Geography Reference
In-Depth Information
Box 5.20 (Distortion energy of six polar azimuthal projections over a spherical cap).
General representation of the distortion energy:
J
:=
d
S
1
2
tr[C
l
G
−
l
]=
d
S
1
2
(
Λ
1
+
Λ
2
)
,
J
=2
πR
2
Δ
0
sin
x
1
2
(
Λ
1
(
x
)+
Λ
2
(
x
))d
x.
(5.123)
(i) Equidistant (Postel) polar azimuthal projection:
Δ
Δ
2
πR
2
=
1
J
sin
x
d
x
+
1
x
2
J
1
:=
sin
x
d
x.
(5.124)
2
2
0
0
1st integral:
Δ
K
sin
x
d
x
=
Δ
2
x
2
1)
k
+1
2(2
2
k−
1
−
1)
(2 + 2
k
)(2
k
)!
B
2
k
Δ
2+2
k
.
+ lim
K→∞
(
−
(5.125)
2
0
k
=1
Bernoulli numbers:
1
2
,B
2
=+
1
30
,B
6
=+
1
1
1
30
,
B
0
=1
,B
1
=
−
6
,B
4
=
−
42
,B
8
=
−
(5.126)
B
10
=+
5
2730
,B
14
=+
7
691
3617
510
.
66
,B
12
=
−
6
,B
16
=
−
2nd integral:
Δ
cos
x
]
0
=1
sin
x
d
x
=[
−
−
cos
Δ.
(5.127)
0
J
1
:
K
(
−
1)
k
+1
(2
2
k−
1
−
1)
J
1
=
1
(2 + 2
k
)(2
k
)!
B
2
k
Δ
2+2
k
+
1
4
Δ
2
+ lim
2
(1
−
cos
Δ
)
.
(5.128)
K→∞
k
=1
(ii) Conformal polar azimuthal projection (UPS):
2
πR
2
=
Δ
d
x
=2
Δ
0
sin
2
cos
3
2
J
sin
x
cos
4
2
J
2
:=
d
x,
0
(5.129)
cos
2
2
cos
2
2
J
2
/
2=[1
/
cos
2
x
cos
2
2
−
1=
1
1
−
,J
2
=2tan
2
Δ
2
]
0
=
2
.
(iii) Equiareal (Lambert) polar azimuthal projection:
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