3 Coordinates
Coordinates (direct, transverse, oblique aspects), coordinate transformations, charts (complete 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.
Question: “Why has the notion of the manifold and the chart, U → φ(U), been axiomatically
introduced?” Question: “What are those coordinates as collectors items in mega data sets
of the order of 10
10 coordinate data?”
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 M. Hotine
(1947a–e) or Hotine Oblique Mercator Projection (HOM) by J. P. 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.
3-1 Coordinates relating to manifolds
Coordinates relating to manifolds, in particular, differential manifolds, elements of topology (Hausdorff
topological space, open and closed domains).
Let us here first consider manifolds and their charts, the complete atlas, and the minimal atlas. Let
us here first define the term chart according to Definition 3.1.
Definition 3.1 (Manifold, chart).
An n-dimensional manifold M
n is locally an n-dimensional Hausdorff topological space. That is to
any element of M
n , called “point”, there is a connected open neighborhood U and a homeomorphism
Φ : U → φ(U) to an open set φ(U) of R
n . M
n is locally homeomorph to R
n . Any homeomorphism φ
is called a chart of M
n . A one-dimensional manifold is called curve, a two-dimensional manifold with
Riemann metric and without Cartan torsion is called surface, and a compact, connected manifold
without boundary is called closed manifold.
End of Definition.
Indeed, we implemented many unknown notions from the theory of morphism, in particular, from
topology, but we do not hope to lose you, the map maker. Therefore, just follow us to stroll along
Hausdorff Street, Bonn (Germany) to meditate over Haussdorff’s axiom of separation (T
2 ) within
Listing’s topoploy, shortly reviewed in the Appendix. What is more important here is the question
that follows.
Coordinates (direct, transverse, oblique aspects), coordinate transformations, charts (complete 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.
Question: “Why has the notion of the manifold and the chart, U → φ(U), been axiomatically
introduced?” Question: “What are those coordinates as collectors items in mega data sets
of the order of 10
10 coordinate data?”
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 M. Hotine
(1947a–e) or Hotine Oblique Mercator Projection (HOM) by J. P. 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.
3-1 Coordinates relating to manifolds
Coordinates relating to manifolds, in particular, differential manifolds, elements of topology (Hausdorff
topological space, open and closed domains).
Let us here first consider manifolds and their charts, the complete atlas, and the minimal atlas. Let
us here first define the term chart according to Definition 3.1.
Definition 3.1 (Manifold, chart).
An n-dimensional manifold M
n is locally an n-dimensional Hausdorff topological space. That is to
any element of M
n , called “point”, there is a connected open neighborhood U and a homeomorphism
Φ : U → φ(U) to an open set φ(U) of R
n . M
n is locally homeomorph to R
n . Any homeomorphism φ
is called a chart of M
n . A one-dimensional manifold is called curve, a two-dimensional manifold with
Riemann metric and without Cartan torsion is called surface, and a compact, connected manifold
without boundary is called closed manifold.
End of Definition.
Indeed, we implemented many unknown notions from the theory of morphism, in particular, from
topology, but we do not hope to lose you, the map maker. Therefore, just follow us to stroll along
Hausdorff Street, Bonn (Germany) to meditate over Haussdorff’s axiom of separation (T
2 ) within
Listing’s topoploy, shortly reviewed in the Appendix. What is more important here is the question
that follows.