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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