“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 193

Eigenspace analysis of the matrix pair { C l , G l } :

Λ 2 = f ( Δ ) ,f ( Δ )= d f

d Δ = c

1
cos 2 2 ⇒ Λ 2 =

c
2 R

1
cos 2 2

,

R

2

c
2 R

1
cos 2 2

Λ 2 = , lim

Δ→ 0 Λ 2 ( Δ )=1

⇒ c =2 R, (5.34)

d f
d Δ =

R
cos 2 2

,r = f ( Δ )=2 R tan Δ

2 .

Fig. 5.6. A spherical vertical section, a conformal mapping of the sphere to the tangential plane at the North
Pole, UPS (Universal Polar Stereographic Projection). This stereographic projection is highlighted by the radial
coordinate r = f ( Δ )=2 R tan Δ/ 2=N p

Note that the UPS (Universal Polar Stereographic Projec-
tion) is already found in Hipparch's works. Nowadays, it is
applied for charts in aerial navigation, for the World Map
of polar regions at latitudes larger than +80 ◦ , namely sub-
stituting the UTM (Universal Transverse Mercator Projec-
tion).

In order to complete the considerations, we present to you Fig. 5.7 , which shows a sample of the
UPS (Universal Polar Stereographic Projection) of the sphere.