“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 211

Fig. 5.18. Line-of-sight, normal perspective mapping of the sphere to the tangential plane at the South Pole,
projection plane at maximal distance

For later reference, Box 5.14 outlines the corresponding data for the line-of-sight and the line-
of-contact: compare with Figs. 5.18 and 5.19 . Note that the special case where the projection
plane is placed in the center

O of the sphere S

2 R is discussed at the end of this section.

Box 5.14 (Data for the line-of-sight and the line-of-contact, critical spherical latitude, center
of perspective over the North Pole).

Tangential plane at the South Pole Tangential plane at the North Pole

sin Φ r = R

R
R + H

sin Φ r = R

R
R + H ,

D =

versus

D =

r max
R + D =

r max
2 R + H

tan Φ r = r max
H

tan Φ r = versus , (5.79)

r max =(2 R + H )tan Φ r versus r max = H tan Φ r ;

sin x R

R + H

1 R

(2 R + H ) H ; tan x = 1 , tan Φ r = 1 = (5.80)

sin 2 x

−

R
( R + H ) 2

−

r max = R 1+2 R
H

R
1+2 H

versus r max = . (5.81)