# [Proj] +towgs84 approximation error

Thomas Knudsen knudsen.thomas at gmail.com
Thu Mar 23 06:43:58 EST 2017

```i.e. your transformation is *exact by definition*: It defines the new
system in terms
of the old (or probably more likely the other way round, but in that case we
will - formally - have to talk about a re-definition of the old system).

I wrote a related comment about the reverse situation (the full matrix
formulation
not being strictly valid) in the code for PJ_helmert.c:

If the option "approximate" is set, small angle approximations are used:
The matrix elements are approximated by expanding the trigonometric
functions to linear order (i.e. cos(x) = 1, sin(x) = x), and discarding
products of second order.

This was a useful hack when calculating by hand was the only option,
but in general, today, should be avoided because:

1. It does not save much computation time, as the rotation matrix
is built only once, and probably used many times.

2. The error induced may be too large for ultra high accuracy
applications: the Earth is huge and the linear error is
approximately the angular error multiplied by the Earth radius.

However, in many cases the approximation is necessary, since it has
been used historically: Rotation angles from older published datum
shifts may actually be a least squares fit to the linearized rotation
approximation, hence not being strictly valid for deriving the full
rotation matrix.

In principle it would not be a problem to switch Proj.4 to the full matrix
convention
even for the +towgs84 case, but in practice it would, for exacly the same
reason
that you need it the other way round: The approximate versions being correct
*by definition*. We would need to introduce an "+noapproximation" option, to
keep the classic behaviour intact by default.

Feel free to submit a pull request implementing this!

/Thomas

2017-03-23 11:52 GMT+01:00 Jochem <jochem.lesparre at kadaster.nl>:

> Hello Antoine,
>
> My coordinates in the old frame have precision of a few cm (obtained by
> triangulation a hundred years ago) and the coordinates in the new frame 1
> cm
> (obtained by static GPS measurements 15 years ago). Both networks are tied
> to WGS84 with an precision of a few metres.
>
> However the two frames are tied to each other far more precisely. The
> relative transformation parameters have a precision of a few cm, since
> these
> are based on the measurements. However these parameters have been in use
> for
> some time now in software with strict formulas, and by that they have
> become
> more or less official. Therefore, I need Proj.4 to produce the exact same
> result (+/- 1 mm) as the other software with the strict formulas.
>
> Regards, Jochem
>
>
>
>
>
> --
> View this message in context: http://osgeo-org.1560.x6.nabble.com/towgs84-
> approximation-error-tp5313738p5313833.html
> Sent from the PROJ.4 mailing list archive at Nabble.com.
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