“Sphere to Tangential Plane”: Transverse Aspect - Map Projections: Cartographic Information Systems - page 236

Fig. 6.1. Mapping the sphere to a tangential plane: transverse aspect. Point-of-contact: meta-North Pole at
Λ 0 = 300 ◦ ,Φ 0 =0 ◦

x
y

= r cos α = f ( B )cos A ,

(6.1)

r sin α f ( B )sin A

tan A = sin( Λ

Λ 0 )
− tan Φ

−

, (6.2)

sin B =cos Φ cos( Λ − Λ 0 ) .

These equations result from ( 3.51 )and( 3.53 ) by setting the position of the meta-North Pole to
Φ 0 =0 ◦ . As a matter of course, the polar coordinate α , usually called azimuth, is not anymore
identical to the spherical longitude Λ . The images of (conventional) meridians and (conventional)
parallels lose their typical behavior of being radial straight lines and equicentric circles. Since r
equals f ( B ), parallel circles Φ = constant are mapped as a function of longitude Λ and longitude
Λ 0 of the meta-North Pole. Likewise, the image of a meridian Λ = constant becomes a complicated
curve satisfying the equation y =

− sin( Λ − Λ 0 ) / tan Φx , i.e. y is a linear function of x but with

a longitude and latitude dependent slope.

6-2 Special Mapping Equations

Setting up special equations of the mapping “sphere to plane”: the meta-azimuthal projec-
tions in the transverse aspect. Equidistant mapping (transverse Postel projection), conformal
mapping (transverse stereographic projection), equal area mapping (transverse Lambert pro-
jection).