Geography Reference
In-Depth Information
12
“Sphere to Cylinder”: Oblique Aspect
Mapping the sphere to a cylinder: meta-cylindrical projections in the oblique aspect. Equidis-
tant, conformal, and equal area mappings.
Cylindrical projections in the oblique aspect are mainly used to display regions which have a
predominant extent in the oblique direction, neither East-West nor North-South. In addition,
they form the most general cylindrical projections because mapping equations for projections
in the polar and the transverse aspect can easily be derived from it. This is done by setting the
corresponding latitude of the meta-North Pole
Φ
0
to a specific value:
Φ
0
=90
◦
generates cylindrical
projections in the polar aspect,
Φ
0
=0
◦
result in cylindrical projections in the transverse aspect.
As an introductory part, we present the equations for general cylindrical mappings together with
the equations for the principal stretches, before derivations for specific cylindrical map projections
of the sphere (oblique equidistant projection, oblique conformal projection and oblique equal area
projection) are given. For a first impression, have a look at Fig.
12.1
.
12-1 General Mapping Equations
Setting up general equations of the mapping “sphere to cylinder”: projections in the oblique
aspect. Meta-longitude, meta-latitude.
The general equations for mapping the sphere to a cylinder in the transverse aspect are based on
the general equation (
10.1
) of Chap.
10
, but spherical longitude
Λ
and spherical latitude
Φ
being
replaced by their counterparts meta-longitude and meta-latitude, which are indicated here by
capital letters
A
and
B
. In order to treat simultaneously both the transverse tangent cylinder and
the transverse secant cylinder, we introduce
B
0
as the meta-latitude of the meta-parallel circles
B
=
B
0
which shall be mapped equidistantly. In consequence, the general equations for this
case are given by the very general vector relation (
12.1
), taking into account the constraints (
3.51
)
and (
3.53
), namely (
12.2
). For the distortion analysis, the left principal stretches result to (
12.3
).
±
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