Geography Reference
In-Depth Information
coordinates
have gained new interest, namely under the postulate of eciency and speed of compu-
tation. Accordingly, we derive here a set of new formulae for the strip transformation of conformal
coordinates of type Gauss-Krueger and of type Universal Transverse Mercator Projection (UTM)
with an optimal dilatation factor different from one.
Section
15-61
has its objective in the derivation of transformation formulae of conformal coordi-
nates
of a strip of
ellipsoidal longitude
L
02
. A two-step-approach is proposed which generates the solution (
15.123
)
and (
15.124
) of the strip transformation problem. Section
15-612
focuses on two examples of strip
transformations relating to (i) the
Bessel reference ellipsoid
and (ii) the
World Geodetic Refer-
ence System
1984 (WGS84). In particular, we compare the strip transformation results with those
produced by a direct transformation of ellipsoidal longitude/latitude of a point on the reference
ellipsoid (ellipsoid-of-revolution) into conformal coordinates in the first and second strip.
{
x
1
,y
1
}
of a strip of ellipsoidal longitude
L
01
to conformal coordinates
{
x
2
,y
2
}
15-61 Two-Step-Approach to Strip Transformations
Here, we outline the two-step-approach which leads us by inversion technology of bivariate homo-
geneous polynomials to the strip transformation
x
2
=
X
(
x
1
,y
1
)and
y
2
=
Y
(
x
1
,y
1
) of conformal
coordinates
{x
1
,y
1
}
of the first
L
01
-strip into conformal coordinates
{x
2
,y
2
}
of the second
L
02
-
strip, namely for conformal coordinates of type Gauss-Krueger (GK) and UTM.
Fig. 15.17.
Commutative diagram for a strip transformation of conformal coordinates of type Gauss-Krueger
or of type UTM
in the first Gauss-Krueger or UTM strip system
of ellipsoidal longitude
L
01
to be given. We also refer to
L
01
as the ellipsoidal longitude of the
meridian of reference which is mapped equidistantly (or up to an optimal dilatation factor) under
a conformal mapping of Gauss-Krueger type (or of UTM type). The minimal distance mapping
of a topographic point on the Earth surface onto the ellipsoid-of-revolution
Assume the conformal coordinates
{
x
1
,y
1
}
2
A
1
,A
2
of semi-major
axis
A
1
and semi-minor axis
A
2
as outlined by
Grafarend and Lohse
(
1991
) identifies the point
{
E
2
A
1
,A
2
. The prob-
lem of a strip transformation may be formulated as following: given the conformal coordinates
{x
1
,y
1
}
with respect to a first strip system
L
01
of a point
{L, B}
on
E
ellipsoidal longitude, ellipsoidal latitude
}
=
{
L, B
}
of surface normal type on
E
4
1
,A
2
, find its conformal
coordinates
{x
2
,y
2
}
with respect to a second strip system
L
02
. An illustration of the involved
transformations is presented in the commutative diagram of Figs.
15.17
and
15.18
. The trans-
formation
x
2
(
x
1
,y
1
)arid
y
2
(
x
1
,y
1
) to which we refer as the strip transformation of conformal
coordinates of type Gauss-Krueger or of type UTM is generated as following.
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