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MathGroup Archive 1998

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Re: Re: Rotate3D bug solution



> Mark Evans wrote:
> 
> > Paul Abbott wrote:
> > >
> > > The use of Eulerian angles for specifying rotations in 3D IS
> > standard
> > > (at least in maths and physics -- especially in quantum mechanics,
> > > crystallography, and angular momemntum theory).  Note that the
> > eulerian
> > > angle parametrization avoids the singularities that arise in other
> > > parametrizations.
> > >
> > In the same sense, you could say that sea shells are standard legal
> > tender if you live in a certain part of the world.
> >
> > Paul is right that there is nothing technically wrong with this kind
> > of
> > rotation.  My point was that Mathematica packages should be written
> > for
> > a wider audience.  It seems intuitive that the most common
> > understanding of a rotation matrix is one that rotates sequentially
> > about each of the three coordinate axes.  The fact that Mathematica
> > does not offer this rotation by default is a slip-up in my mind.
> 
> Right on, Mark!
> 
> I've never heard of Eulerian angles before encountering them in
> Mathematica. Maybe they're the usual tricks-of-the-trade to a few
> quantum physicists, but you'll be hard-pressed to find a reference to
> them in any but the most esoteric mathematics literature.
> 
> Dr. Selwyn Hollis
> Associate Professor of Mathematics
> Armstrong Atlantic State University
> Savannah, GA 31419 USA
> <http://www.math.armstrong.edu/faculty/hollis/>

If you haven't heard of Euler angles, and you have a need for 3D
rotation matrices, then I would certainly encourage you to take a look
at Euler angles.  They are very useful, and provide a convenient,
time-tested, and widely-used parameterization of 3D rotation matrices.

To double-check that I wasn't just making this up, I walked over to my
local university library just now and found that I wasn't at all
hard-pressed to find references to Euler angles in general technical
literature.  I found hundreds of references, simply by looking up
"euler angles" on their computer.  Nearly all of those references were
from various areas of engineering: mechanical engineering (e.g. 
"Mechanical Engineering Essentials Reference Guide", Harold A.
Rothbardt (ed), McGraw-Hill (1988) -- which has a nice section on Euler
angles written by a professor in the mechanical engineering department
from UC Santa Barbara), aerospace engineering (e.g. "On the nonlinear
deformation geometry of Euler-Bernoulli beams", Dewey H. Hodges and
Robert A.  Ormiston, National Aeronautics and Space Administration
(1980) -- an article on the design and construction of helicopter
rotors), and so forth.

In fact, although I'm sure something would turn up if I took a closer
look (probably Paul Abbott could find something), I didn't see any
items on this list from pure mathematics or quantum physics.  This also
matches my personal experience.  Although I have a formidable
background in quantum physics, all of the dozen or so encounters that I
have had with Euler angles have been in other fields, such as robotics,
structural dynamics, and classical mechanics.

If the technical literature is any indication, Euler angles are standard
just about everywhere.  This usefulness is reflected in the general
engineering literature, where Euler angles are very common. That is why
the RotationMatrix3D function uses Euler angles.  While you could
certainly point to other features of Mathematica that show a bias
toward physics, the use of Euler angles is not an example of that bias.

I would be strongly supportive of adding other common parameterizations
of 3D rotation matrices in Mathematica.  This suggestion about adding
other parameterizations is a very good suggestion, and I hope that it
will not be lost.

I also hope, however, that Euler angles will not be lost amidst claims
that they are just an obscure trick used by some tiny group of
scientists.  That claim is simply not true.

If there are other parameterizations of 3D rotation matrices that you
would like to see added to Mathematica, I'm sure that the people at
Wolfram Research would be glad to hear your suggestions.  If the
parameterizations that you want are as widely used as Euler angles,
that would of course be a strong reason to include them. 

Dave Withoff
Wolfram Research



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