Geoscience Reference
In-Depth Information
flattening
f
) and (2) its position with respect to the earth or the geoid.
This relative position is most simply defined by the coordinates
x
0
,y
0
,z
0
of the center of the reference ellipsoid with respect to the geocenter. Since
the geocenter was not accessible to classical geodetic measurements before
the satellite era, a
fundamental
or
initial point P
1
on the earth surface was
chosen, such as Meades Ranch for North America and Potsdam for Central
Europe. It turns out that a convenient but conventional choice of the el-
lipsoidal coordinates
ϕ
1
,λ
1
,h
1
of the fundamental point
P
1
is equivalent to
x
0
,y
0
,z
0
of the geocenter.
Thus, we have 5 defining parameters:
•
2 parameters
a
(semimajor axis) and
f
(flattening) as
form parameters
,
and
3 parameters
x
0
,y
0
,z
0
or
ϕ
1
,λ
1
,h
1
as
position parameters
.
Later on we shall also admit a scale factor and small rotations around the
three coordinate axes.
A (geodetic) datum transformation defines the relationship between a
global (geocentric) and a local (in general nongeocentric) three-dimensional
Cartesian coordinate system; therefore, a datum transformation transforms
one coordinate system of a certain type to another coordinate system of the
same type. This is one of the primary tasks when combining GPS data with
terrestrial data, i.e., the transformation of geocentric WGS 84 coordinates
to local terrestrial coordinates. The terrestrial system is usually based on a
locally best-fitting ellipsoid, e.g., the Clarke ellipsoid or the GRS-80 ellipsoid
in the U.S. and the Bessel ellipsoid in many parts of Europe. The local
ellipsoid is linked to a nongeocentric Cartesian coordinate system, where the
origin coincides with the center of the ellipsoid.
•
5.7.2
Three-dimensional transformation in general form
Consider two arbitrary sets of three-dimensional Cartesian coordinates form-
ing the vectors
X
and
X
T
(Fig. 5.7). The 7-parameter transformation, also
denoted as Helmert transformation or similarity transformation in space,
between the two sets can be formulated by the relation
X
T
=
x
0
+
µ
RX
,
(5-41)
where
x
0
is the translation (or shift) vector,
µ
is a scale factor, and
R
is a
rotation matrix.
The components of the shift vector
⎡
⎤
x
0
y
0
z
0
⎣
⎦
x
0
=
(5-42)
