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In-Depth Information
distortion
Λ
1
(
Δ
)
Λ
2
(
Δ
) and the maximal angular shear 2 arcsin[
|
Λ
1
(
Δ
)
−
Λ
2
(
Δ
)
|
/
(
Λ
1
(
Δ
)+
Λ
2
(
Δ
))]
0
◦
,
30
◦
,
60
◦
,
90
◦
}
as functions of colatitude (polar distance
Δ
), namely for
Δ
given by
Δ
∈{
and
six typical polar azimuthal projections.
Tab l e 5 . 2
Distortion data of spherical mappings: “sphere to plane”, azimuthal projections, normal aspect (polar,
direct)
2arcsin
|
Λ
1
(
Δ
)
−
Λ
2
(
Δ
)
|
(
Λ
1
(
Δ
)+
Λ
2
(
Δ
))
Name
Δ
=
π/
2
−
Φ
Λ
1
Λ
2
Λ
1
Λ
2
(parallel circle)
(meridian)
(area distortion)
(max. ang. distortion)
0
◦
0
◦
00
Equidistant
1.000
1
1.000
30
◦
2
◦
38
(Postel)
1.047
1
1.047
60
◦
10
◦
52
1.209
1
1.209
90
◦
25
◦
39
1.571
1
1.571
0
◦
0
◦
Conformal
1.000
1.000
1.000
30
◦
0
◦
(UPS)
1.072
1.072
1.149
60
◦
0
◦
1.333
1.333
1.778
90
◦
0
◦
2.000
2.000
4.000
0
◦
0
◦
00
Equiareal
1.000
1.000
1.000
30
◦
3
◦
58
1.035
0.966
1.000
60
◦
16
◦
26
1.155
0.866
1.000
90
◦
38
◦
57
1.414
0.707
1.000
0
◦
0
◦
00
Gnomonic
1.000
1.000
1.000
30
◦
8
◦
14
1.155
1.333
1.540
60
◦
38
◦
57
2.000
4.000
8.000
90
◦
∞
∞
∞
180
◦
00
0
◦
0
◦
00
Orthographic
1
1.000
1.000
30
◦
8
◦
14
1
0.866
0.866
60
◦
38
◦
57
1
0.500
0.500
90
◦
180
◦
00
1
0
0
0
◦
0
◦
Lagrange
0.500
0.500
0.250
30
◦
0
◦
conformal
0.536
0.536
0.287
60
◦
0
◦
0.667
0.667
0.445
90
◦
0
◦
1.000
1.000
1.000
In addition, a collection of the distortion energy density tr[C
l
G
−
l
]
/
2=(
Λ
1
(
Δ
)+
Λ
2
(
Δ
))
/
2, the
arithmetic mean of the left principal stretches squared, is presented in Box
5.18
. The distortion
energy density has been given both as a function of colatitude
Δ
and latitude
Φ
.Next,bymeans
of Box
5.19
, we outline the computation of the total surface element
S
of a spherical cap between
a parallel circle of latitude
Φ
(colatitude
Δ
)and
Φ
=
π/
2 (North Pole). Finally, we are prepared
to compute by means of Box
5.20
the distortion energy over a spherical cap, relatively to the six
typical polar azimuthal projections. Note that all integral formulae were taken from
Grabner and
Hofreiter
(
1973
), in particular, 331.10
k
(p. 119) 331.11
k
(p. 120), and 333.8
b
(p. 130).
Box 5.18 (Distortion energy density tr[C
l
G
−
l
]
/
2=(
Λ
1
(
Δ
)+
Λ
2
(
Δ
))
/
2 for various azimuthal
map projections of the sphere, normal aspect (polar aspect)).
Equidistant (Postel):
sin
2
Δ
+
Δ
2
sin
2
Δ
cos
2
Φ
+(
2
−
Φ
)
2
1
2
(
Λ
1
+
Λ
2
)=
1
=
1
2
.
(5.114)
2
cos
2
Φ
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