“Sphere to Cylinder”: Polar Aspect - Map Projections: Cartographic Information Systems - page 309

10-22 Conformal Mapping (Mercator Projection)

The requirement for conformality leads to the postulate ( 10.9 ). Again, the integration constant is
determined from the additional constraint that for Φ = 0 the coordinate y should vanish, namely
y =0 ⇒ const. = 0. Therefore, the mapping equations are provided by ( 10.10 ). The left principal
stretches are provided by ( 10.11 ). The parallel circle Φ = ±Φ 0 is mapped free from any distortion.
Compare with Fig. 10.5 .

Λ 1 = Λ 2
⇒

cos Φ 0

cos Φ = 1

d f

d Φ ⇒ d f = R cos Φ 0 d Φ

(10.9)

R

cos Φ

⇒

d f = f ( Φ )= R cos Φ 0 d Φ

cos Φ = R cos Φ 0 ln cot π

+const .,

Φ
2

4 −

x
y

= R cos Φ 0 Λ = R cos Φ 0 Λ ,

(10.10)

ln cot( 4 −

Φ

ln tan( 4 + 2 )

2 )

Λ 1 = Λ 2 = cos Φ 0

cos Φ .

(10.11)

Fig. 10.5. Mapping the sphere to a cylinder: polar aspect, conformal mapping, Φ 0 =0 ◦ : tangent cylinder
(Mercator projection)