Geography Reference
In-Depth Information
Lemma 21.1 (Local conformal coordinates are transformed into global conformal coordinates of
type Gauss-Krueger or UTM).
Let there be given conformal coordinates
{
x, y
}
of type Gauss-Krueger or UTM of a local ref-
a
1
,a
2
erence ellipsoid-of-revolution
. Then, under a curvilinear datum transformation (
21.67
)and
(
21.66
) represented by three parameters
E
{
t
x
,t
y
,t
z
}
of translation, three parameters
{
α,β,γ
}
of
rotation, and one scale parameter
s
, the conformal coordinates
{
X,Y
}
of type Gauss-Krueger or
UTM of a global reference ellipsoid-of-revolution
E
A
1
A
2
are represented by the bivariate polyno-
mial
x
mn
x
ρ
m
∞
X
=
X
(
x, y, ρ, t
x
,t
y
,t
z
,α,β,γ,s,δE
2
)=
ρ
m
=0
,n
=0
,m
+
n
=
N
y
ρ
− y
00
n
,
y
mn
x
m
∞
Y
=
Y
(
x, y, ρ, t
x
,t
y
,t
z
,α,β,γ,s,δE
2
)=
ρ
ρ
m
=0
,n
=0
,m
+
n
=
N
y
ρ
−
y
00
n
.
(21.85)
X
and
Y
are given by (
21.82
)inBox
21.25
up to order three. The coecients
x
mn
and
y
mn
,
which are given in Boxes
21.26
and
21.27
, are product sums of the coecients
U
MN
and
V
MN
of
Boxes
21.20
and
21.21
and of the coecients
l
mn
and
b
mn
of Boxes
21.23
and
21.24
.
y
00
indicates
the arc length of the meridian-of-reference
l
0
in the interval [0
,b
0
].
End of Lemma.
21-42 Inverse Transformation of Global Conformal into Local
Conformal Coordinates
The software attached to a satellite Global Positioning System (GPS) allows the direct conversion
of global ellipsoidal coordinates
{L − L
0
,B− B
0
}
into global conformal coordinates
{X,Y }
of
type Gauss-Krueger or UTM, namely with reference to a global reference ellipsoid-of-revolution
E
A
1
,A
2
, i.e. WGS 84. In order to locate an observer with first hand information of global conformal
coordinates
{X,Y }
of type Gauss-Krueger or UTM in a Gauss-Krueger or UTM chart given in a
local reference system (regional, National, European), we are left with the problem of transforming
global conformal coordinates
{X,Y }
into local conformal coordinates of type Gauss-Krueger or
UTM, the chart coordinates. The problem is solved by the inverse representation of the bivariate
polynomials
: such bivariate polynomials are inverted by means of the GKS
algorithm presented by
Grafarend
(
1996
). Box
21.28
contains the inverse bivariate polynomials
{
{
X
(
x, y
)
,Y
(
x, y
)
}
with respect to the coecients of Boxes
21.29
and
21.30
, where the datum
parameters are included in the coecients
x
(
X,Y
)
,y
(
X,Y
)
}
x
MN
,y
MN
{
}
.
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