170 5 “Sphere to tangential plane”: polar (normal) aspect
N
S
T N S
2
R
P
p
O
∆
2
=
π
4
−
Φ
2
∆ =
π
2
− Φ
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 (∆) = 2R tan ∆/2 = Np.
Historical
aside.
Note that the UPS (Universal Polar Stereographic Projection) 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 substituting the UTM (Universal Transverse
Mercator Projection).
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.
Fig. 5.7. Conformal map (UPS) of the sphere S
2
R onto the tangent space T N S
2
R , Tissot ellipses, polar aspect,
graticule 15
◦ , shorelines.