal cylin-
dric mapping of the biaxial ellipsoid
2
E
A,B
).
The left principal stretches
{Λ
1
,Λ
2
}
of the coordinates
c
AB
of the left Cauchy-Green deformation
tensor normalized with respect to the coordinates
G
A,B
of the left metric tensor are represented
by (
F.47
) if the mapping equations (
F.32
) for the vertical mean and (
F.43
)-(
F.46
) for the hori-
zontal mean of mixed equiareal mappings of the biaxial ellipsoid apply.
1
2
(tr[C
l
G
−
l
]) +
(tr[C
l
G
−
l
])
2
4
,
Λ
1
=+
−
(F.47)
Λ
2
=+
1
2
(tr[C
l
G
−
l
])
4
.
(tr[C
l
G
−
l
])
2
−
−
First, for the vertical mean holds (
F.48
).
tr[C
l
G
−
l
]=
c
11
G
11
+
c
22
G
22
=
=
A
4
(
α
+
β
)
4
[
A
2
Λ
2
β
2
sin
2
Φ
+(
αL
+
βA
)
2
]+(
αL
+
βA
)
6
A
2
(
α
+
β
)
2
(
αL
+
βA
)
4
.
(F.48)
Second, for the horizontal mean holds (
F.49
).
tr[C
l
G
−
l
]=
c
11
C
11
+
c
22
G
22
=
=
(
αA
+
βL
)
4
+
Λ
2
β
2
sin
2
ΦL
2
(
αA
+
βL
)
2
+
L
4
(
α
+
β
)
4
(
α
+
β
)
2
L
2
(
αA
+
βL
)
2
.
(F.49)
End of Corollary.
Proof (of (F.47)).
For the proof of (
F.47
), we depart from the general eigenvalue problem (
F.15
), whose charac-
teristic equation is solved under the postulate of equal area
Λ
1
Λ
2
= 1, which is equivalent to
det[C
l
G
−
l
]= 1.
Λ
S
G
AB
|
c
AB
−
|
=0
⇒
Λ
S
−
Λ
S
tr[C
l
G
−
l
]+det[C
l
G
−
l
]=0
,
(F.50)
(tr[C
l
G
−
l
])
±
−
4(det[C
l
G
−
l
])
(tr[C
l
G
−
l
])
2
Λ
1
,
2
=
1
2
⇒
(F.51)
det[C
l
G
−
l
]=1
.
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