Geography Reference
In-Depth Information
X
(
Λ, Φ
)=
E
1
F
(
Φ
)cos
Λ
+
E
2
F
(
Φ
)sin
Λ
+
E
3
G
(
Φ
)
.
Inverse parameterization:
{
X,Y,Z
}→{
Λ, Φ
}
,
(14.41)
Λ
(
X
) = arctan
YX
−
1
, Φ
(
X
) : the general form is not representable .
Coordinates of the left metric tensor (rotationally symmetric figure):
G
l
=
F
2
(
Φ
)
.
0
(14.42)
0
F
2
(
Φ
)+
G
2
(
Φ
)
Coordinates of the right metric tensor (cylinder):
G
r
=I
2
.
(14.43)
Parameterized mapping:
x
=
F
(0)
Λ, y
=
f
(
Φ
)
.
(14.44)
Left Jacobi matrix:
J
l
=
D
Λ
xD
Φ
x
=
F
(0)
.
0
(14.45)
D
Λ
yD
Φ
y
0
f
(
Φ
)
Left Cauchy-Green matrix:
C
l
=J
l
G
1
J
l
=
F
2
(0)
.
0
(14.46)
f
2
(
Φ
)
0
Left principal stretches:
Λ
1
=
c
11
/G
11
=
F
(0)
F
(
Φ
)
, Λ
2
=
c
22
/G
2
=
f
(
Φ
)
F
2
(
Φ
)+
G
2
(
Φ
)
.
(14.47)
Structure of the coordinate lines:
(i) :
x
=
F
(0)
Λ.
(14.48)
(Straight line through the origin for
Λ
=const
.
)
(ii) :
y
=
f
(
Φ
)
.
(14.49)
(Straight line through the origin for
Φ
=const
.
)
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