5-2 Special mapping equations 199
Box 5.20 (Distortion energy of six polar azimuthal projections over a spherical cap).
General representation of the distortion energy:
J :=
Z
dS
1
2
tr
ˆ
C l G
−1
l
˜
=
Z
dS
1
2
`
Λ
2
1 + Λ
2
2
´
, J = 2πR
2
Z ∆
0
sin x
1
2
`
Λ
2
1 (x) + Λ
2
2 (x)
´
dx . (5.123)
(i) Equidistant (Postel) polar azimuthal projection:
J 1 :=
J
2πR 2 =
1
2
Z ∆
0
x
2
sin x
dx +
1
2
Z ∆
0
sin xdx .
(5.124)
1st integral:
Z ∆
0
x
2
sin x
dx =
∆
2
2
+ lim
K→∞
K
X
k=1
(−1)
k+1 2
`
2
2k−1 − 1
´
(2 + 2k)(2k)!
B 2k ∆
2+2k .
(5.125)
Bernoulli numbers:
B 0 = 1 , B 1 = −
1
2
, B 2 = +
1
6
, B 4 = −
1
30
, B 6 = +
1
42
, B 8 = −
1
30
,
B 10 = +
5
66
, B 12 = −
691
2730
, B 14 = +
7
6
, B 16 = −
3617
510
.
(5.126)
2nd integral:
Z ∆
0
sin xdx = [− cos x]
∆
0 = 1 − cos ∆ .
(5.127)
J 1 :
J 1 =
1
4
∆
2 + lim
K→∞
K
X
k=1
(−1)
k+1
`
2
2k−1 − 1
´
(2 + 2k)(2k)!
B 2k ∆
2+2k +
1
2
(1 − cos ∆) .
(5.128)
(ii) Conformal polar azimuthal projection (UPS):
J 2 :=
J
2πR 2 =
Z ∆
0
sin x
cos 4 x
2
dx = 2
Z ∆
0
sin
x
2
cos 3 x
2
dx ,
J 2 /2 = [1/ cos
2 x
2
]
∆
0 =
1
cos 2 ∆
2
− 1 =
1 − cos
2 ∆
2
cos 2 ∆
2
, J 2 = 2 tan
2 ∆
2
.
(5.129)
(iii) Equiareal (Lambert) polar azimuthal projection:
J 3 :=
J
2πR 2 =
Z ∆
0
sin x
1 + cos
4 x
2
cos 2 x
2
dx = 2
Z ∆
0
sin
x
2
1 + cos
4 x
2
cos
x
2
dx =
= 2
Z ∆
0
tan
x
2
“
1 + cos
4 x
2
”
dx ,
x/2 := y
⇒
Z
tan
x
2
“
1 + cos
4 x
2
”
dx = 2
Z
tan y
`
1 + cos
4 y
´
dy =
= 2
Z
tan ydy + 2
Z
sin y cos
3 ydy = −2 ln cos y −
1
2
cos
4 y = −2 ln cos
x
2
−
1
2
cos
4 x
2
,
h
ln cos
x
2
i ∆
0
= ln cos
∆
2
,
h
cos
4 x
2
i ∆
0
= cos
4 ∆
2
− 1 ,
J 3 = 1 − cos
4 ∆
2
− 4 ln cos
∆
2
.
(5.130)