Geoscience Reference
In-Depth Information
P
1
rotation axis
ellipsoid
geoid
Fig. 5.16. The reference ellipsoid is tangent to the geoid at
P
1
N
1
at the initial point
P
1
(in our case,
N
1
= 0), we can finally compute the
geoidal heights
N
of any point of the triangulation net by repeated applica-
tion of (5-114). These geoidal heights refer to the ellipsoid that was fixed by
prescribing
ξ
1
,η
1
,N
1
, and, of course, its semimajor axis
a
and its flattening
f
. To employ a frequently used term, they refer to the given
astrogeodetic
datum
(
a, f
;
ξ
1
,η
1
,N
1
).
By means of
N
and the orthometric height
H
, the height
h
above the
ellipsoid is obtained via
h
=
H
+
N
, so that the rectangular spatial coordi-
nates
X, Y, Z
can be computed by (5-27). But unless
ξ
and
η
are absolute
(geocentric) deflections, the origin of the coordinate system will not be at
the center of the earth (see Sect. 5.7).
A flaw in the procedure described above apparently is that
N,ξ,η
are
already needed for the reduction of the measured angles and distances to
the ellipsoid. However, for this purpose approximate values of
N,ξ,η
are
sucient. These are obtained by performing the process just explained with
unreduced angles and distances. We can also get suitable values for
N,ξ,η
in other ways, for instance, by Stokes' formula.
Use and misuse of Laplace's equation
It should be mentioned that in practice the component
η
has been often
obtained from azimuth measurements using (5-102) in rearranged form, that
is,
η
=(
A
−
α
)cot
ϕ,
(5-117)
because astronomical measurements of azimuth are simpler than those of
longitude. This is a misuse which may lead to a systematic distortion of the
