Geography Reference
In-Depth Information
6
“Sphere to Tangential Plane”: Transverse Aspect
Mapping the sphere to a tangential plane: meta-azimuthal projections in the transverse
aspect. Equidistant, conformal (stereographic), and equal area (transverse Lambert) map-
pings.
Azimuthal projections may be classified by reference to the point-of-contact of the plotting surface
with the Earth. While Chap.
5
treated the case of a polar azimuthal projections (azimuthal
projection in the polar aspect), this section concentrates on meta-azimuthal mappings of the
Earth onto a plane in the transverse aspect, which are often also called equatorial: the point-of-
contact (meta-North Pole) may be any point on the (conventional) equator of the reference sphere.
According to Chap.
3
, its spherical coordinates, referring to the equatorial frame of reference, are
specified through
Λ
0
∈
[0
◦
,
360
◦
]
.Φ
0
=0
◦
. In the special case
Λ
0
= 270
◦
, the meta-North Pole is
located in the West Pole and the meta-equator (then called transverse equator) agrees with the
Greenwich meridian of reference. For a first impression, consult Fig.
6.1
.
6-1 General Mapping Equations
Setting up general equations of the mapping “sphere to plane”: the meta-azimuthal projec-
tions in the transverse aspect. Meta-longitude, meta-latitude.
The general equations for meta-azimuthal projections are based on the general equation (
5.9
)of
Chap.
5
, but spherical longitude
Λ
and spherical latitude
Φ
being replaced by their counterparts
meta-longitude and meta-latitude. In order to distinguish the polar coordinate
α
in the plane from
the meta-longitude
α
as introduced in Chap.
3
,see(
3.51
)and(
3.53
), we here refer to
A
and
B
as the meta-coordinates
meta
-
longitude
and
meta
-
latitude
. In consequence, the general equations
of a meta-azimuthal mapping in the transverse aspect are provided by the vector relation (
6.1
)
taking into account the constraints (
6.2
):
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