Geography Reference
In-Depth Information
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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