Geography Reference
In-Depth Information
3
Coordinates
Coordinates (direct, transverse, oblique aspects), coordinate transformations, charts (com-
plete atlas, minimal atlas), homeomorphism, Killing vectors of symmetry, universal transverse
Mercator projection, universal oblique Mercator projection.
Coordinates are in the heart of curves and surfaces, left and right, one- and two-dimensional
Riemann manifolds. As parameters of curves and surfaces, they can be experimentally determined.
For instance, by the satellite
Global Positioning System
(“global problem solver”: GPS), we obtain
{
ellipsoidal longitude, ellipsoidal latitude, ellipsoidal height
}
as
Gauss surface normal coordinates
with respect to the
International Reference Ellipsoid
in order to coordinate points (so-called
“bench marks”) on the Earth's
topographic surface
. These coordinates are collected in
National
Data Files
, each of the order of 10
10
coordinate data. Here, we answer the following questions.
Question: “Why has the notion of the manifold and the
chart,
), been axiomatically introduced?” Ques-
tion: “What are those coordinates as collectors items in
mega data sets of the order of 10
10
coordinate data?”
U
→
φ
(
U
First, we present you a more careful explanation of the notion of a manifold, its chart, minimal
and complete atlas, in particular, the change from one chart to another chart (“Kartenwechsel”).
Second, we highlight the direct aspect, the transverse aspect, in general, the
oblique aspect
of
a surface. Such a notion is needed (i) to understand the popular optimal
Universal Transverse
Mercator Projection
(UTM) with respect to the International Reference Ellipsoid and (ii) to
understand the so-called
Universal Oblique Mercator Projection
(UOM), also called “rectified
skew orthomorphic” by
Hotine
(
1947a
,
b
,
c
,
d
)or
Hotine Oblique Mercator Projection
(HOM) by
Snyder
(
1982
, p. 76), which has been used for casting the
Heat Capacity mapping Mission
(HCMM)
imagery since 1978, particularly suitable for mapping
Landsat type data
.
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