From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 77

λ r =cos 4 π

.

A 1 cos 2 Φ
1 − E 2 sin 2 sin 2 Φ

1
f 2 ( Φ ) .

φ
2

λ l =

4 −

Conformally flat left manifold: Conformally flat right manifold:

d S 2 = λ l (d P 2 +d Q 2 ) . d s 2 = λ r (d p 2 +d q 2 ) .

(1.250)

Box 1.35 (Representation of the factors of conformality in terms of conformal coordinates).

Left factor of conformality:
P ( Λ, Φ )= f ( Φ )cos Λ, Q ( Λ, Φ )= f ( Φ )sin Λ,

A 1 cos 2 f − 1 ( P 2 + Q 2 ) A 1 cos 2 Φ 1

f 2 ( Φ ) =

1
P 2 + Q 2 ,

λ l =

E 2 sin 2 f − 1 ( P 2 + Q 2 )

(1.251)

E 2 sin 2 Φ

1 −

1 −

E 2 sin 2 f − 1 ( P 2 + Q 2 )
A 1 cos 2 f − 1 ( P 2 + Q 2 )

Λ l = 1

−

( P 2 + Q 2 ) .

Right factor of conformality:
p ( λ, φ )=2 r tan π

cos λ, q ( λ, φ )=2 r tan π sin λ,

φ
2

φ
2 4 − 4 −

tan( α/ 2) = 1 − cos α
1+cos α

φ = 1 tan π

=tan 1
2 1+sin α , 2 r tan π

=

π
2 −

φ
2

− sin α φ

2

4 − 4 −

2 r 1 − sin α

1+sin α = p 2 + q 2
⇒

4 r p 2 + q 2

sin φ = 4 r 2

( p 2 + q 2 )
4 r 2 +( p 2 + q 2 ) , cos φ =

−

4 r 2 + p 2 + q 2 ,

(1.252)

tan λ

q
p 2 + q 2 , cos λ =

p
p 2 + q 2 ,

1

sin λ =

√ 1+tan 2 λ = √ 1+tan 2 λ =

λ r =cos 4 ( π

φ

16 r 4

2 )= 1 4 (1 + sin φ ) 2 = 4 − (4 r 2 + p 2 + q 2 ) 2 ,

(1 + sin φ ) 2 = (4 r 2 + p 2 + q 2 ) 2

1
cos 4 ( 4 −

4

λ r =

2 ) =

.

φ

16 r 4