Geography Reference
In-Depth Information
E
2
A
1
,A
2
⊂
R
3
and the spherical image of the surface
Fig. J.5.
Vertical section of the ellipsoid-of-revolution
∈
S
2
, position vectors
x
1
,...,
x
5
and associated surface normal vectors
normal vector (Gauss map)
ν
(
l,b
)
ν
(
x
1
)=
ν
1
,...,
ν
(
x
5
)=
ν
5
The four steps that are outlined in Box
J.4
are needed to derive the desired representation.
Comparing the spherical representation and the Cartesian representation in terms of the surface
normal vector (
J.32
)or(
J.33
) under the side condition (
J.34
), we are able to derive an
isometric
embedding
2
3
.
{
x
(
l, b
)
,y
(
l, b
)
,z
(
l, b
)
}
of
E
A
1
,A
2
⊂
R
ν
(
l, b
)=
ν
(
x, y, z
)
,
(J.32)
⎡
⎤
⎡
⎤
cos
b
cos
l
cos
b
sin
l
sin
b
x
y
E
2
1
−
⎣
⎦
=
⎣
⎦
(1
E
2
)
2
(
x
2
+
y
2
)+
z
2
,
(J.33)
−
E
2
)
z/
(1
−
f
(
x, y, z
)=
x
2
+
y
2
A
1
+
z
2
z
2
1or
x
2
+
y
2
+
E
2
=
A
1
.
A
2
−
(J.34)
1
−
2
3
).
Box J.4 (Four steps towards an isometric embedding
{
x
(
l,b
)
,y
(
l,b
)
,z
(
l,b
)
}
of
E
A
1
,A
2
⊂
R
The first operation. Use(
J.33
), the first and second equation, and add
x
2
+
y
2
cos
2
b
=(1
E
2
)
2
−
(J.35)
(1
−
E
2
)
2
(
x
2
+
y
2
)+
z
2
⇒
x
2
+
y
2
=[
x
2
+
y
2
+
z
2
/
(1
E
2
)
2
]cos
2
b.
−
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