470 21 Datum problems
Second, in contrast, surface normal ellipsoidal coordinates of type ellipsoidal longitude, ellipsoidal
latitude, ellipsoidal height {L, B, H} and {l, b, h} replace the Cartesian coordinates as user coordinates.
{L, B, H} refer to a global datum, in particular, to a global reference ellipsoid-of-revolution E
2
A 1 ,A 2
(semi-major axis A 1 , semi-minor axis A 2 , relative eccentricity squared E
2 := (A
2
1 − A
2
2 )/A
2
1 ), while
{l, b, h} refer to a local reference ellipsoid-of-revolution E
2
a 1 ,a 2
(semi-major axis a 1 , semi-minor axis a 2 ,
relative eccentricity squared e
2 := (a
2
1 −a
2
2 )/a
2
1 . Accordingly, a transformation of ellipsoidal coordinates
from {l, b, h} to {L, B, H} or vice versa is called a curvilinear datum transformation, i. e. close to the
identity, expressed as a linear function represented by three parameters {t x , t y , t z } of translation, three
parameters {α, β, γ} of rotation, and one scale parameter s. Such a curvilinear datum transformation
(user oriented) has been investigated by A. Leick and B. H. W. van Gelder (1975), T. Soler (1976),
R. Schreiber (1991), E. Grafarend, F. Krumm, and F. Okeke (1995) as well as F. Okeke (1997). Third,
the target of our contribution is the datum transformation of conformal coordinates {X, Y } of type
Gauss–Krueger or UTM from a local datum to global one and vice versa. The first subsection is
devoted to the derivation of the direct equations of the datum transformation {x, y} → {X, Y },
where we take advantage of computer-aided bivariate polynomial inversion pioneered by H. Glasmacher
and K. Krack (1984), G. Joos and K. Joerg (1991), and E. Grafarend, T. Krarup and R. Syffus (1996).
The second subsection collects the inverse equations {X, Y } → {x, y} of a datum transformation
of conformal coordinates of type Gauss–Krueger or UTM. Both transformations, direct and inverse
equations, respectively, amount to bivariate polynomials with coefficients which depend on the datum
transformation parameters {t x , t y , t z , α, β, γ, s} and the change of the form parameter δE
2 := E
2
− e
2 .
Some remarks to our notation have to be made. We already mentioned that all quantities which refer
to a global datum are written in capital letters, while those with reference to a local datum are written
in small letters.
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 reference
ellipsoid-of-revolution E
2
A 1 ,A 2
in terms of Gauss–Krueger or UTM conformal coordinates {x, y} of a
local (regional, National, European) reference ellipsoid-of-revolution E
2
a 1 ,a 2
under a curvilinear datum
transformation is solved here by a three-step-procedure according to the commutative 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−L 0 , B−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 transform the global ellipsoidal longitude/ellipsoidal latitude
increments into local ellipsoidal longitude/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−L 0 , B−B 0 } → {l−l 0 , b−b 0 } into the representation {X(l−l 0 , b−b 0 ), Y (l−l 0 , b−b 0 )}, in particular,
including the curvilinear datum transformation for polynomial coefficients, 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 )}
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
Conformal coordinates of an ellipsoid-of-revolution E
2
A 1 ,A 2
in a global frame of reference (semi-major
axis A 1 , semi-minor axis A 2 , relative eccentricity squared E
2 := (A
2
1 −A
2
2 )/A
2
1 ) of type 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.
Second, in contrast, surface normal ellipsoidal coordinates of type ellipsoidal longitude, ellipsoidal
latitude, ellipsoidal height {L, B, H} and {l, b, h} replace the Cartesian coordinates as user coordinates.
{L, B, H} refer to a global datum, in particular, to a global reference ellipsoid-of-revolution E
2
A 1 ,A 2
(semi-major axis A 1 , semi-minor axis A 2 , relative eccentricity squared E
2 := (A
2
1 − A
2
2 )/A
2
1 ), while
{l, b, h} refer to a local reference ellipsoid-of-revolution E
2
a 1 ,a 2
(semi-major axis a 1 , semi-minor axis a 2 ,
relative eccentricity squared e
2 := (a
2
1 −a
2
2 )/a
2
1 . Accordingly, a transformation of ellipsoidal coordinates
from {l, b, h} to {L, B, H} or vice versa is called a curvilinear datum transformation, i. e. close to the
identity, expressed as a linear function represented by three parameters {t x , t y , t z } of translation, three
parameters {α, β, γ} of rotation, and one scale parameter s. Such a curvilinear datum transformation
(user oriented) has been investigated by A. Leick and B. H. W. van Gelder (1975), T. Soler (1976),
R. Schreiber (1991), E. Grafarend, F. Krumm, and F. Okeke (1995) as well as F. Okeke (1997). Third,
the target of our contribution is the datum transformation of conformal coordinates {X, Y } of type
Gauss–Krueger or UTM from a local datum to global one and vice versa. The first subsection is
devoted to the derivation of the direct equations of the datum transformation {x, y} → {X, Y },
where we take advantage of computer-aided bivariate polynomial inversion pioneered by H. Glasmacher
and K. Krack (1984), G. Joos and K. Joerg (1991), and E. Grafarend, T. Krarup and R. Syffus (1996).
The second subsection collects the inverse equations {X, Y } → {x, y} of a datum transformation
of conformal coordinates of type Gauss–Krueger or UTM. Both transformations, direct and inverse
equations, respectively, amount to bivariate polynomials with coefficients which depend on the datum
transformation parameters {t x , t y , t z , α, β, γ, s} and the change of the form parameter δE
2 := E
2
− e
2 .
Some remarks to our notation have to be made. We already mentioned that all quantities which refer
to a global datum are written in capital letters, while those with reference to a local datum are written
in small letters.
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 reference
ellipsoid-of-revolution E
2
A 1 ,A 2
in terms of Gauss–Krueger or UTM conformal coordinates {x, y} of a
local (regional, National, European) reference ellipsoid-of-revolution E
2
a 1 ,a 2
under a curvilinear datum
transformation is solved here by a three-step-procedure according to the commutative 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−L 0 , B−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 transform the global ellipsoidal longitude/ellipsoidal latitude
increments into local ellipsoidal longitude/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−L 0 , B−B 0 } → {l−l 0 , b−b 0 } into the representation {X(l−l 0 , b−b 0 ), Y (l−l 0 , b−b 0 )}, in particular,
including the curvilinear datum transformation for polynomial coefficients, 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 )}
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
Conformal coordinates of an ellipsoid-of-revolution E
2
A 1 ,A 2
in a global frame of reference (semi-major
axis A 1 , semi-minor axis A 2 , relative eccentricity squared E
2 := (A
2
1 −A
2
2 )/A
2
1 ) of type 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.