Geography Reference
In-Depth Information
J
Gauss Surface Normal Coordinates in Geometry and Gravity
Three-dimensional geodesy, minimal distance mapping, geometric heights. Reference plane,
reference sphere reference ellipsoid-of-revolution, reference triaxial ellipsoid.
With the advent of artifical satellites in an Earth-bound orbit, geodesists succeeded to position
points of the topographic surface
3
coordinates in a three-dimensional
reference frame at the mass center of the Earth oriented along the equatorial axes at some reference
epoch
t
0
2
by a set of
T
{
X, Y, Z
}∈
R
. In particular, global positioning systems (“global problem solver”: GPS), were
responsible for the materialization of
three-dimensional geodesy
in an Euclidean space. Based
upon a triple
∈
R
3
of coordinates new concepts for converting these coordinates into
heights with respect to a reference surface have been developed.
{
X, Y, Z
}∈
R
In the geometry space, the triplet
{X, Y, Z}∈
T
2
is trans-
formed by a geodesic projection into geometric heights with
respect to (i) a
reference plane
P
2
, (ii) a
reference sphere
S
r
, (iii) a
reference ellipsoid-of-revolution
E
A
1
,A
2
.or(iv)a
reference triaxial ellipsoid
E
A
1
.A
2
,A
3
.
First, the geodesic projection is performed by a straight line
as the geodesic in flat geometry space. Second, the special
geodesic passing the point
{X, Y, Z}∈
T
has been chosen
which has minimal distance
S
to the reference surface. The
length of the geodesic from
2
{
X, Y, Z
}∈
T
to
{
x, y, z
}∈
P
r
or
A
1
,A
2
A
1
,A
2
,A
3
or
being deter-
mined by the
minimal distance mapping
, constitute the pro-
jective height in geometry space, namely of type (i) planar,
(ii) spherical, (iii) ellipsoidal, or (iv) triaxial ellipsoidal.
S
E
or
E
, in short,
{
x, y, z
}
Section J-1.
By algebraic mean, Sect.
J-1
outlines various step procedures to establish projection heights in
geometry space. By means of
minimal distance mapping
, various computational steps, either for-
ward or backwards, are reviewed depending on the nature of the projection surface. The projection
Search WWH ::
Custom Search