“Sphere to Tangential Plane”: Oblique Aspect - Map Projections: Cartographic Information Systems - page 247

7-23 Equal Area Mapping (Oblique Lambert Projection)

The oblique equal area mapping of the sphere to a tangential plane is the generalization of
equations derived earlier. The results are stated more precisely in Box 7.3 . Figure 7.4 gives an
impression of the famous oblique conformal mapping of the sphere to a tangential plane with the
meta-North Pole located at Perth ( Λ 0 = − 115 ◦ 52 ,Φ 0 = − 31 ◦ 57 ).

Box 7.3 (Oblique equal area mapping of the sphere to a plane at the meta-North Pole
Λ 0

[0 ◦ , 360 ◦ ] ,Φ 0 90 ◦ , 90 ◦ ]).

∈ ∈ [ −

Parameterized mapping:

α = A, r = f ( B )=2 R tan π

, x =2 R sin π cos A,

B
2

B
2

4 − 4 −

y =2 R sin π

sin A,

B
2

4 −

(7.14)

cos Φ sin( Λ − Λ 0 )

tan A =

sin Φ cos Φ 0 ,

cos Φ sin Φ 0 cos( Λ − Λ 0 ) −

sin B =cos Φ cos Φ 0 cos( Λ − Λ 0 )+sin Φ sin Φ 0 .

Left principal stretches:

2 , Λ 2 =cos π

.

1 B

2 cos 4 −

Λ 1 =

4 −

(7.15)

B

Left eigenvectors:

2 , C 2 Λ 2 = E B cos π

.

1
cos 4 −

B
2

C 1 Λ 1 = E A

4 −

(7.16)

B

Parameterized inverse mapping:

x , sin π

=

2 R x 2 + y 2 ,

tan A = y

B
2

1

4 −

(7.17)

sin A
tan B cos Φ 0 +cos A sin Φ 0 ,

tan( Λ − Λ 0 )=

sin Φ = − cos B cos A cos Φ 0 +sin B sin Φ 0 .