Ellipsoid-of-Revolution to Sphere and from Sphere to Plane - Map Projections: Cartographic Information Systems - page 289

Box 9.1 (The surface normal vector G 3 , the surface tangent vectors G 1 and G 2 ).

G 1 × G 2

G 3 =

=cos Φ cos Λ E 1 +cos Φ sin Λ E 2 +sin Φ E 3 ,

(9.10)

G 1 × G 2

∂U L G K ( U 1 ,U 2 ) , det[ G KL ]= A 1 (1

∂ 2 X
∂U K ∂U L =

E 2 )cos Φ

∂ −

, (9.11)

E 2 sin 2 Φ ) 2

(1 −

3

∂ 2 X J
∂Λ 2 E J =

∂ G 1
∂U 1 =

A 1 cos Φ
(1 − E 2 sin 2 Φ ) 1 / 2 (cos Λ E 1 +sin Λ E 2 ) ,

− (9.12)

J =1

3

∂ G 1 ∂ 2 X J

∂Λ∂Φ E J =

A 1 (1 − E 2 )

∂U 2 =

E 2 sin 2 Φ ) 3 / 2
(sin Φ sin Λ E 1 − sin Φ cos Λ E 2 )
versus

(1 −

J =1

3 3 3

∂ 2 X J
∂Φ∂Λ E J =

∂ 2 X J
∂Λ∂Φ E J ,

∂ 2 X J
∂Φ 2 E J

∂ G 2
∂U 1 =

∂ G 2
∂U 2 =

J =1 J =1 J =1

A 1 (1 − E 2 )

E 2 sin 2 Φ ) 5 / 2 ×

=

(1 −

× −

cos Φ cos Λ (1 + 2 E 2 sin 2 Φ ) E 1 − cos Φ sin Λ (1 + 2 E 2 sin 2 Φ ) E 2 −

− sin Φ (1 + 2 E 2 sin 2 Φ − 3 E 2 ) E 3

(9.13)

A 1 (1 − E 2 )(1 + 2 E 2 sin 2 Φ )
(1

= − [cos Φ cos Λ E 1 +cos Φ sin Λ E 2 +

E 2 sin 2 Φ ) 5 / 2

−

1 sin Φ E 3 .

3 E 2
1+2 E 2 sin 2 Φ

−

Box 9.2 (The matrix H KL ).

A 1 cos 2 Φ

∂ G 1 /∂U 1 H 11 = G 3 | = G 3 | ∂ G Λ /∂Λ = −

E 2 sin 2 Φ ) 1 / 2 ,

(1 −

∂ G 1 /∂U 2

H 12 = G 3 | = G 3 | ∂ G Λ /∂Φ =

E 2 )

A 1 (1 −

=

E 2 sin 2 Φ ) 3 / 2 (sin Φ cos Φ sin Λ cos Λ

− sin Φ cos Φ sin Λ cos Λ )=0 ,

(1 −

H 22 = G 3 |∂ G 2 /∂U 2

= G 3 |∂ G Φ /∂Φ =