Geography Reference
In-Depth Information
11
“Sphere to Cylinder”: Transverse Aspect
Mapping the sphere to a cylinder: meta-cylindrical projections in the transverse aspect.
Equidistant, conformal, and equal area mappings.
Among cylindrical projections, mappings in the transverse aspect play the most important role.
Although many worldwide adopted legal map projections use the ellipsoid-of-revolution as the
reference figure for the Earth, the spherical variant forms the basis for the Universal Transverse
Mercator (UTM) grid and projection. In the subsequent chapter, we first introduce the general
concept of a cylindrical projection in the transverse aspect. Following this, three special map
projections are presented: (i) the equidistant mapping (transverse Plate Carree projection), (ii) the
conformal mapping (transverse Mercator projection), and (iii) the equal area mapping (transverse
Lambert projection). The transverse Mercator projection is especially appropriate for regions with
a predominant North-South extent. As in previous chapters, the two possible cases of a tangent
and a secant cylinder are treated simultaneously by introducing the meta-latitude
B
=
±B
1
of a
meta-parallel circle which is mapped equidistantly. For a first impression, have a look at Fig.
11.1
.
11-1 General Mapping Equations
Setting up general equations of the mapping “sphere to cylinder”: projections in the trans-
verse aspect. Meta-spherical longitude, meta-spherical 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 the transverse
tangent cylinder
and
the transverse
secant cylinder
, we introduce
B
0
as the meta-latitude of those 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 (
11.1
), taking into account the constraints (
3.51
)
and (
3.53
)for
Φ
0
=0
◦
.namely(
11.2
). For the distortion analysis, the left principal stretches
result to (
11.3
).
±
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