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(vi) Lagrange conformal polar azimuthal projection:
Δ
Δ
sin
2
cos
3
2
2
πR
2
=
1
J
sin
x
cos
4
2
d
x
=
1
2
J
6
:=
d
x,
4
0
0
sin
x
=2sin
x
2
cos
x
2
,x/
2=
y
:d
x
=2d
y,
(5.133)
cos
2
2
−
1
,
J
6
=
Δ/
2
0
cos
3
y
d
y
=
1
sin
y
=
1
2
1
2
[1
/
cos
2
y
]
Δ/
2
0
2
tan
2
Δ
J
6
=
1
2
.
The portrait of the distortion energy density and of the total distortion energy over a spherical cap
(0
◦
≤
60
◦
) is given by Figs.
5.27
and
5.28
, and Table
5.3
for six polar azimuthal projections
of type (i) equidistant (Postel), (ii) conformal (UPS), (iii) equiareal (Lambert), (iv) gnomonic, (v)
orthographic, and (vi) Lagrange conformal. Contact Appendix
A
in order to enjoy the ordering
Δ
≤
J
6
<J
5
<J
1
<J
2
<J
4
<J
3
for
<
49
◦
,
248502
and
(5.134)
J
6
<J
5
<J
1
<J
2
<J
3
<J
4
for
>
49
◦
,
248502
.
Denote for a moment the symbol
<
by “better”. Then we can make a most important qualitative
statement about the six polar azimuthal projections based upon the ordering of the respective
total distortion energies over a spherical cap, namely
conformal (Lagrange)
<
orthographic
<
equidistant (Postel)
<
conformal (UPS)
<
<
gnomonic
<
equal area (Lambert) for
<
49
◦
,
248502
and
(5.135)
conformal (Lagrange)
<
orthographic
<
equidistant (Postel)
<
conformal (UPS)
<
<
equal area (Lambert)
<
gnomonic for
>
49
◦
,
248502
.
Of course, in practice, decision makers for azimuthal map projections do not follow objective
criteria: they prefer the equiareal (Lambert) projection.
5-3 The Pseudo-Azimuthal Projection
Setting up general equations of the mapping “sphere to plane”: the pseudo-azimuthal pro-
jection in the normal aspect (polar aspect).
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