Hi Tilman

That's great!  Your derivation for the 'rotating axes' case is
essentially the same as one I derived for myself and also agrees with
that of Jorge Navaza (ITC Vol F, sect A13.2.1.1 p.273), but it's nice to
have it all readily accessible on a webpage.  BTW I note that you hide
the full horror of the complexity of the matrix algebra for the rotating
axes case with a link which some people may not have noticed at first
glance:

http://www.biozentrum.unibas.ch/personal/schirmer/modtrafo_html/analogou
s.html .

Personally I would show the gory details all up front to let people know
what they are letting themselves in for if they insist on using this
description, i.e. a concatenated matrix product of no less that
*thirteen* simple axial rotations if written out in full, compared with
only *three* for the 'fixed-axes' description!  Personally I avoid the
'moving axes' description like the plague, simply because it's the
source of so much complexity and confusion!

A similar derivation is also given in Wikipedia:

http://en.wikipedia.org/wiki/Euler_angles#Equivalence_of_the_movements

(note that this page is currently in a state of flux, and do check
everything because some of it is just plain wrong - these facts are
probably not unconnected!).

However there's still a problem, and it's by no means a trivial one!
Most sources on this subject give a different result but are all
consistent with each other, e.g. see here:

http://mathworld.wolfram.com/EulerAngles.html

and here:

http://www.sp.uconn.edu/~ch351vc/pdfs/euler2.pdf

and here: Goldstein, Classical Mechanics (this is the original 'gold'
standard on this subject, pun not intended!),

and most surprising of all here: ITC Vol. B, sect. 2.3.6.2, p.252-3
(Rossmann). Note that Rossmann's matrix here is identical to the
MathWorld one.

Note also that the matrix given on the Wikipedia page is completely
wrong and not consistent with anywhere else (e.g. for ZYZ it seems to
have alpha & gamma swapped in the matrix even though they are defined
identically - unless someone has fixed it by the time you read this), so
ignore it!

On the MathWorld page we see the rotations phi (= our alpha) around z,
theta (= beta) around *rotated* x, psi (= gamma) around *rotated* z and
the product matrix is given as:

	A        = B  .  C  .  D
or
	R(a,b,g) = Rz(g).Rx(b).Rz(a)

where note the matrices B=Rz(g), C=Rx(b) & D=Rz(a) are axial and
therefore rotations around the old *fixed* axes.  This is clearly
different from the CCP4 convention (even taking into account that CCP4
uses ZYZ):

	R(a,b,g) = Rz(a).Ry(b).Rz(g)

So we have two completely different answers using apparently identical
definitions, except that some people use the ZYZ convention, and some
use ZXZ: but that doesn't account for the inconsistency (though it does
tend to obscure the fact that there is an inconsistency!).  What's more,
I can assure you that CCP4 and the ROTW (rest of the world) are BOTH
correct - so how is that possible?

I believe the answer goes to the root of all the historical confusion
about Eulerian angles.  The answer is that people are assuming an
absolutely fundamental convention (without usually even taking the
trouble to state it explicitly!), which is different for CCP4 and the
ROTW.  Actually this is not quite true: in fact computer gaming software
seems to use the same convention as CCP4 - e.g. see here:
http://www.opengl.org/sdk/docs/man/xhtml/glRotate.xml (this of course
puts us in a clear majority by far over ROTW!).

A clue to the answer actually lies in the apparently trivial change of
sign in the axial matrix used, e.g. CCP4 and OpenGL uses for rotation
about z:

		[  cos(a) -sin(a)  0 ]
		[  sin(a)  cos(a)  0 ]
		[    0       0     1 ]

whereas the ROTW uses:

		[  cos(a)  sin(a)  0 ]
		[ -sin(a)  cos(a)  0 ]
		[    0       0     1 ]

However this is much more than a simple change of sign of the angles.
In fact if anyone cares to read Goldstein (thoroughly!) he makes it
quite clear: there are actually two fundamental conventions for
describing rotating bodies in use, one we can call the "physicists'
convention" (particularly quantum mechanicists), where the object whose
co-ordinates are to be transformed is nevertheless considered to be
stationary and the observer is imagined to move around the object
instead (you can understand why non-physicists have trouble with this!).
Obviously this convention has been used by people who have traditionally
been able to comprehend what they're doing purely via the equations of
motion and who therefore apparently don't need graphical visualisation
aids(!).  Goldstein writes the equation:

		(r)' = Ar

(the brackets indicate that the vector r *doesn't* move, only its
components change value).

The other convention we can call the "chemists' convention" since it is
used by people who do need to visualise things, where the observer is
considered to be stationary and the object is moved, but note in the
opposite direction which explains the changes of sign in the matrix, so
the net result is the same.  Also it's pretty obvious why you have to
use this convention for computer graphics!  Goldstein writes this as:

		r' = Ar

(now the vector r does move and becomes the vector r').

So one could call this a fixed vs. moving object convention, but note
that it is completely separate from the fixed vs. moving axes convention
we have all been talking about, and I believe that mixing up these two
concepts has caused additional confusion in the past.

Cheers

-- Ian


> -----Original Message-----
> From: [log in to unmask] 
> [mailto:[log in to unmask]] On Behalf Of Tilman Schirmer
> Sent: 13 December 2007 18:47
> To: [log in to unmask]
> Cc: Tilman Schirmer
> Subject: Re: CCP4 rotation convention
> 
> Dear all,
> 
> I was not aware of this discussion, when I also ran into 
> confusion. Finally, I figured it out myself and 
> wrote a summary. Perhaps it is useful for somebody:
> http://www.biozentrum.unibas.ch/personal/schirmer/modtrafo_htm
> l/rotation.html
> 
> I fully agree with the final comments by Ian and Sacha.
> 
> The comment is part of the documentation for my program 
> MODTRAFO, a collection of jiffy programs 
> dealing basically with models and transformations. The 
> platform allows e.g. to create easily rotation 
> matrices from polar or Eulerian angles or, vice versa, to 
> analyze a transformation for its angular 
> components.
> 
> Best,
> Tilman
> 
> 
> 


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