Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 283

Fig. 8.15. The ratio r/ X P − X c − X P − X c | n n = H 0 / | X P − X c | n |

In the next phase, we compute the rectangular coordinates x p and y p as well as the angular
parameters r =( x ∗ p + y ∗ p ) 1 / 2 and α = arctan y p /x p , taking advantage of the ray condition. In
Fig. 8.15 , the principle situation is illustrated.

x p = − H 0 E 1 ∗ | X P − X 0

x p = E 1 ∗ | X p − X 0
y p =

X P − X 0 | n −H 0
y p = − H 0 E 2 ∗ | X P − X 0

⇒ , (8.108)

E 2 ∗ | X p − X 0

X P − X 0 | n −H 0

α = arctan

E 2 ∗ | X p − X 0

, (8.109)

E 1 ∗ | X p − X 0

r H 0

= = | λ |

X P − X c − X P − X c | n , n | X P − X c | n |

⇒

(8.110)

r = H 0

X P − X c

2

− X P − X c | n

2

.

| X P − X c | n |

In Box 8.10 , the computational products are listed. As it is sketched in Box 8.10 , the angular
parameter r can be rewritten as ( 8.111 ).

r = H 0

X P − X 0

2

− X P − X 0 | n

2

. (8.111)

| X P − X 0 | n −H 0 |