Geography Reference
In-Depth Information
The procedure of how to set up special equations of the mapping sphere to cylinder in the oblique
aspect (oblique equidistant mapping, oblique conformal mapping, oblique equal area mapping)
can be easily deduced from the preceding chapters. Far easier, in the mapping equations as well
in the equations for the left principal stretches defined in Chap.
10
, conventional coordinates
spherical longitude Λ
and
spherical latitudes Φ
and
Φ
0
are simply replaced by their corresponding
items
meta-spherical longitude A
and
meta-spherical latitudes B
and
B
0
. Transformations of
conventional spherical coordinates to meta-spherical coordinates is performed using (
12.2
).
12-2 Special Mapping Equations
Setting up special equations of the mapping “sphere to cylinder”: meta-cylindrical projections
in the
oblique aspect
. Equidistant mapping (oblique Plate Carree projection), conformal map-
ping (oblique Mercator projection), equal area mapping (oblique Lambert cylindrical equal
area projection).
12-21 Equidistant Mapping (Oblique Plate Carree Projection),
Compare with Fig.
12.2
x
y
=
R
A
cos
B
0
,
(12.4)
B
Λ
1
=
cos
B
0
cos
B
, Λ
2
=1
.
12-22 Conformal Mapping (Oblique Mercator Projection), Compare
with Fig.
12.3
x
y
=
R
cos
B
0
=
R
cos
B
0
=
A
A
ln cot
4
−
2
ln tan
4
+
2
B
=
R
cos
B
0
=
A
(12.5)
1
2
ln
1+sin
B
1
−
sin
B
=
R
cos
B
0
A
,
artanh(sin
B
)
Λ
1
=
Λ
2
=
cos
B
0
cos
B
.
(12.6)
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