From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 74

− f ( Φ cos Λ,

G 22

1 − E 2 1

cos Φ [ −f ( Φ )sin Λ ] , G 11 P Λ =

(1.233)

E 2 sin 2 Φ 1 −

1 − E 2 sin 2 Φ
1

1
cos Φ

Δ Λ,Φ P = −f ( Φ )cos Λ −

f ( Φ )cos Λ =

− E 2

E 2 E 2

1 − 1

cos Φ f ( Φ )cos Λ

1 − 1

cos Φ f ( Φ )cos Λ =0 q . e . d

=+ −

E 2 sin 2 Φ E 2 sin 2 Φ

1 − 1 −

(2nd) G 11

f ( Φ )sin Λ =

E 2 sin 2 Φ
1

E 2
1 − E 2 sin 2 Φ G 22 Q Φ =cos Φ 1

− 1 − 1

cos Φ

−

− E 2

− f ( Φ )sin Λ,

G 22
G 11 Q Λ =

E 2

1 − 1

cos Φ [+ f ( Φ )cos Λ ] ,

(1.234)

E 2 sin 2 Φ

1 −

E 2 sin 2 Φ
1 − E 2

1
cos Φ

1 −

f ( Φ )sin Λ

Δ Λ,Φ Q = − − f ( Φ )sin Λ =

E 2 E 2 1 − 1

cos Φ f ( Φ )sin Λ

1 − 1

cos Φ f ( Φ )sin Λ =0

=+ −

E 2 sin 2 Φ E 2 sin 2 Φ 1 − 1 −

q. e. d.
Left orientation:

P Λ P Φ
Q Λ Q Φ

ff (sin 2 Λ +cos 2 Λ )= f ( Φ ) f ( Φ ) =

= P Λ Q Φ − P Φ Q Λ = − − (1.235)

1 − E 2
1 − E 2 sin 2 Φ

1

cos Φ f 2 ( Φ ) > 0 due to

= − π/ 2 <Φ< + π/ 2 ⇒ cos Φ> 0

q. e. d.

r , harmonicity, orientation).

Box 1.33 (Right Korn-Lichtenstein equations, UPS of S

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

g 22

g 22 q φ , φ = − g 11 q λ (2nd) ,

(1.236)

g 11

g 22 p φ , φ =+ g 22

(1st) q λ = −

g 11 p λ (2nd);

g 11
g 22 =cos φ,

g 22
g 11 =

1
cos φ ;

(1.237)

(UPS right) q φ = g ( φ )sin λ, p λ = −g ( φ )sin λ,
(UPS right) q λ = g ( φ )cos λ, p φ = −g ( φ )cos λ,

gφ
cos φ

g ( φ )=

− (1.238)