Guidelines for the use of GNSS in surveying and mapping

6.3.1 Ellipsoidal to cartesian
transformations and reverse
The following formulae are suitable for conversion
of ellipsoidal co-ordinates (φ and λ) to cartesian coordinates
(x, y and z). In these formulae, the x-axis
is defined as being parallel to the conventional zero
meridian of Greenwich, the z-axis parallel to the
CIO and the y-axis at right angles to these two
(eastwards). The cartesian system may be
geocentric or referred to the vertical at some
specified point.
6.1
where:
6.6
(e’) 2 the second
eccentricity 2 6.6a
6.6b
6.6c
6.2
6.7
6.3
Or
(using Bomford’s notation)
6.7a
Notice that the height of the point is (N+h) above
the ellipsoid. Thus, equations 6.1, 6.2 and 6.3
presuppose knowledge of the geoid/ellipsoid
separation (N).
The reverse transformation, from cartesian to
ellipsoidal co-ordinates, does not produce such
closed formulae. The formulae for this reverse
transformation are:
Approximate values of φ and λ are used and
iteration gives rapid convergence.
6.4
6.5
Another alternative is the closed formula due to
Bowring:
It is important to note that again N needs to be
known (see paragraph 6.3.4), and that (N+h) is
determined when using equation 6.7. (N+h) isthe
ellipsoidal height and is not a height obtained by
spirit levelling.
In summary, to convert from:
+ ellipsoidal to cartesian: use equations 6.1, 6.2
and 6.3
+ cartesian to ellipsoidal: use equations 6.4, 6.5
and 6.7.
Evidently, different mathematical models can exist
for the transformations, hence the small differences
that may occur when using different manufacturers’
methods of transformation. These differences will
manifest as small systematic offsets between two
groups of local co-ordinates when transformed by
two different methods. The most important issue
when working on high-accuracy projects is to be
consistent and use the same method of
transformation and software. Thus any small
differences in co-ordinates will not be propagated
into a project and distort the geometry of a survey
undertaken using GNSS.
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