166 5 “Sphere to tangential plane”: polar (normal) aspect
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 (∆) = R∆. 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 systematically 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 (∆) = R∆ ,
x = r cos α = R∆ cos Λ = R
“ π
2
− Φ
”
cos Λ , y = r sin α = R∆ sin Λ = R
“ π
2
− Φ
”
sin Λ .
(5.18)
Left principal stretches:
Λ 1 =
∆
sin ∆
=
π
2
− Φ
cos Φ
, Λ 2 = 1 .
(5.19)
Left eigenvectors:
C 1 Λ 1 = E Λ
∆
sin ∆
(Easting) , C 2 Λ 2 = E Φ (Northing) .
(5.20)
Parameterized inverse mapping:
tan Λ =
y
x
, ∆ =
r
x 2 + y 2
R 2
.
(5.21)
Left maximal angular distortion:
Ω l = 2 arcsin
˛
˛
˛
˛
Λ 1 − Λ 2
Λ 1 + Λ 2
˛
˛
˛
˛ = 2 arcsin
˛
˛
˛
˛
∆ − sin ∆
∆ + sin ∆
˛
˛
˛
˛ .
(5.22)
Lemma 5.1 (EPM, equidistant mapping of the sphere to the tangential plane at the North Pole).
The equidistant mapping of the sphere to the tangential plane at the North Pole, in short, EPM
(Equidistant Polar Mapping), is parameterized by
x = R∆ cos Λ = R
π
2
− Φ
cos Λ , y = R∆ sin Λ = R
π
2
− Φ
sin Λ ,
(5.23)
subject to the left Cauchy–Green eigenspace
E Λ
∆
sin ∆ , E Φ
.
End of Lemma.