276 10 “Sphere to cylinder”: polar aspect
10-2 Special mapping equations
Setting up special equations of the mapping “sphere to cylinder”. Equidistant mapping (Plate Carr´ ee
projection), conformal mapping (Mercator projection), equal area mapping (Lambert cylindrical equal
area projection).
10-21 Equidistant mapping (Plate Carr´ ee projection)
For the first mapping of the sphere to a cylinder, we postulate that all meridians shall be mapped
equidistantly, namely
Λ 2 = 1 ⇒
f
(Φ)
R
= 1 ⇒ df = RdΦ ⇒ f (Φ) = RΦ + const.
(10.6)
The integration constant is determined from the additional constraint that for Φ = 0 the coordinate
y should vanish, y = 0 ⇒ const. = 0. We end up with the most simple mapping equations (10.7). The
left principal stretches are provided by (10.8). For the parallel circle Φ = ±Φ 0 , we experience isometry,
conformality Λ 1 = Λ 2 = 1, and no area distortion Λ 1 Λ 2 = 1. Compare with Fig. 10.4.
x
y
= R
Λ cos Φ 0
Φ
,
(10.7)
Λ 1 =
cos Φ 0
cos Φ
, Λ 2 = 1 .
(10.8)
Fig. 10.4. Mapping the sphere to a cylinder: polar aspect, equidistant mapping, Φ 0 = 0
◦ : tangent cylinder
(Plate Carr´ ee projection, quadratische Plattkarte).