From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 64

A 1 ,A 1 ,A 2

Box 1.25 (Left Korn-Lichtenstein equations, UMP of E , harmonicity, orientation).

Left Korn-Lichtenstein equations:
(1st) P Λ =+ G 11

G 22
G 11 Q Λ (2nd) ,

G 22 Q Φ , P Φ =

−

G 11

G 22 P Φ , Q Φ =+ G 22

(1st) Q Λ = −

G 11 P Λ (2nd);

(1.196)

G 11 G 22

E 2 sin 2 Φ
1

E 2

G 22 =cos Φ 1

− 1

cos Φ

1 −

⇔

G 11 =

E 2 sin 2 Φ ;

(1.197)

− E 2

1 −

A 1 (1 − E 2 )
1

1
cos Φ , P Λ = A 1 (KL 2nd) ,
(UMP left) Q Λ =0 , P Φ =0 (KL1st) .

(UMP left) Q Φ =

E 2 sin 2 Φ −

(1.198)

Left integrability conditions:

G 22 P Φ G 11 P Λ

G 11 G 22

Δ Λ,Φ P = + =0 , (1.199)

Φ

G 22 Q Φ G 11 Q Λ

Λ

G 11 G 22

Δ Λ,Φ Q = + =0;

Φ Λ

G 11

G 22 P Φ =0 , G 22

E 2 )
1 − E 2 sin 2 Φ

A 1 (1 − 1

cos Φ

(1st)

G 11 P Λ =

⇒

G 22 P Φ G 11 P Λ

Δ Λ,Φ P = G 11

G 22

= A 1 (1

E 2 ) − 1

cos Φ

+ = 0 (1.200)

E 2 sin 2 Φ 1 −

Λ

Φ Λ

q.e.d.

(2nd) G 11

G 22 Q Φ = A 1 , G 22 G 11 Q Λ =0

⇒

G 22 Q Φ G 11 Q Λ

G 11

+ G 22

Δ Λ,Φ Q = = 0 (1.201)

Φ Λ

q. e. d.

Left orientation: