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G
11
G
22
y
B
=
A
1
,
G
22
G
11
y
L
=0
,
G
11
=0
G
22
B
Orientation preserving conformeomorphism:
x
L
x
B
E
2
)
(1
− E
2
sin
2
B
)cos
B
>
0
,
A
1
(1
−
=(
x
L
y
B
−
x
B
y
L
)=
(15.16)
y
L
y
B
due to
− π/
2
<B<
+
π/
2
→
cos
B>
0
.
End of Example.
The UMP solution of the Korn-Lichtenstein equations subject to the vectorial Laplace-Beltrami
equations as integrability conditions and the condition of orientation conservation is based upon
the constraint of the following type: map the equator equidistantly, for instance,
x
(
B
=0)=
A
1
Λ
.
Example 15.2 (Universal Polar Stereographic Projection (UPS)).
E
2
1
−
E
E/
2
tan
π
1+
E
sin
B
1
E/
2
2
A
1
B
2
x
=
√
1
4
−
cos
L,
1+
E
−
E
sin
B
−
(15.17)
1
E/
2
tan
π
1+
E
sin
B
1
E/
2
2
A
1
E
1+
E
−
B
2
√
1
− E
2
y
=
4
−
sin
L,
−
E
sin
B
2
A
1
,A
1
,A
2
The matrix of the metric of the ellipsoid-of-revolution
E
is represented by
⎡
⎤
⎦
=
A
1
cos
2
B
.
G
11
G
12
0
⎣
1
−E
2
sin
2
B
G
MN
=
(15.18)
A
1
(1
−E
2
)
2
0
G
12
G
22
E
2
sin
2
B
)
3
(1
−
The mapping equations of type UPS imply
x
L
=
f
(
B
)sin
L, x
B
=
f
(
B
)cos
L,
−
(15.19)
y
L
=
f
(
B
)cos
L, y
B
=
f
(
B
)sin
L,
subject to
f
(
B
):=
1
E/
2
tan
π
1+
E
sin
B
1
E/
2
2
A
1
√
1
− E
2
E
1+
E
−
B
2
:=
−
4
−
,
−
E
sin
B
f
(
B
) :=
(15.20)
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