[Proj] rotated ellipsoid (towgs84)

Oscar van Vlijmen ovv at hetnet.nl
Fri Oct 20 19:13:10 EDT 2006

> From: Heinrich Schewe
> Date: Fri, 20 Oct 2006 23:08:26 +0200
> Subject: [Proj] rotated ellipsoid (towgs84)
> I try the following datum transformation from WGS84 to a rotated Bessel
> ellipsoid. The back-transformation gives a height difference of 0.01 m.
> Is this due to the approximate rotation computation (assuming cos = 1,
> sin = rad and sin*sin = 0) ?

Short answer: yes.

Long answer:
I have no idea if I understand the cs2cs command line correctly, but going
from wgs84 to bessel 1841 gives with my software (set to approximate
position vector rotation), starting from
lat=9, lon=48, h=0
the following bessel coordinates:
09d 00m 01.8004s, 47d 59m 52.0176s, 738.30632 m
and back with the same procedure:
09d 00m 00.0002s, 48d 00m 00.0000s, 0.01023 m

If I use the mathematically exact to/from transform (position vector
rotation, order of the rotation XYZ), then I get:
to bessel:
09d 00m 01.8002s, 47d 59m 52.0176s, 738.30120 m
back to wgs:
09d 00m 00.0000s, 48d 00m 00.0000s, 1.05e-9 m

This shows anyway that you can expect a 1 cm difference in height using PROJ
in this case.

> Are these rotation values too big?
Normal rotations are some tenths of an arcsecond.
But there are exceptions with tens of arcseconds. See e.g.

> What can I do to get the same values or smaller differences for the
> back-transformation?
Use other software.
The Helmert transform on an ellipsoid is merely a first order approximation.
The procedure is exact for a sphere, but even using the mathematically exact
spherical rotations on an ellipsoid is a bit silly.
Furthermore it is rather unlikely that transformation parameters can be
measured so accurately that a geocentric rotation (Helmert transform) or
even a local rotation (Molodensky-Badekas transform) will produce centimeter

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