From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 75

⇒

g ( φ )
cos φ sin λ q. e. d.
(KL 1st) g ( φ )cos λ = − cos φg ( φ )cos λ q. e. d.
Right integrability conditions:

(KL 2nd) g ( φ )sin λ = −

Δ λ,φ p = g 11

g 22 p φ

+ g 22

g 11 p λ

=0 , (1.239)

φ λ

Δ λ,φ q = g 11

g 22 q φ

+ g 22

g 11 q λ

=0 .

φ λ

(1st) g 11

g 22 p φ =cos φg ( φ )cos λ = −g ( φ )cos λ,

g 22

1
cos φ [ − g ( φ )sin λ ]= −

g ( φ )
cos φ sin λ g 11 p λ =

(1.240)

⇒

g ( φ ) g ( φ )

cos φ ⇒ Δ λ,φ p =0 q.e.d.

Δ λ,φ p = −g ( φ )cos λ −

cos φ cos λ, g ( φ )= −

g 11

g 22 q φ =cos φg ( φ )sin λ =

(2nd) − g ( φ )sin λ,

g 22

1
cos φ g ( φ )cos λ

g 11 q λ =

(1.241)

⇒

g ( φ ) g ( φ )

cos φ ⇒

g ( φ )sin λ

cos φ sin λ, g ( φ )=

Δ λ,Φ q = − − − Δ λ,φ q =0 q.e.d.

Right orientation:

g ( φ ) g ( φ )= g 2 ( φ )

p λ p φ
q λ q φ

= p λ q φ − p φ q λ = −

cos φ > 0

(1.242)

due to

− π/ 2 <φ< + π/ 2 ⇒ cos φ> 0 q. e. d. (1.243)

Box 1.34 (Conformally flat left and right manifold, ellipsoid-of-revolution versus sphere).

Left manifold, Right manifold,

{Λ, Φ} left coordinates: {λ, φ} right coordinates: