Geography Reference
In-Depth Information
space”) subject to the
A. C. Clairaut
constant for a rotational symmetric surface like the ellipsoid-
of-revolution are solved by means of the
Lie recurrence
(“Lie series”), in particular, initial value
problem versus boundary value problem, by the technique of
standard series inversion
.
Section 20-3.
Section
20-3
treats the elaborate
Soldner coordinates
or
geodetic parallel coordinates
.Thesecoor-
dinates compete with Gauss-Krueger coordinates and Riemann normal coordinates. The way of
construction is illustrated by Fig.
20.4
. As the first problem of computing such a
geodetic projec-
tion
, we treat the case (1). In contrast, the second problem in computing Soldner coordinates may
be summarized by (2). As an example, we introduce the
Soldner map
centered at the Tubingen
Observatory.
Input:
at the initial point, longitude/latitude are
given as well as the azimuth of the
orthogonal projection (geodetic
projection) of a point
P
(
L,B
) onto the
reference meridian: {
L
0
,B
0
,A
c
, y
c
}.
(1)
Þ
Output:
longitude/latitude of the point {
}as
well as the meridian convergence
L,B
γ
.
(2)
Þ
Input:
at the initial point {
L
0
,B
0
} as well as at
the moving point {
L,B
}, the geodetic
coordinates are given.
Output:
the Soldner coordinates are computed:
{
x, y
}
= {
y
c
, x
c
.
}
Section 20-4.
Section
20-4
focuses on the celebrated
Fermi coordinates
which extend the notion of a geodetic
projection. An initial point
{
L
0
,B
0
}
is chosen. A moving point
{
L, B
}
is projected at right
angles onto the point
P
F
, which is fixed. The two-step solution from
{
L
0
,B
0
,u,v,u
F
,v
F
}
to
L, B, A
c
{
PF
}
is given.
{
u, v
}
are the Fermi coordinates of the moving point and
{
u
F
,v
F
}
are the
Fermi coordinates of the geodetic projections
{
L, B
}
onto
{
u
F
,v
F
}
. The geodetic projections are
fixed by identifying the coordinates of the point
{
u
F
,v
F
}
.
Section 20-5.
Section
20-5
reviews all the details of
Riemann coordinates
, compares them with the Soldner coor-
dinates, and additionally compares them with the Gauss-Krueger coordinates. First, we intro-
duce the left deformation analysis or distortion analysis of the Riemann mapping, namely by
outlining the
additive measure of deformation
, called the left
Cauchy-Green deformation tensor
.
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