216 7 “Sphere to tangential plane”: oblique aspect
7-2 Special mapping equations
Setting up special equations of the mapping “sphere to plane”: the meta-azimuthal projections in the
oblique aspect. Equidistant mapping (oblique Postel projection), conformal mapping (oblique stereographic projection, UPS), equal area mapping (oblique Lambert projection).
7-21 Equidistant mapping (oblique Postel projection)
The oblique equidistant mapping of the sphere to a tangential plane is the generalization of equations
derived earlier. The results are stated more precisely in Box 7.1. Figure 7.2 gives an impression of the
famous oblique equidistant mapping of the sphere to a tangential plane with the meta-North Pole
located in Stuttgart/Germany (Λ 0 = 9
◦ 11
, Φ 0 = 48
◦ 46
).
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) = Rarc(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 ,
tan A =
cos Φ sin(Λ − Λ 0 )
cos Φ sin Φ 0 cos(Λ − Λ 0 ) − sin Φ cos Φ 0
, sin B = cos Φ cos Φ 0 cos(Λ − Λ 0 ) + sin Φ sin Φ 0 .
(7.4)
Left principal stretches:
Λ 1 =
π/2 − B
cos B
, Λ 2 = 1 .
(7.5)
Left eigenvectors:
C 1 Λ 1 = E A
π/2 − B
sin (π/2 − B)
, C 2 Λ 2 = E B .
(7.6)
Parameterized inverse mapping:
tan A =
y
x
, B =
π
2
−
r
x 2 + y 2
R 2
,
tan(Λ − Λ 0 ) =
sin A
tan B cos Φ 0 + cos A sin Φ 0
, sin Φ = − cos B cos A cos Φ 0 + sin B sin Φ 0 .
(7.7)
Left maximum angular distortion:
Ω l = 2 arcsin
˛
˛
˛
˛
Λ 1 − Λ 2
Λ 1 + Λ 2
˛
˛
˛
˛ = 2 arcsin
˛
˛
˛
˛
π
2
− B − cos B
π
2
− B + cos B
˛
˛
˛
˛ .
(7.8)