250 8 “Ellipsoid-of-revolution to tangential plane”
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/2 and Λ 0 are
not specified. g, h, and k are defined by (8.104).
X P ∈ S
2
R : X P = R cos Φ cos ΛE 1 + R cos Φ sin ΛE 2 + R sin ΦE 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 ∆ ,
h = H 0 = 2R ,
k = X 0 − X P = R
cos 2 Φ + (1 − sin Φ) 2 = R
√
2
√
1 − sin Φ = R
√
2
√
1 − cos ∆ .
(8.104)
At this point we specialize α = Λ and r = r(Φ). Note that the analogous Φ representation is obtained
by 4g
2 h
2 = 32R
4 (1+sin Φ) = 32R
4 (1+cos ∆) and (g
2 +h
2
−k
2 )
2 = 16R
4 (1+sin Φ)
2 = 16R
4 (1+cos ∆)
2 .
α = Λ ,
r = 2R
cos Φ
1 + sin Φ
= 2R
sin ∆
1 + cos ∆
= 2R tan ∆/2 = 2R tan
π
4
−
Φ
2
.
(8.105)
N = p 0 = P 0
S = P c
δ
g
H 0 = h
k
P
r
p = π(P )
T M P 0
Φ
R
Fig. 8.13. Maximal distance mapping “sphere to tangential plane”.