> -----Original Message-----
> From: Bernhard Rupp [mailto:[log in to unmask]] 
> Sent: 13 August 2007 22:14
> To: Ian Tickle
> Subject: step 1
> 
> I think I understand what you sent last, but
> but I am slow and need this in simple words:

My first point which I would stress is that I believe we are actually
all agreed on the correct *equation* to use, namely it's the one in
Phil's article.  I certainly never intended to imply that Phil's
equations are wrong!  So there's no point going to the source code and
comparing it with the equation, you won't find any inconsistency.  My
point is that it all depends on precisely what you *mean* by the symbols
in the equations, particularly the rotation matrix symbols R(a), R(b) &
R(g).  As they say "define your terms, sir, or we'll never agree!".  The
net rotation can be written in several different ways (probably a very
large number, but let's just stick to the ones that people may be
familiar with!).  I think this is the point that many people have not
yet latched onto, that the same equation can be described in words in
different ways.

To summarise these, the *SAME* net rotation can be expressed in (at
least) 3 ways:

1. Rotations about *fixed* axes (i.e. attached to the laboratory frame,
or 'old'): first gamma, then beta, then alpha:

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

The matrices R are precisely the ones in Phil's article: however he
refers to them as applied w.r.t. 'new' axes (i.e. attached to the
rotating frame) in the order gamma, beta, alpha, which is the exact
statement that I take issue with.  In other words, Phil's verbal
description does not match his equation.  Note that the matrices are
applied (i.e. pre-multiplied) onto the co-ordinates in the order as
given but writing them from *right to left* (because the co-ordinate
vectors go on the right of the equation).

Note that this is by far the easiest description to understand and the
one I recommend that everyone sticks with!  However as people seem to
insist on talking about the more difficult to comprehend rotated axes:

2. Rotations about *rotated* ('new') axes (i.e. attached to the
laboratory frame), first alpha, then beta, then gamma:

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

The equation is absolutely identical to the last one and the matrices R
are again precisely the ones in Phil's article even though my words
describing them are now different!  Note now that the matrices appear in
order from *left to right*.  The reason this happens is because the
individual matrices do not actually represent rotations about rotated
axes as described, they are still the matrices for fixed axis rotations,
so they are actually the wrong matrices!  However it turns out that if
you also write the matrices in the *wrong* order, then two wrongs do
indeed make a right - as can be seen the correct equation as in (1) is
obtained!  This is probably the hardest point for many people to
appreciate, yet the result is so simple!  Note that this is not
mentioned in Jorge's article, for more info see this ancient MR tutorial
I wrote (probably out of date in most other respects!):

http://www.ccp4.ac.uk/dist/ccp4i/help/modules/appendices/mr-bathtutorial
/mrbath2.html

3. Rotations about *rotated* axes, first alpha, then beta, then gamma:

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

Now my description is the same as (2), but the equation is now
completely different from the others (the order of the angles is
reversed), though I repeat that the net rotation is still the same, as
Jorge proves in his IT(F) article.  The matrices R are also not the same
as the ones in Phil's article, even though my words describing them are
the same as Phil's!  Ry'(b) is the rotation beta about the y axis
*after* rotation by alpha so it will depend on alpha as well as beta,
similarly Rz'(g) is a function of alpha, beta and gamma (I won't try to
write them down and prove that you end up with the same result - I'll
leave that as an exercise for the reader, as they say!).  So you have to
say precisely what you mean by Ry'(be) etc because that makes all the
difference as to whether the equation is right or wrong!

> Step 1 (Please check my ?marks - would be great to get this 
> right once and
> for good):
> 
> 1) Eleanor authoritatively states that   
>    "CCP4: ROTATING reference axis system: 
>       leads to matrix R=[R_alpha_about Z0] {R_beta_about_Y1]
> [R_gamma_about_Z2]"
>       = Rzo(al)Ry'(be)Rz"(ga)
>       meaning: rotate first about ga(z") then be(y') then 
> al(zo) <--- is
> this true (Y/N)?

I can't answer this without seeing the matrices!  It appears to be the
same as my statement (2), so it may or may not be correct depending on
what you mean by Ry'(be) and Rz"(ga).  I would assume they are the same
simple matrices as in Phil's article each a function of only one angle,
not the more complicated ones involving 2 or 3 angles that you worked
out for yourself, so by my definitions they would be Ry(b) not Ry'(be)
and Rz(g) not Rz"(ga).  Note that I reserve the primes for the
complicated axial rotation matrices involving more than one angle that
you need when you're talking about rotated axes, and as I keep saying,
try to avoid talking about rotated axes if at all possible!  However as
you are referring to rotated axes then the first rotation to be applied
to the co-ordinates must be alpha, so that is wrong.  If the first
rotation were gamma then you would have to be referring to fixed axes.

>       i.e., R = R(last)R(be)R(first) - 
>       *first applied* rotation written *last in matrix* (if Euler
> convention) <--- is this true (Y/N)?
>       as Phil says"
> Because if step 1 were true,
>       meaning: rotate first about ga(z") then be(y') then al(zo) <---
is
> this true?
>       as Phil says"

Again the equation appears to be correct, but its truth or falsity all
hangs on the definitions you are using!

>       Hmmmm....
>       Phil says: 
>       "rotate by gamma around z(zo, ed.), then by beta around the new
y, 
>        then by alpha around the new z(z", ed.) again, R =
Rz(al)Ry(be)Rz(ga)"
>      i.e., R = Rz"(al)Ry(be)Rzo(ga)
>      This seems consistent with Eleanor only if z and z" are reversed
in
> Phil <--- is this true?

Again this description doesn't look right because the first rotation for
rotated axes (i.e. what Phil is talking about) has to be alpha, but the
equation could well be right, again depending on your definition of
Rz"(al) etc.

HTH

-- Ian


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