10-3 Optimal cylinder projections 281
Starting from the above relations, we obtain
(Λ 1 − 1)
2 + (Λ 2 − 1)
2
2
=
cos Φ 0
cos Φ
− 1
2
=
1
cos 2 Φ
(cos
2 Φ 0 − 2 cos Φ cos Φ 0 + cos
2 Φ) , (10.19)
I A (conformal) =
1
sin Φ
Φ
0
dΦ
∗
1
cos Φ ∗ (cos
2 Φ 0 − 2 cos Φ
∗ cos Φ 0 + cos
2 Φ
∗ )
=
1
sin Φ
cos
2 Φ 0 ln tan
π
4
+
Φ
2
− 2Φ cos Φ 0 + sin Φ
.
(10.20)
Auxillary integrals:
dΦ
cos Φ
= ln tan
π
4
+
Φ
2
=
1
2
ln
1 + sin Φ
1 − sin Φ
,
(10.21)
dΦ = Φ ,
dΦ cos Φ = sin Φ .
(10.22)
I n 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
⇔
dI A /dΦ 0 = 0 ,
(10.23)
−2 sin
Φ 0 cos
Φ 0 ln tan
π
4
+
Φ
2
+ 2Φ sin
Φ 0 = 0 ,
sin
Φ 0 = 0
⇒
(10.24)
cos
Φ 0 =
Φ
ln tan
π
4 +
Φ
2
,
(10.25)
Φ = 85
◦ ,
Φ 0
= 6 1 .72
◦ ,
I A = 0.5426 .
(10.26 )
End of Proof.