[Proj] Left- or right-handed rotation convention: which is which?

Clifford J Mugnier cjmce at lsu.edu
Sun Oct 1 22:10:09 EDT 2006

Of course, my way is correct and everyone else is wrong!  :-)

Dr. Tomas Soler of the U.S. National Geodetic Survey gave an elaborate
exposition in Bulletin Geodesique regarding the left-handed vs.
right-handed rotations of the various transforms.  Apparently, somebody
misunderstood the distinction in Paris, and ergo, the "European" way of
doing things.  All of this became possible because of the American SECOR,
then TRANSIT, and currently NAVSTAR (GPS)geodetic satellite systems.  If
the Americans that invented the contraptions and put the stuff into orbit
for everyone else to get access to define it a particular way ... guess
which way is correct?

Whether it's infinitessimal rotations with arc seconds, radians, or
whatever ... there's two basic 7-parameter datum transformations - the
Molodensky that uses the geocentric coordinates of the datum origin and the
Bursa-Wolfe that does not use the datum origin.  The sign of the rotations
is right-handed as the Americans do it or left-handed as the continental
Europeans do it.

As long as a test point is published, it's pretty easy to distinguish what
the sign of the rotations are.  Without a test point, you're going to have
to rely on the nomenclature whether you like it or not.

C. Mugnier

> Hello,
> I am confused.
> At my work, I have tried to document the rotation convention used by
> our software for seven-parameter datum shifts.  Since it was originally
> based on publications from the Swedish Land Survey, it uses the
> rotation convention of USA/Sweden/Luxembourg/Australia, as opposed to
> the opposite European/IAG/ISO19111 convention.
> Well, I suppose that everyone wants to be right, and no one wants to
> be left!  But are there any simple, unambiguous names for these two
> conventions?  Would the following be correct:
> American convention = "coordinate frame rotation";
> European convention = "position vector transformation".

Probably the best way to document a rotation method is: use the accepted
terms "coordinate frame" and "position vector".
These terms are also used in other disciplines like kinematics (robotics).
But there are more difficulties.

One datum transformation method is laid down in the ISO 19111 standard.
is an approximated 7-parameter Helmert transformation with position vector
rotation. See also ISO/IEC 18026 - Annex B.
PROJ uses about the same method, only the scaling method differs. PROJ does
a scalar multiplication, ISO a matrix multiplication.
With the commonly used parameter ranges, the differences between the
methods are less than microns, so not important.

As far as I know, the Bursa-Wolfe transform is an approximation to the
Helmert transform. The Helmert transform has sines and cosines in the
rotation matrices, whereas Bursa-Wolfe (and ISO 19111) use the angles
themselves (since sin(a) ~ a, sin(a)*sin(b) ~ 0, and cos(a) ~ 1 for small

If you read section B.6 of ISO/IEC 18026, then you'll notice that a
Bursa-Wolfe transform can be done with a position vector rotation model OR
with a coordinate frame rotation model. Just what one likes the best; be
sure to use the correct sign of the rotation angles. Therefore: Bursa-Wolfe
is NOT equivalent with this or that rotation model.

A well known expert repeatedly states that the Australians use the same
datum transform rotation model as the Americans.
This is NOT true!
The order of the rotations differs (XYZ vs. ZYX). See the Australian GDA
Technical Manual. By the way, if the rotations are approximated, then the
order is not important.
Again, the differences in rotation order for real life numbers are

This, and your (mr. Rittri) remarks merely demonstrate how silly it is to
refer to a datum transformation method as an American, Australian,
whatever regional model.

If you want to document the way for instance an application transforms,
the complete formulae, not just an ill-defined name.

Why not referring to the EPSG coordinate transformation method numbers?
These clearly define the most used datum transformation methods and
projection methods.

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