246 8 “Ellipsoid-of-revolution to tangential plane”
P c
P 0 = p 0
p = π(P )
P
k
g
H 0 = h
δ
G3
projection line
T M P 0
E A 1 ,A 2
Fig. 8.11. Fundamental perspective graph. P ∈ E
2
A 1 ,A 2 , P 0 P ∈ E 2 ∪ P P c P 0 P .
The point P c (Λ c , Φ c , H 0 = h) is defined as follows:
X c =
= E 1
A 1
1 − E 2 sin
2 Φ 0
+ H 0
Λ c , Φ c
cos Φ 0 cos Λ 0 +
+ E 2
A 1
1 − E 2 sin
2 Φ 0
+ H 0
Λ c , Φ c
cos Φ 0 sin Λ 0 +
+ E 3
A 1 (1 − E
2 )
1 − E 2 sin
2 Φ 0
+ H 0
Λ c , Φ c
sin Φ 0 .
(8.88)
Given the coordinates of the point X c , we derive h, g, and k as follows:
g :=
X c − X P
2 +
Y c − Y P
2 +
Z c − Z P
2 ,
h :=
X c − X 0
2 +
Y c − Y 0
2 +
Z c − Z 0
2 ,
k :=
X P − X 0
2 +
Y P − Y 0
2 +
Z P − Z 0
2 .
(8.89)