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Fig. 21.4. The basic commutative diagram, rectangular, curvilinear, and conformal datum transformation, with
three parameters of translation, three parameters of rotation, and one scale parameter, local reference ellipsoid-
of-revolution
E a 1 ,a 2 , 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 1
a 2 ) /a 1 . Accord-
ingly, a transformation of ellipsoidal coordinates from
or vice versa is called
a curvilinear datum transformation , i.e. close to the identity, expressed as a linear function repre-
sented by three parameters
{
l,b,h
}
to
{
L, B, H
}
of rotation, and
one scale parameter s . Such a curvilinear datum transformation (user oriented) has been inves-
tigated by Leick and van Gelder ( 1975 ), Soler ( 1976 ), Schreiber ( 1991 ), Grafarend and Syffus
( 1995 )aswellas Okeke ( 1997 ). Third, the target of our contribution is the datum transformation
of conformal coordinates
{
t x ,t y ,t z }
of translation, three parameters
{
α,β,γ
}
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
}
, where we take advantage of computer-aided bivariate
polynomial inversion pioneered by Glasmacher and Krack ( 1984 ), Joos and Joerg ( 1991 ), and Gra-
farend ( 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 coe cients 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
{
x, y
} →{
X,Y
}
−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.
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