“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 188

D Λ X

C 1 = E Λ =

D Λ X

(Easting) ,

(5.17)

D Φ X

C 2 = E Φ =

D Φ X

(Northing) .

Next, we specialize the general azimuthal mapping to generate an equidistant mapping, a series
of conformal mappings (stereographic projections) and an equiareal mapping.

5-2 Special Mapping Equations

Setting up special mappings “sphere to plane”: azimuthal projections in the normal aspect
(polar aspect). Equidistant Polar Mapping (EPM), Universal Polar Stereographic Projection
(UPS). Conformal mapping, equiareal mapping, normal projective mapping.

5-21 Equidistant Mapping (Postel Projection)

Let us postulate an equidistant mapping of the family of meridians Λ = constant, namely the
mapping r = f ( Δ )= RΔ . Indeed, R arc ( π/ 2 Φ )= r generates such a simple equidistant

mapping, which we illustrate by means of Fig. 5.4 . The corresponding distortion analysis is sys-
tematically presented in Box 5.3 .The EPM ( Equidistant Polar Mapping ) is finally summarized
in Lemma 5.1 .

−

Box 5.3 (Equidistant mapping of the sphere to the tangential plane at the North Pole).

Parameterized mapping:
α = Λ, r = f ( Δ )= RΔ,
x = r cos α = RΔ cos Λ = R π Φ cos Λ,

2 −

(5.18)

y = r sin α = RΔ sin Λ = R π Φ sin Λ.

2 −

Left principal stretches: