“Sphere to Cylinder”: Polar Aspect - Map Projections: Cartographic Information Systems - page 314

Proof (equiareal cylindrical mapping).

Next, we deal with the mapping equations of equiareal type constrained to the equidistance postu-
late on two parallel circles. Again, we enjoy the condition of an equiareal mapping of type ( 10.28 ).
x
y

= R Λ cos Φ 0 ,

(10.27)

sin Φ/ cos Φ 0

Λ 1 =1 /Λ 2 =cos Φ 0 / cos Φ,

(10.28)

Λ 2 =1 /Λ 1 =cos Φ/ cos Φ 0 .
Starting from the above relations, we obtain
( Λ 1 1) 2 +( Λ 2 1) 2

− −

=

2

= (cos Φ 0 / cos Φ − 1) 2 +(cos Φ/ cos Φ 0 − 1) 2
2

= (10.29)

cos 4 Φ 0 − 2cos Φ cos 3 Φ 0 +2cos 2 Φ cos 2 Φ 0 − 2cos 3 Φ cos Φ 0 +cos 4 Φ

= 1
2

,

cos 2 Φ 0 cos 2 Φ

Φ

1
2sin Φ

1
cos Φ ∗ cos 2 Φ 0 ×

d Φ ∗

I A (equiareal) =

0

(10.30)

(cos 4 Φ 0 − 2cos Φ ∗ cos 3 Φ 0 +2cos 2 Φ ∗ cos 2 Φ 0 − 2cos 3 Φ ∗ cos Φ 0 +cos 4 Φ ∗ ) . ×

Auxillary integrals:

d Φ ∗

cos Φ ∗ =lntan π

= 1

4 + Φ ∗ 2 ln 1+cos Φ ∗ cos Φ ∗ ,

(10.31)

2 1 −

d Φ ∗ = Φ ∗ ,

d Φ ∗ cos 2 Φ ∗ = Φ ∗
2

+ sin 2 Φ ∗
4

= Φ ∗

2 + 1 2 sin Φ ∗ cos Φ ∗ ,

(10.32)

d Φ ∗ cos Φ ∗ =sin Φ ∗ ,
d Φ ∗ cos 3 Φ ∗ = 1

3 sin Φ ∗ (2 + cos 2 Φ ∗ ) .

For the unknown parameter Φ 0 , we shall compute the Airy distortion energy for the given region
of Φ = ± 85 ◦ .

I A (equiareal) = min
⇔ (10.33)

d I A / d Φ 0 =0 ,