“Sphere to Cylinder”: Polar Aspect - Map Projections: Cartographic Information Systems - page 308

10-21 Equidistant Mapping (Plate Carree Projection)

For the first mapping of the sphere to a cylinder, we postulate that all meridians shall be mapped
equidistantly, namely

f ( Φ )
R

Λ 2 =1 ⇒ =1 ⇒ d f = R d Φ ⇒ f ( Φ )= RΦ +const . (10.6)

The integration constant is determined from the additional constraint that for Φ = 0 the coor-
dinate y should vanish, y =0 ⇒ const. = 0. We end up with the most simple mapping equa-
tions ( 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 ,

(10.7)

Φ

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 Carree projection, quadratische Plattkarte)