Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 284

Box 8.10 (Computational products).

X c = X 0 + H 0 n ,

2 = X P − X 0 − H 0 n | X P − X 0 − H 0 n

X P − X c

=

2

+ H 0 , = X P − X 0 − 2 H 0 X P − X 0 | n

2 = 2 =(

H 0 ) 2 =

X P − X c | n X P − X 0 − H 0 n | n X P − X 0 | n −

) 2 + H 0 =( X P − X 0 | n − 2 H 0 X P − X 0 | n (8.112)

⇒

2 2 = X P − X 0 2 2

X P − X c − X P − X c | n − X P − X 0 | n

⇒

r = H 0

X P − X 0

2

− X P − X 0 | n

2

.

| X P − X 0 | n − H 0 |

Finally, we have to represent the five vectors n , X P , X 0 , E 1 ∗ ,and E 2 ∗ in the fixed reference
frame { E 1 , E 2 , E 3 } in order to be able to compute the projections onto X P − X 0 .InBoxes 8.11
and 8.12 , the respective relations are collected.

Box 8.11 (Representation of the vectors n , X P , X 0 , E 1 ∗ ,and E 2 ∗ in the fixed reference
frame).

⎡ ⎤

cos Φ 0 cos Λ 0
cos Φ 0 sin Λ 0
sin Φ 0

⎣ ⎦ ,

n =[ E 1 , E 2 , E 3 ] (8.113)

⎡ ⎤

N 0 cos Φ 0 cos Λ 0
N 0 cos Φ 0 sin Λ 0
N 0 (1

⎣ ⎦ ,

X 0 =[ E 1 , E 2 , E 3 ]

E 2 )sin Φ 0 −

⎡ ⎤

( N + H )cos Φ cos Λ
( N + H )cos Φ sin Λ
([ N (1

⎣ ⎦ ,

X P =[ E 1 , E 2 , E 3 ] (8.114)

E 2 )+ H ]) sin Φ

−

⎡ ⎤

sin Φ 0 cos Λ 0
sin Φ 0 cos Λ 0
−

⎣ ⎦

E 1 ∗ =[ E 1 , E 2 , E 3 ] (South) ,

cos Φ 0

⎡ ⎤

− sin Λ 0

cos Λ 0
0

⎣ ⎦

E 2 ∗ =[ E 1 , E 2 , E 3 ] (East) . (8.115)