“Sphere to Cylinder”: Polar Aspect - Map Projections: Cartographic Information Systems - page 313

= cos Φ 0 1 2

( Λ 1 − 1) 2 +( Λ 2 − 1) 2

cos Φ − 2

1

cos 2 Φ (cos 2 Φ 0 − 2cos Φ cos Φ 0 +cos 2 Φ ) ,

= (10.19)

Φ

1
sin Φ

1
cos Φ ∗

d Φ ∗ (cos 2 Φ 0 2cos Φ ∗ cos Φ 0 +cos 2 Φ ∗ )

I A (conformal) = −

0

(10.20)

cos 2 Φ 0 ln tan π

2 Φ cos Φ 0 +sin Φ .

1
sin Φ 4 + Φ

= −

2

Auxiliary integrals:

cos Φ =lntan π

= 1

d Φ

4 + Φ 2 ln 1+sin Φ 1 − sin Φ ,

(10.21)

2

d Φ = Φ,

(10.22)

d Φ cos Φ =sin Φ.

In order to determine the unknown parameter Φ 0 , we restrict the Airy distortion energy integral
to the region between Φ =

85 ◦ .

±

I A (conformal) = min
⇔ (10.23)

d I A / d Φ 0 =0 ,

2sin Φ 0 cos Φ 0 ln tan π

+2 Φ sin Φ 0 =0 ,

4 + Φ

−

2

sin Φ 0 = 0 (10.24)

⇒

Φ
ln tan ( 4 + 2 ) ,

cos Φ 0 = (10.25)

Φ =85 ◦ ,

Φ 0

=61 . 72 ◦ , I A =0 . 5426 .

(10.26)

End of Proof.