“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 210

O ∗ south on the NS line, projection plane at the North
Pole), in contrast, Fig. 5.16 illustrates the second chart (

While Fig. 5.10 illustrates the first chart (

O ∗ north on the SN line, projection plane

at the South Pole).

Maximal extension of a normal central perspective of the sphere, the functions √ 1+ x and 1 / √ 1+ x , Tab l e 5 . 1

domain 0 ≤ x ≤ 5 ,x := 2 R/H

√ 1+ x

1 / √ 1+ x x

0.0 1.0 1.0

0.5 1.225 0.816

1.0 1.414 0.707

1.5 1.581 0.632

2.0 1.732 0.57

2.5 1.871 0.535

3.0 2.0 0.50

3.5 2.121 0.471

4.0 2.236 0.47

4.5 2.345 0.426

5.0 2.49 0.408

The normal perspective mapping, which corresponds to Fig. 5.12 , but where the perspective
center

O ∗ is placed north on the SN line and the projection plane is identified as the tangent
plane at the North Pole, is presented in Fig. 5.17 .

Fig. 5.17. Spherical vertical section, general normal perspective mapping of the sphere to the tangential plane
which is at minimal distance to O ∗ : D = R + H