Geography Reference
In-Depth Information
8-32 The Special Case “Sphere to Tangential Plane”
Let us here specialize to the mapping “sphere to tangential plane”, namely to the case where
P
0
is located at the North pole and
P
c
at the South pole, and we only treat the case where
P
0
is
at maximal distance from
P
c
. Consult Fig.
8.13
for more details. We here begin with identifying
the points
X
0
and
X
c
, respectively, by their coordinates
{
0
,
0
,Z
}
and
{
0
,
0
,
−
Z
}
, respectively.
Φ
0
=
π/
2and
Λ
0
are not specified.
g, h
,and
k
are defined by (
8.104
).
2
X
P
∈
S
R
:
X
P
=
R
cos
Φ
cos
Λ
E
1
+
R
cos
Φ
sin
Λ
E
2
+
R
sin
Φ
E
3
,
(8.103)
g
=
X
c
−
X
P
=
R
2
cos
2
Φ
cos
2
Λ
+
R
2
cos
2
Φ
sin
2
Λ
+
R
2
(1 + sin
Φ
)
2
=
=
R
√
2
√
1+sin
Φ
=
R
√
2
√
1+cos
Δ,
(8.104)
h
=
H
0
=2
R,
=
R
cos
2
Φ
+(1
sin
Φ
)
2
=
R
√
2
√
1
sin
Φ
=
R
√
2
√
1
k
=
X
0
−
X
P
−
−
−
cos
Δ.
At this point we specialize
α
=
Λ
and
r
=
r
(
Φ
). Note that the analogous
Φ
representation is
obtained by 4
g
2
h
2
=32
R
4
(1 + sin
Φ
)=32
R
4
(1 + cos
Δ
)and(
g
2
+
h
2
k
2
)
2
=16
R
4
(1 + sin
Φ
)
2
=
−
16
R
4
(1 + cos
Δ
)
2
.
α
=
Λ,
1+cos
Δ
=2
R
tan
Δ/
2=2
R
tan
4
−
2
.
(8.105)
r
=2
R
cos
Φ
sin
Δ
Φ
1+sin
Φ
=2
R
Fig. 8.13.
Maximal distance mapping “sphere to tangential plane”
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