[Proj] +towgs84 approximation error

[Jochem]

Kristian,

I did that, I'll show you in a bit. I know what is going wrong already. The
formulas in cs2cs are the approximate ones instead of the strict ones: 

Lookup the method with EPSG code 1032 on  http://epsg-registry.org/
<http://epsg-registry.org/>  

This refers to  Guidance Note 7, part 2
<http://www.iogp.org/pubs/373-07-2.pdf>  
On page 134 with page number 132 of that publication the strict and
approximate formulas are mentioned (see image)
<http://osgeo-org.1560.x6.nabble.com/file/n5313844/Untitled.png>  
This publication also states: "For this small angle approximation to be
valid to better than 3mm at the Earth’s surface, the rotations must be under
10 arc-seconds". My rotations are upto 40 arc-seconds (causing a errors of
several cm).

cs2cs uses these approximate formulas: 
http://proj.maptools.org/gen_parms.html
<http://proj.maptools.org/gen_parms.html>  

Example:
cs2cs +proj=geocent +ellps=WGS84
+towgs84=-294.1985,231.0171,-285.2462,-11.34441,40.54620,-9.73364,15.7901
+to +proj=geocent +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 -f %.4f 

input: 
3194419.1451 -3194419.1451 4487348.4089

output:
3194906.7475 -3194142.5106 4486681.7637

This is exactly identical to what I get if I program the approximate
formulas in Excel too.

What it should be is what I produced by programming the strict formulas in
Excel and what I also got (within few mm) with 3 other software packages
(Trimble, Topcon en Move3):
3194906.7284 -3194142.5438 4486681.6702

The difference is: -0.0191 -0.0332 -0.0934
The 3D difference: 0.1010 m

Regards, Jochem



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