“Sphere to Tangential Plane”: Oblique Aspect - Map Projections: Cartographic Information Systems - page 243

7-1 General Mapping Equations

Setting up general equations of the mapping sphere to plane: meta-azimuthal projections in
the oblique aspect. Meta-longitude, meta-latitude.

The general equations for mapping the sphere to the plane using a meta-azimuthal projection
in the oblique aspect involve the most general equations of meta-azimuthal mappings ( 7.1 )in
connection with the constraints ( 7.2 ) for oblique frames of references:
x
y

= r cos α = f ( B )cos A ,

(7.1)

r sin α f ( B )sin A

cos Φ sin( Λ − Λ 0 )

tan A =

sin Φ cos Φ 0 ,

cos Φ sin Φ 0 cos( Λ − Λ 0 ) −

(7.2)

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

In order not to mix up the polar coordinate α in the plane and meta-longitude α as introduced in
Chap. 3 ,see( 3.51 )and( 3.53 ), we here refer to A and B as the meta-coordinates meta - longitude
and meta - latitude . In contrast to previous sections, the latitude Φ 0 of the meta-North Pole is not
restricted to Φ 0 =90 ◦ (polar aspect) and Φ 0 =0 ◦ (transverse aspect), respectively, but can take
all values between Φ 0 = 90 ◦ and Φ 0 =90 ◦ , i.e. Λ 0 ∈ [0 ◦ , 360 ◦ ]and Φ 0 ∈ 90 ◦ , 90 ◦ ]. − [ −

7-2 Special Mapping Equations

Setting up special equations of the mapping “sphere to plane”: the meta-azimuthal pro-
jections 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 ).