“Sphere to Cylinder”: Polar Aspect - Map Projections: Cartographic Information Systems - page 315

I A (equiareal) = I A ( Φ 0 )=

cos 2 Φ 0 lntan π

1
2sin Φ 4 + Φ

= − 2 Φ cos Φ 0 +2sin Φ −

2

(10.34)

cos 2 Φ 0 (2 + cos 2 Φ ) ,
d I A / d Φ 0 =0
⇒
− 2sin Φ 0 cos Φ 0 ln tan π

cos Φ 0 ( Φ + 1

1

2 sin 2 Φ )+ 1

sin Φ

−

3

+2 Φ sin Φ 0 −

4 + Φ 2

Φ + 1

2 sin 2 Φ + 2

sin Φ 0
cos 2 Φ 0

sin Φ 0
cos 3 Φ 0

sin Φ (2 + cos 2 Φ )=0 ,

− (10.35)

3

sin Φ 0 =0 , cos Φ 0 =0

⇒

3cos 4 Φ 0 ln tan π

+3 Φ cos 3 Φ 0

3cos Φ 0 Φ

4 sin 2 Φ +(2+cos 2 Φ )sin Φ =0 . 4 + Φ 2 + 1

− −

2

The result is an algebraic equation of fourth order in terms of cos 4 Φ 0 = x ,namely

x 4 + ax 3 + bx + c =0 .

ln tan ( 4 + 2 ) + (2 Φ + sin 2 Φ )cos Φ 0

Φ cos 3 Φ 0

(2 + cos 2 Φ )sin Φ
3lntan( 4 + 2 )

cos 4 Φ 0 −

4lntan( 4 + 2 ) −

=0 ,

(10.36)

ln tan ( 4 + 2 ) ,b =+ (2 Φ + sin 2 Φ )

Φ

a = −

4lntan( 4 + 2 ) ,

(10.37)

(2 + cos 2 Φ )sin Φ
3lntan( 4 + 2 ) ,

c = −

Φ =85 ◦ ,

Φ 0

=49 . 31 ◦ , I A =0 . 5248 . (10.38)

End of Proof.