From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 70

Here, we follow the commutative diagram of Fig. 1.26 and identify the left conformal coordinates
{

A 1 ,A 1 ,A 2

P,Q } with the Universal Stereographic Projection (UPS) of E , and the right conformal

coordinates with the Universal Stereographic Projection (UPS) of S r , which is outlined in

Box 1.31 . In addition, we adopt the left and right matrices of the metric

{ p, q }

{ G l , G r } of Example 1.3 .

End of Example.

We pose five problems. (i) Do the left and right conformal maps that are parameterized by
{ as “UPS left” and “UPS right” fulfil the Korn-

Lichtenstein equations, the integrability conditions (vector-valued Laplace-Beltrami equations
of harmonicity, the condition “orientation preserving conformeomorphism”? (ii) Derive the left
and right factors of conformality, Λ 2 = Λ 1 = Λ 2 and λ 2 = λ 1 = λ 2 . Do the factors of conformality
fulfill a special Helmholtz equation? (iii) Prove that under “UPS left” as well as “UPS right”
both the ellipsoidal North Pole and the spherical North are mapped isometrically. (iv) Derive a
“simple conformal mapping” E

P ( Λ, Φ ) ,Q ( Λ, Φ ) } and { p ( λ, φ ) ,q ( λ, φ ) }

A 1 ,A 1 ,A 2 → S r . (v) Why is the conformal mapping “UPS” called

stereographic ?

S r , shorelines of the northern hemisphere,
Tissot ellipses of distortion. Graticule: 30 ◦ in longitude, 15 ◦ in latitude. Domain:

Fig. 1.26. Universal Polar Stereographic Projection of the sphere

{ 0 <λ< 2 π, 0 <φ<π/ 2 }