Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 275

Fig. 8.8. Perspective mappings of a perspective center P c to the plane which is at the minimal distance from
an ellipsoid-of-revolution E 2 A 1 ,A 2

In the passages that follow, we use the representation of the distances g, h ,and k in surface
normal ellipsoidal coordinates which are supported by E

A 1 ,A 2

.

The point P c ( Λ c ,Φ c ,H 0 = h )isdefinedasfollows:

X c =

+ H 0 ( Λ c ,Φ c ) cos Φ 0 cos Λ 0 +

A 1

= E 1 1

E 2 sin 2 Φ 0 −

+ E 2 + H 0 ( Λ c ,Φ c ) cos Φ 0 sin Λ 0 +

A 1

1

(8.88)

E 2 sin 2 Φ 0

−

+ E 3 A 1 (1 + H 0 ( Λ c ,Φ c ) sin Φ 0 .

E 2 )

−

1

E 2 sin 2 Φ 0

−

Given the coordinates of the point X c , we derive h, g ,and k as follows:
g := ( X c

− X P ) 2 +( Y c − X P ) 2 +( Z c − Z P ) 2 ,

h := ( X c

− X 0 ) 2 +( Y c − X 0 ) 2 +( Z c − Z 0 ) 2 , (8.89)

k := ( X P − X 0 ) 2 +( Y P − Y 0 ) 2 +( Z P − Z 0 ) 2 .