“Sphere to Cylinder”: Pseudo-Cylindrical Projections - Map Projections: Cartographic Information Systems - page 330

13-21 Sinusoidal Pseudo-Cylindrical Mapping ( Cossin 1570 ,
N. Sanson 1675, J. Flamsteed 1646-1719), Compare
with Fig. 13.1

The mapping equations ( 13.12 ) are derived from ( 13.1 ) in connection with ( 13.8 ) and the special
instruction f ( Φ )= Φ . The left Jacobi matrix is given by ( 13.13 ) and the left Cauchy-Green
matrix (G r =I 2 )by( 13.14 ). The left principal stretches are defined by ( 13.15 ). The structure of
the coordinate lines are defined by ( 13.16 ).
x
y

= R Λ cos Φ ,

(13.12)

Φ

J l = R cos Φ

,

− Λ sin Φ

(13.13)

0 1

C l =J l G r J l = R 2

,

cos 2 Φ

−Λ sin Φ cos Φ

(13.14)

Λ sin Φ cos ΦΛ 2 sin 2 Φ +1

−

√ 2
2

2+ Λ 2 sin 2 Φ + Λ sin Φ 4+ Λ 2 sin 2 Φ,

( Λ S ) 1 , 2 = ±

(13.15)

√ 2
2

2+ Λ 2 sin 2 Φ

Λ sin Φ 4+ Λ 2 sin 2 Φ,

( Λ S ) 3 , 4 = ± −

Φ = y

R ⇒ x = RΛ cos y R ,

(13.16)

x
RΛ ⇒

x
RΛ .

cos Φ = y = R arccos

13-22 Elliptic Pseudo-Cylindrical Mapping (C. B. Mollweide),
Compare with Fig. 13.2

Starting from the general equation of an ellipse, i.e. x 2 /a 2 + y 2 /b 2 = 1, with constant minor axis
b and major axis a = a ( Λ ) being a function of spherical longitude, we fix the size of b in such a
way that a hemisphere − π/ 2 ≤ Λ ≤ π/ 2 is mapped onto a circle of the same area.