“Sphere to Tangential Plane”: Transverse Aspect - Map Projections: Cartographic Information Systems - page 237

6-21 Equidistant Mapping (Transverse Postel Projection)

Let us formulate a transverse equidistant mapping of the sphere to a plane by the postulate
that for the family of meta-meridians A = constant the relations ( 6.3 ) hold true. The mapping
equations and the corresponding distortion analysis are systematically presented in Box 6.1 .A
sketch of this mapping with the choice of Λ 0 = 270 ◦ is given in Fig. 6.2 .

r = R arc( π/ 2 − B ) , (6.3)

sin B =cos Φ cos( Λ − Λ 0 ) .

Fig. 6.2. Mapping the sphere to a tangential plane: transverse aspect, equidistant mapping. Point-of-contact:
meta-North Pole at Λ 0 = 270 ◦ ,Φ 0 =0 ◦

Box 6.1 (Transverse equidistant mapping of the sphere to a plane at the meta-North Pole.
Parameters: Λ 0 ∈ [0 ◦ , 360 ◦ ] ,Φ 0 =0).

Parameterized mapping:

α = A, r = f ( B )= R arc( π/ 2 − B ) , (6.4)

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

(6.5)

tan A = sin( Λ

− Λ 0 )

, sin B =cos Φ cos( Λ − Λ 0 ) .

− tan Φ