Geography Reference
In-Depth Information
21-41 Direct Transformation of Local Conformal into Global
Conformal Coordinates
The problem of generating Gauss-Krueger or UTM conformal coordinates
{
X,Y
}
of a global
2
A
1
,A
2
reference ellipsoid-of-revolution
E
in terms of Gauss-Krueger or UTM conformal coordinates
2
a
1
,a
2
{
under a
curvilinear datum transformation is solved here by a three-step-procedure according to the com-
mutative diagram of Fig.
21.5
. The first step is a representation of global conformal coordinates
of type Gauss-Krueger or UTM in terms of conformal bivariate polynomials
X
(
L
x, y
}
of a local (regional, National, European) reference ellipsoid-of-revolution
E
B
0
)
and
Y
(
L − L
0
,B− B
0
) with respect to surface normal ellipsoidal longitude/ellipsoidal latitude
increments
{L − L
0
,B− B
0
}
. The second step is divided into two sub-steps. First we trans-
form the global ellipsoidal longitude/ellipsoidal latitude increments into local ellipsoidal longi-
tude/ellipsoidal latitude increments
{l−l
0
,b−b
0
}
by means of a curvilinear datum transformation
(i.e. a linear function of the three parameters of translation, three parameters of rotation, and one
scale parameter). Second, we implement the transformation
−
L
0
,B
−
{
L
−
L
0
,B
−
B
0
} →{
l
−
l
0
,b
−
b
0
}
into the representation
, in particular, including the curvi-
linear datum transformation for polynomial coecients, too. Finally, the third step is split into
two sub-steps. First, we repeat the representation of local conformal coordinates of type Gauss-
Krueger or UTM in terms of conformal bivariate polynomials
{
X
(
l
−
l
0
,b
−
b
0
)
,Y
(
l
−
l
0
,b
−
b
0
)
}
{
x
(
l
−
l
0
,b
−
b
0
)
,y
(
l
−
l
0
,b
−
b
0
)
}
with respect to surface normal ellipsoidal longitude/ellipsoidal latitude increments
{
l
−
l
0
,b
−
b
0
}
,
namely in order to construct the inverse polynomials
{
l
−
l
0
(
x, y
)
,b
−
b
0
(
x, y
)
}
by bivariate series
inversion. Second, we transfer the bivariate polynomial representation
{
l
−
l
0
(
x, y
)
,b
−
b
0
(
x, y
)
}
to
the power series
{
X
(
l
−
l
0
)
,Y
(
b
−
b
0
)
}
in order to achieve the final bivariate general polynomials
{
X
(
x, y
)
,Y
(
x, y
)
}
.
21-411 The First Step: Conformal Coordinates in a Global Datum
A
1
,A
2
Conformal coordinates of an ellipsoid-of-revolution
in a global frame of reference (semi-
major axis
A
1
, semi-minor axis
A
2
, relative eccentricity squared
E
2
:= (
A
1
−
E
A
2
)
/A
1
)oftype
Gauss-Krueger or UTM are generated by a polynomial representation in terms of surface normal
ellipsoidal longitude
L
and ellipsoidal latitude
B
with respect to an evaluation point
{L
0
,B
0
}
:
see Box
21.11
in connection with Boxes
15.4
and
15.5
.
L
0
is also called the
ellipsoidal longitude
of the meridian of reference. While the coecient
Y
00
represents the arc length of the meridian
L
0
of reference, the coecients are generated by
solving the vector-valued boundary value problem of the Korn-Lichtenstein equations of conformal
mapping subject to the integrability conditions, the vector-valued Laplace-Beltrami equations.
The constraint of the vector-valued boundary value problem is the equidistant mapping of the
meridian
L
0
of reference and outlined by
Grafarend
(
1995
)and
Grafarend and Ardalan
(
1997
)
(Figs.
21.6
and
21.7
).
Box 21.11 (Polynomial representation of conformal coordinates of type Gauss-Krueger or
UTM, ellipsoid-of-revolution
(Easting
X
, Northing
Y
), surface normal ellipsoidal lon-
gitude
L
and ellipsoidal latitude
B
, evaluation point
E
A
1
,A
2
{
L
0
,B
0
}
, optimal factor of conformality
3
.
5
◦
,
+3
.
5
◦
]
[80
◦
S
,
84
◦
N]; coe-
ρ
=0
.
999578 (UTM) for a strip [
−
l
E
,
+
l
E
]
×
[
B
S
,B
N
]=[
−
×
cients are given in Boxes
15.4
and
15.5
).
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