From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 65

E 2
1 − E 2 sin 2 Φ

1 − 1

cos Φ > 0

P Λ P Φ
Q Λ Q Φ

P Φ Q Λ = A 1

= P Λ Q Φ − (1.202)

due to

− π/ 2 <Φ< + π/ 2 ⇒ cos Φ> 0 (1.203)

q. e. d.

2

Box 1.26 (Right Korn-Lichtenstein equations, UMP of S

r , harmonicity, orientation).

Right Korn-Lichtenstein equations:
(1st) p λ =+ g 11

g 22

g 22 q φ , φ =

−

g 11 q λ (2nd) ,

(1.204)

g 11

g 22 p φ , φ =+ g 22

(1st) q λ = −

g 11 p λ (2nd);

g 11 g 22

g 11 =

1
cos φ ; g 22 =cos φ,

(1.205)

r
cos φ ,p λ = r (KL 2nd) ,

(UMP right) q φ =

(1.206)

(UMP right) q λ =0 ,p φ =0 (KL1st) .

Right integrability conditions:
Δ λ,φ p = g 11

g 22 p φ

+ g 22

g 11 p λ

=0 , (1.207)

φ λ

Δ λ,φ q = g 11

g 22 q φ

+ g 22

g 11 q λ

=0;

φ λ

g 11

g 22 p φ =0 , g 22

r
cos Φ

g 11 p λ =

(1st)

⇒

Δ λ,φ p = g 11

g 22 p φ

+ g 22

g 11 p λ

= r
cos φ

= 0 (1.208)

λ

φ λ

q.e.d