21-4 Gauss–Krueger/UTM coordinates: from a local to a global datum 481
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 reference
ellipsoid-of-revolution E
2
a 1 ,a 2
. Then, under a curvilinear datum transformation (21.67) and (21.66)
represented by three parameters {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
2
A 1 ,A 2
are represented by the bivariate polynomial
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
¯
x mn
x
ρ
m
y
ρ
− y 00
n
,
Y = Y (x, y, ρ, t x , t y , t z , α, β, γ, s, δE
2 ) = ρ
∞
m=0,n=0,m+n=N
¯
y mn
x
ρ
m
y
ρ
− y 00
n
.
(21.85)
X and Y are given by (21.82) in Box 21.25 up to order three. The coefficients ¯
x mn and ¯
y mn , which
are given in Box 21.26 and Box 21.27, are product sums of the coefficients U MN and V MN of Box 21.20
and Box 21.21 and of the coefficients l mn and b mn of Box 21.23 and Box 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
2
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
{X(x, y), Y (x, y)}: such bivariate polynomials are inverted by means of the GKS algorithm presented
by E. Grafarend, T. Krarup and R. Syffus (1996). Box 21.28 contains the inverse bivariate polynomials {x(X, Y ), y(X, Y )} with respect to the coefficients of Box 21.29 and Box 21.30, where the datum
parameters are included in the coefficients {x
MN , y
MN
}.
Important!
The direct and inverse equations for a datum transformation of conformal coordinates of type
Gauss–Krueger or UTM from a local datum (regional, National, European) to a global datum
(i. e. WGS 84) are given in terms of a bivariate polynomial representation. The polynomial
coefficients depend on the datum transformation parameters, namely three parameters of
translation, three parameters of rotation, one scale parameter, and one form parameter
change, in total eight parameters. The form parameter change accounts for the variation of
the eccentricity of the reference ellipsoid-of-revolution under the change from one geodetic
datum to another one, namely from local to global or vice versa. The equations generating
the transformation of local conformal coordinates of type Gauss–Krueger or UTM to global
conformal coordinates of the same type enable us to transform mega data sets stored in
data bases or in charts from the local datum (the datum of the data base, the datum of
the chart) to the global datum (the datum of satellite derived coordinates by means of the
Global Positioning System, i.e. WGS 84) and vice versa.
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 reference
ellipsoid-of-revolution E
2
a 1 ,a 2
. Then, under a curvilinear datum transformation (21.67) and (21.66)
represented by three parameters {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
2
A 1 ,A 2
are represented by the bivariate polynomial
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
¯
x mn
x
ρ
m
y
ρ
− y 00
n
,
Y = Y (x, y, ρ, t x , t y , t z , α, β, γ, s, δE
2 ) = ρ
∞
m=0,n=0,m+n=N
¯
y mn
x
ρ
m
y
ρ
− y 00
n
.
(21.85)
X and Y are given by (21.82) in Box 21.25 up to order three. The coefficients ¯
x mn and ¯
y mn , which
are given in Box 21.26 and Box 21.27, are product sums of the coefficients U MN and V MN of Box 21.20
and Box 21.21 and of the coefficients l mn and b mn of Box 21.23 and Box 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
2
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
{X(x, y), Y (x, y)}: such bivariate polynomials are inverted by means of the GKS algorithm presented
by E. Grafarend, T. Krarup and R. Syffus (1996). Box 21.28 contains the inverse bivariate polynomials {x(X, Y ), y(X, Y )} with respect to the coefficients of Box 21.29 and Box 21.30, where the datum
parameters are included in the coefficients {x
MN , y
MN
}.
Important!
The direct and inverse equations for a datum transformation of conformal coordinates of type
Gauss–Krueger or UTM from a local datum (regional, National, European) to a global datum
(i. e. WGS 84) are given in terms of a bivariate polynomial representation. The polynomial
coefficients depend on the datum transformation parameters, namely three parameters of
translation, three parameters of rotation, one scale parameter, and one form parameter
change, in total eight parameters. The form parameter change accounts for the variation of
the eccentricity of the reference ellipsoid-of-revolution under the change from one geodetic
datum to another one, namely from local to global or vice versa. The equations generating
the transformation of local conformal coordinates of type Gauss–Krueger or UTM to global
conformal coordinates of the same type enable us to transform mega data sets stored in
data bases or in charts from the local datum (the datum of the data base, the datum of
the chart) to the global datum (the datum of satellite derived coordinates by means of the
Global Positioning System, i.e. WGS 84) and vice versa.