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=
A
1
ΛPP
g
2
+
ΛP
2
gg
Λ
2
P
2
g
2
+
Λ
2
P
2
g
2
+2
Λ
2
PP
gg
+
f
2
.
P
2
g
2
ΛPP
g
2
+
ΛP
2
gg
{
c
AB
}
(G.14)
The infinitesimal distance d
S
between two points
X
and
X
+d
X
, both elements of the biaxial
ellipsoid
2
A
1
,A
2
, is in terms of the first chart represented by (
G.15
). Simultaneous diagonalization
of the two matrices
E
being positive-definite leads to the general eigenvalue
problem (
G.16
) with the eigenvalues (principal distortions)
{
c
AB
}
and
{
G
AB
}
{
Λ
1
,Λ
2
}
given by (
G.17
).
=
N
2
(
Φ
)cos
2
Φ
=
A
1
P
2
0
M
2
,
0
0
d
S
2
=
G
AB
d
U
A
d
U
B
∀{
G
AB
}
(G.15)
M
2
(
Φ
)
0
|c
AB
− Λ
S
G
AB
|
=0
,
(G.16)
Λ
S
[
Λ
2
(
P
g
+
Pg
)
2
+
f
2
+
g
2
M
2
]+
g
2
f
2
=0
⇔
Λ
S
=
1
2
(
Λ
2
P
2
g
2
+
Λ
2
P
2
g
2
+2
ΛPP
gg
+
f
2
+
g
2
M
2
)
±
(G.17)
1
4
(
Λ
2
P
2
g
2
+
ΛPP
gg
+
f
2
+
g
2
M
2
)
2
+1
⇔
±
Λ
S
=
a
Λ
1
=
a
+
b, Λ
2
=
a
±
b
⇔
−
b.
The postulate of an equiareal mapping
Λ
1
Λ
2
=1leadsto(
a
+
b
)(
a
b
2
=1or
gf
2
=1
or
g
=
f
−
1
or
g
−
1
=
f
. Obviously, the postulate of a conformal mapping
Λ
1
=
Λ
2
cannot be
fulfilled since
a
+
b
b
)=
a
2
−
−
=
a
−
b, b
= 0 holds, in general.
End of Proof.
In summarizing, we are led to the
equiareal pseudo-cylindrical mapping equations
(
G.18
)withthe
principal distortions
(
G.19
).
cos
Φ
f
(
Φ
)
=
N
(
Φ
)
Λ
cos
Φ
1
1
x
=
A
1
Λ
1
f
(
Φ
)
E
2
sin
2
Φ
−
cos
Φ
1
d
y
d
φ
=
A
1
Λ
1
,
(G.18)
E
2
sin
2
Φ
−
y
=
A
1
f
(
Φ
)
,
1
4
(
Λ
2
P
2
+
Q
2
+ 1) + 1
Λ
S
=
1
2
(
Λ
2
P
2
+
Q
2
+1)
±
(G.19)
sin
Φ
E
2
sin
Φ
cos
2
Φ
(1
∀ P
(
Φ
)=
−
1
E
2
sin
2
Φ
)
3
/
2
.
+
−
E
2
sin
2
Φ
−
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