Geography Reference
In-Depth Information
l
1
)
,
l
1
l
2
(3
q
1
b
1
−
−
1
(
q
1
b
1
+
l
1
)
3
×
β
2
=
(16.54)
×
2
q
1
b
1
l
1
(
q
1
b
1
+
l
1
)
s
2
+
s
1
(
q
1
b
2
+
q
2
b
1
)(
−
3
q
1
b
1
+
l
1
)
l
1
+(
3
l
1
+
q
1
b
1
)
q
1
b
1
l
1
.
−
End of Theorem.
The coecients
{
q
1
,q
2
}
,
{
s
1
,s
2
}
,
{
b
1
,b
2
}
,and
{
l
1
,l
2
}
are collected in the following Boxes
16.2
-
16.5
.
Box 16.2 (Isometric latitude
q
(
b
) as a function of latitude
b
).
Power series expansion
Δq
=
N
r
=1
q
r
Δb
r
up to order
N
=2
(higher-order terms are given by
Engels and Grafarend 1995
):
E
2
1
−
q
1
:=
E
2
sin
2
B
0
)
,
(16.55)
cos
B
0
(1
−
sin
B
0
2cos
2
B
0
(1
E
2
sin
2
B
0
)
2
[1 +
E
2
(1
3sin
2
B
0
)+
E
4
(
2+3sin
2
B
0
)]
.
q
2
:=
−
−
−
Box 16.3 (Arc length of the oblique meta-equator).
Power series expansion
Δs
=
N
r
=1
s
r
Δα
r
up to order
N
=2:
s
1
(
α
0
):=
A
1
1
−
E
2
cos
2
α
0
,
A
1
E
2
sin
α
0
cos
α
0
s
2
(
α
0
):=
1
2
√
1
E
2
cos
2
α
0
.
(16.56)
−
Box 16.4 (Latitude
B
(
α
) as a function of meta-longitude
α
).
Power series expansion
Δb
=
N
r
=1
b
r
Δα
r
uptoorderN
=2:
A
2
A
1
(1
E
2
)sin
i
cos
α
0
−
b
1
:=
−
E
2
)
2
+
E
2
A
2
sin
2
i
sin
2
α
0
]
[
A
1
+(
A
2
cos
2
i
[
A
1
(1
−
A
1
)sin
2
α
0
]
−
1
/
2
,
−
A
2
A
1
(1
− E
2
)sin
i
2
b
2
:=
−
E
2
)
2
+
E
2
A
2
sin
2
i
sin
2
α
0
]
2
[
A
1
+(
A
2
cos
2
i
[
A
1
(1
−
A
1
)sin
2
α
0
]
−
3
/
2
−
×
(16.57)
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