“Sphere to Cylinder”: Pseudo-Cylindrical Projections - Map Projections: Cartographic Information Systems - page 328

C l =J l G r J l =

= R 2 ⎡

⎤

g 2 cos 2 Φ Λg 2 sin Φ cos Φ + Λg g cos 2 Φ

−

⎣ ⎦ .

Λg 2 sin Φ cos Φ + Λg g cos 2 ΦΛ 2 g 2 sin 2 Φ + f 2 + Λ 2 g 2 cos 2 Φ − − 2 Λ 2 gg sin Φ cos Φ

(13.3)

The left principal stretches are determined from the characteristic equation det[C l − Λ S G l ]=0
and G l =diag[ R 2 cos 2 Φ, R 2 ], which leads to the biquadratic equation ( 13.4 ), the solution of which
is provided by ( 13.5 ).

Λ S − Λ S ( Λ 2 g 2 sin 2 Φ + f 2 + g 2 + Λ 2 g 2 cos 2 Φ 2 Λ 2 gg sin Φ cos Φ )+ g 2 f 2 =0 ,

− (13.4)

Λ S = 1

2 ( Λ 2 g 2 sin 2 Φ + f 2 + g 2 + Λ 2 g 2 cos 2 Φ

2 Λ 2 gg sin Φ cos Φ ) − ±

1
4 ( Λ 2 g 2 sin Φ + f 2 + g 2 + Λ 2 g 2 cos 2 Φ

± − 2 Λ 2 gg sin Φ cos Φ ) − g 2 f 2 =: (13.5)

=: a ± b.

The four roots are then given by ( 13.6 ). The postulate of “no area distortion”, i.e. ( 13.7 )now
determines the relationship between the unknown functions f and g as ( 13.8 ).

( Λ S ) 1 =

√ a + b,

( Λ S ) 1 , 2 = ± ±

(13.6)

( Λ S ) 2 =

± √ a ( Λ S ) 3 , 4 = ± − b,

( Λ S ) 1 , 2 ( Λ S ) 3 , 4 =

(13.7)

= √ a + b √ a

b 2 ! =1 ,

b = a 2

− −

f = g − 1 g = f − 1 . ⇔ (13.8)

We therefore end up with the general mapping equations ( 13.9 ) and the left principal
stretches ( 13.10 ). For the special case f ( Φ ) = 1, the left principal stretches can easily calculated
as ( 13.11 ), which shows that on the equator, Φ =0 ◦ , we experience isometry (conformality).