Geography Reference
In-Depth Information
The second operation. Use(
J.33
), the third equation:
z
2
sin
2
b
=
(1
−
E
2
)
2
(
x
2
+
y
2
)+
z
2
⇒
(J.36)
z
2
/
(1
E
2
)=(1
E
2
)[
x
2
+
y
2
+
z
2
/
(1
E
2
)
2
]sin
2
b.
−
−
−
The third operation. Replace the terms in(
J.34
)by(
J.35
)and(
J.36
):
A
1
=
x
2
+
y
2
+
z
2
/
(1
E
2
)=[
x
2
+
y
2
+
z
2
/
(1
E
2
)
2
](1
E
2
sin
2
b
)
−
−
−
(J.37)
A
1
x
2
+
y
2
+
z
2
/
(1
− E
2
)
2
=
E
2
sin
2
b
1
−
x
2
+
y
2
+
z
2
/
(1
− E
2
)
2
=
A
1
/
1
− E
2
sin
2
b.
The fourth operation. Solve (
J.33
)for
{x, y, z}
and replace the square root by (
J.37
):
x
=
x
2
+
y
2
+
z
2
/
(1
−
E
2
)
2
cos
b
cos
l,
y
=
x
2
+
y
2
+
z
2
/
(1
−
E
2
)
2
cos
b
sin
l,
(J.38)
E
2
)
x
2
+
y
2
+
z
2
/
(1
z
=(1
−
−
E
2
)
2
sin
b.
The result is summarized in Lemma
J.3
.
2
A
1
,A
2
Lemma J.3 (
E
surface normal coordinates).
Let the surface normal vector
ν
(
l,b
) of an ellipsoid-of-revolution be the spherical image (Gauss
map) represented by (
J.25
). Then (
J.39
)and(
J.40
)holdand(
J.41
) is the parameter representation
of the ellipsoid-of-revolution
E
A
1
,A
2
in terms of
surface normal coordinates
, namely in terms of
surface normal longitude and surface normal latitude.
A
1
A
1
ν
(
l,b
)=
e
1
1
cos
b
cos
l
+
e
2
1
cos
b
sin
l
+
(J.39)
E
2
sin
2
b
E
2
sin
2
b
−
−
A
1
(1
− E
2
)
+
e
3
1
sin
b,
E
2
sin
2
b
−
⎡
⎤
cos
b
cos
l
cos
b
sin
l
(1
A
1
⎣
⎦
x
(
l,b
)=[
e
1
,
e
2
,
e
3
]
1
,
(J.40)
E
2
sin
2
b
−
E
2
)sin
b
−
A
1
cos
b
cos
l
A
1
cos
b
sin
l
x
(
l,b
)=
1
,y
(
l,b
)=
1
,
(J.41)
E
2
sin
2
b
E
2
sin
2
b
−
−
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