From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 25

c 11 = 2 R 2

3 π (4

− 3sin | Φ | ) ,

R π Λ cos Φ sign Φ,
c 21 = c 12 ,c 22 = 2 R π

C l =J l G r J l , G r =I 2 C l =J l J l ,

c 12 = −

⇒ (1.80)

cos 2 Φ
4 − 3sin |Φ|

( Λ 2 + π 2 ) .

Box 1.11 (Eckert II, the third problem).
G l = R 2 cos 2 Φ 0

,

C l according to Box 1.10 . (1.81)

01

Left general eigenvalue problem:
|

Λ 2 G l | Λ 1 , 2 = Λ 2 + ,−

C l − =0 ⇔

tr[C l G − l ]

4det[C l G − l ] ,

(tr[C l G − l ]) 2

= 1
2

± − (1.82)

11 =+ 2

4 − 3sin | Φ |
cos 2 Φ

(C l G − l ) 11 = c 11 G − 1

,

3 π

1
π Λ cos Φ sign Φ,

(C l G − l ) 12 = c 12 G − 1

22 = −

1
π Λ cos Φ sign Φ,

(C l G − l ) 21 = c 21 G − 1

11 = −

(1.83)

cos 2 Φ
4 − 3sin |Φ|

22 =+ 3

(C l G − l ) 22 = c 22 G − 1 ( Λ 2 + π 2 ) ,

2 π

cos 2 Φ

det[C l G − l ]=1 , tr[C l G − l ]= 2

4 − 3sin | Φ | 3

2 π

( Λ 2 + π 2 ) .

+ (1.84)

3 π cos 2 Φ 4 − 3sin | Φ |

Λ 1 Λ 2 =1:
Λ 1 Λ 2 =det[C l G − l ]=1 .

(1.85)

Left eigencolumns:
(i) √ := G 11 ( c 22

Λ 1 G 22 ) 2 + G 22 c 12

− ( G 12 =0) ,

F 11
F 22

= 1 2 π R 2 ;

cos 2 Φ
4 − 3sin |Φ|

( Λ 2 + π 2 ) − Λ 1 R 2

(1.86)

√ 1

π R 2 Λ cos Φ sign Φ

(ii) √ := G 22 ( c 22

Λ 2 G 11 ) 2 + G 11 c 12

− ( G 12 =0) ,

F 12
F 21

= 1 .

π − R 2 Λ cos Φ sign Φ

1

(1.87)

√

3 π R 2 (4

2

Λ 2 R 2 cos 2 Φ

− 3sin | Φ | ) −