“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 191

Question 2: “How can we gauge the integration constant?”
Answer 2: “The postulate lim Δ→ 0 Λ 2 ( Δ ) = 1 that is quoted
in Box 5.5 establishes an isometry at the North Pole
of the sphere. Indeed, the limit Δ 0of Λ 2 ( Δ )=

c/ (2 R cos 2 Δ/ 2) fixes c as c =2 R . Accordingly, the polar
coordinate r = f ( Δ )=2 R tan Δ/ 2 leads to the param-
eterized conformal mapping x = r ( Δ )cos Λ and y =
r ( Δ )sin Λ. ”

→

This conformal mapping is called UPS ( Universal Polar Stereographic Projection ) for the following
reason. Figure 5.6 , which illustrates this stereographic projection, focuses on the peripheral angle
Δ/ 2= π/ 4

Φ/ 2 at the South Pole. Obviously, a projection line departing from the perspective
center intersects at P

−

2 2

R . Compare with Lemma 5.2 , which summarizes the UPS
(Universal Polar Stereographic Projection).

∈ S

R and p

∈ T N S

Lemma 5.2 (UPS, conformal mapping of the sphere to the tangential plane at the North Pole).

The conformal mapping of the sphere to the tangential plane at the North Pole, in short, UPS
(Universal Polar Stereographic Projection), is parameterized by

x =2 R tan Δ

2 cos Λ =

=2 R tan π

cos Λ,

Φ
2 4 −

(5.24)

y =2 R tan Δ

2 sin Λ =

=2 R tan π

sin Λ,

Φ
2

4 −

subject to the left Cauchy-Green eigenspace
left CG eigenspace = E Λ

.

1 1

cos 2 4 −

2 , E Φ

cos 2 4 −

2

(5.25)

Φ Φ

End of Lemma.

In the case of UPS, the areal distortion increases fast with colatitude (polar distance Δ ), namely
Λ 1 Λ 2 1=cos − 4 ( π/ 4 − − Φ/ 2) − 1and Λ 1 Λ 2 − 1 →∞ for Δ → π , and this is the reason for the

application of UPS as outlined in the following historical aside.

Box 5.4 (Conformal mapping of the sphere to the tangential plane at the North Pole).

Postulate of a conformeomorphism:
Λ 1 = Λ 2 ,

(5.26)