“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 190

The equidistant mapping of the sphere to the tangen-
tial plane at the North Pole is associated with the name
of G. Postel (1510-1581), though it was already known
to Mercator ( 1569 ). Both used it for mapping the polar
regions. Nowadays, it is applied for plotting stars around
the North Pole, for the World Map 1:2.5 Mio, and for charts
in aerial navigation, remote sensing, and seismology.

In order to complete the considerations, we present to you Fig. 5.5 , which shows a sample of a
polar equidistant map of the sphere.

S 2 R onto the tangent space T N S 2 R , Tissot ellipses, polar aspect,

Fig. 5.5. An equidistant mapping of the sphere

graticule 15 ◦ , shorelines

5-22 Conformal Mapping (Stereographic Projection, UPS)

Let us postulate a conformal mapping by means of the canonical measure of conformality, i.e.
Λ 1 = Λ 2 . Such a conformal mapping of the sphere to the tangential plane of the North Pole is
illustrated by means of Fig. 5.6 that follows after the Boxes 5.4 and 5.5 .

Question 1: “How can we generate the conformal map-
ping equations?” Answer 1: “Following the procedure of
Boxes 5.4 and 5.5 , we here depart from the general repre-
sentation of Λ 1 and Λ 2 . By means of separation of variables,
the relation Λ 1 = Λ 2 leadsustod f/f =d Δ/ sin Δ as the
characteristic differential equations. Integration of the left
side as well as of the right side leads us to the indefinite
mapping equation f ( Δ )= c tan Δ/ 2.”