278 10 “Sphere to cylinder”: polar aspect
10-23 Equal area mapping (Lambert projection)
Λ 1 Λ 2 = 1
⇒
cos Φ 0
cos Φ
f
(Φ)
R
= 1
⇒
df = R
cos Φ
cos Φ 0
dΦ
⇒
df =f (Φ) = R
sin Φ
cos Φ 0
+ const.
(10.12)
As before, the integration constant is determined from the additional constraint that for Φ = 0 the
coordinate y should be zero, namely y = 0 ⇒ const. = 0. Therefore, the mapping equations are
provided by (10.13). The left principal stretches are provided by (10.14). Compare with Fig. 10.6.
x
y
= R
Λ cos Φ 0
sin Φ
cos Φ 0
,
(10.13)
Λ 1 =
cos Φ 0
cos Φ
, Λ 2 =
cos Φ
cos Φ 0
.
(10.14)
Fig. 10.6. Mapping the sphere to a cylinder: polar aspect, equal area mapping, Φ 0 = 0
◦ : tangent cylinder
(normal Lambert cylindrical equal area projection).