Geography Reference
In-Depth Information
A
1
√
1
A
1
√
1
−
E
2
−
E
2
X
=
cos
B
∗
cos
A
∗
, Y
=
cos
B
∗
cos
A
∗
,
√
1
√
1
(H.10)
−
E
2
cos
2
A
∗
−
E
2
cos
2
A
∗
Z
=
A
1
sin
B
∗
,
X
=
A
1
cos
Φ
∗
cos
Λ
∗
, Y
=
A
1
cos
Φ
∗
sin
Λ
∗
, Z
=
A
1
(1
−
E
2
)sin
Φ
∗
1
1
1
.
(H.11)
E
2
sin
2
Φ
∗
E
2
sin
2
Φ
∗
E
2
sin
2
Φ
∗
−
−
−
The third equation of (
H.10
) as well as the second equation of (
H.10
) divided by the first equation
of (
H.10
) subject to (
H.8
) lead to the transformation
Λ
∗
,Φ
∗
}→{
A
∗
,B
∗
}
, namely tan
A
∗
=
{
Y
/X
,sin
B
∗
=
Z
/A
1
and
E
2
)tan
Φ
∗
cos(
Λ
∗
− Ω
)
,
sin
B
∗
=
cos
Φ
∗
sin(
Λ
∗
−
tan
A
∗
=
−
(1
−
Ω
)
1
− E
2
sin
2
Φ
∗
.
(H.12)
Later on, we have to use sin
A
,
cos
A
∗
,
cos
B
∗
, which is derived from (
H.12
) to coincide with
1
√
1+tan
2
A
∗
cos
A
∗
=
,
(H.13)
tan
A
∗
sin
A
∗
=
√
1+tan
2
A
∗
,
cos(
Λ
∗
−
Ω
)
cos
A
∗
=
cos
2
(
Λ
∗
−
,
(H.14)
E
2
)
2
tan
2
Φ
∗
Ω
)+(1
−
E
2
)tan
Φ
∗
−
(1
−
sin
A
∗
=
cos
2
(
Λ
∗
−
,
E
2
)
2
tan
2
Φ
∗
Ω
)+(1
−
cos
B
∗
=
1
E
2
sin
2
Φ
∗
−
cos
2
Φ
∗
sin
2
(
Λ
∗
−
−
Ω
)
1
− E
2
sin
2
Φ
∗
.
(H.15)
For the special choice
Ω
= 270
◦
, the transformation
{Λ
∗
,Φ
∗
}→{A
∗
,B
∗
}
is given by
E
2
)tan
Φ
∗
sin
Λ
∗
cos
Φ
∗
cos
Λ
∗
1
− E
2
sin
2
Φ
∗
tan
A
∗
=
(1
−
,
sin
B
∗
=
.
(H.16)
H-12 The Equiareal Mapping of the Biaxial Ellipsoid onto
a Transverse Tangent Plane
A
1
,A
2
onto the transverse
tangent plane normal to
E
3
which is parameterized either by Cartesian coordinates
We are going to construct the equiareal mapping of the biaxial ellipsoid
E
x
∗
,y
∗
}
{
or by
related by
x
∗
=
r
cos
α,y
∗
=
r
sin
α
. The mapping
A
∗
,B
∗
}
polar coordinates
{
α,r
}
{
is transverse
α
=
A
∗
+
π,r
=
r
(
A
∗
,B
∗
)
azimuthal by means of
{
}
.namely
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