“Sphere to Tangential Plane”: Oblique Aspect - Map Projections: Cartographic Information Systems - page 244

Box 7.1 (Oblique equidistant 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 )= R arc( π/ 2 − B ) , (7.3)

x = r cos α = R ( π/ 2 − B )cos A, y = r sin α = R ( π/ 2 − B )sin A, (7.4)

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:

Λ 1 = π/ 2

B
cos B

−

, Λ 2 =1 . (7.5)

Left eigenvectors:

B
sin( π/ 2 − B ) , C 2 Λ 2 = E B .

π/ 2 −

C 1 Λ 1 = E A (7.6)

Parameterized inverse mapping:

x 2 + x 2
R 2

tan A = y

x , B = π

2 −

, (7.7)

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 .

Left maximum angular distortion:

Ω t =2arcsin =2arcsin

π
2

Λ 1 − Λ 2
Λ 1 + Λ 2

− B − cos B

. (7.8)

π
2

− B +cos B