From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 76

A 1 cos 2 Φ A 1 (1 E 2 ) 2 −

d S 2 =

1 − E 2 sin 2 Φ d Λ 2 + (1 − E 2 sin 2 Φ ) 3 d Φ 2 .

d s 2 = r 2 cos 2 φ d λ 2 + r 2 d φ 2 . (1.244)

Left Cauthy-Green matrix: Right Cauchy-Green matrix:

d P 2 +d Q 2 =( P Λ + Q Λ )d Λ 2 + d p 2 +d q 2 =( p λ + q λ )d λ 2 +

+2( P Λ P Φ + Q Λ Q Φ )d Λ d Φ + +2( p λ p φ + q λ q φ )d λ d φ +

+( P Φ + Q Φ )d Φ 2 = +( p φ + q φ )d φ 2 =

= f 2 ( Φ )d Λ 2 + f 2 ( Φ )d Φ 2 = g 2 ( φ )d λ 2 + g 2 ( φ )d φ 2

(1.245)

⇔ ⇔
l c 11 = f 2 ( Φ ) , l c 12 =0 , l c 22 = f 2 ( Φ );
r c 11 = g 2 ( φ ) , r c 12 =0 , r c 22 = g 2 ( Φ );

Λ l G l | Λ r G r |

| C l − =0 | C r − =0

⇔⇔

f 2 ( Φ ) − Λ l G 11 0 g 2 ( φ ) − Λ r g 11 0

=0; =

f 2 ( Φ ) Λ l G 22 g 2 ( φ )

0 0 − Λ r g 22

0; (1.246)

l Λ 1 f 2 ( Φ )

f 2 ( Φ )
A 1 cos 2 Φ (1

r Λ 1 g 2 ( φ )

g 2 ( φ )
r 2 cos 2 Φ ,

E 2 sin 2 Φ ) ,

= − =

G 11 g 11

l Λ 2 = f 2 ( Φ ) r Λ 2 = g 2 ( φ ) = =

G 22 g 22

f 2 ( Φ )
A 1 (1

= g 2 ( φ )
r 2

E 2 sin 2 Φ ) , =

E 2 ) (1

− , (1.247)

−

E 2 ) 2
(1 − E 2 sin 2 Φ ) 2

f 2 ( Φ )
cos 2 Φ

g 2 ( φ )= g 2 ( φ )
cos 2 φ

(1 −

= f 2 ( Φ )=

⇒⇒
l Λ 1 = l Λ 2 = Λ l ; r Λ 1 = r Λ 2 = Λ r ;

sin 2 2 − φ

= sin 2 ( 4 −

φ

2 )cos 2 ( 4 −

φ

r 2 cos φ
g 2 ( φ )

r 2 cos 2 φ
4 r 2 tan 2 ( 4 −

2 )

2 ) = 1

=

tan 2 4 −

2

=

φ

4

φ φ

tan 2 ( 4 −

2 )

=cos 4 ( π

φ
2 ); 4 −

(1.248)

E 2 sin 2 φ
A 1 cos 2 Φ

Λ l = 1

− 1

cos 4 ( 4 −

f 2 ( Φ ) Λ r =

Φ

2 )

⇔⇔ (1.249)