Just my own amount of salt in the Rotation function soup...
I just want to try defending the poor little Euler angles.
First, Euler invented them... Yes !
Second, only Euler angles yield a very nice interpretation of Rot Funct
symmetry in terms of space group.
See the two venerable papers:
Tollin, Main & Rossmann, Acta Cryst 20 (1966) 404
and Narasinga, Jih & Hartsuck, Acta Cryst A36(1980) 878
Third, only Euler angles yield a very practical and intuitive thing, namely
it does not matter rotating by (alpha, beta,gamma) in the usual way,
or first, prerotating by gamma around Z (but without rotating any axes !)
and then making the alpha rotation around Z (rotating now the axes) followed
by the beta rotation around the new Y. This is most easily seen with a
cylinder that one rotates in the two ways.
Last, the inverse rotation matrix of the Euler matrix defined by (alpha,
beta, gamma) is just the Euler matrix defined by (-gamma, -beta, -alpha).
Ian, isn't worth the effort ?
Philippe Dumas
IBMC-CNRS, UPR9002
15, rue René Descartes 67084 Strasbourg cedex
tel: +33 (0)3 88 41 70 02
[log in to unmask]
-----Message d'origine-----
De : CCP4 bulletin board [mailto:[log in to unmask]]De la part de Ian
Tickle
Envoyé : Monday, August 13, 2007 8:11 PM
À : [log in to unmask]
Objet : Re: [ccp4bb] CCP4 rotation convention
Hi folks
I hate to say this but I think everyone here has got it wrong to some
degree (including myself - and I hereby retract my previous e-mail and
issue the correction below!). If you don't believe me then read &
digest Jorge Navaza's article "Rotation functions" in Int. Tab. Vol. F
(sect 13.2, p. 269), particularly sections 13.2.2 and Appendix
A13.2.1.1.
Phil's article in Acta D57 1355-1359 (2001), i.e. the 2001 S/W
proceedings, states:
"... the convention used in AMoRe (Navaza, 1994) and other CCP4 programs
(Collaborative Computational Project, Number 4, 1994) is to rotate by
gamma around z, then by beta around the new y, then by alpha around the
new z again, R = Rz'(a).Ry'(b).Rz(g)"
Compare this with Jorge's equation 13.2.2.3 which he explicitly states
applies to rotations about fixed axes, not rotated axes (but using my
notation):
R = Rz(a).Ry(b).Rz(g)
i.e. first by gamma about z, then by beta about the *fixed* y axis, then
by alpha about the *fixed* z axis.
The same formula cannot apply to both rotations about fixed and rotated
axes at the same time!
Looking at Jorge's equation 13.2.2.1 it's plain that the correct version
involving rotated axes is (again substituting my own notation which
should be obvious):
R = Rz'(g).Ry'(b).Rz(a)
i.e. the correct statement is that the rotation is generated by rotating
first by alpha about z, then by beta about the rotated y axis (y'), then
by gamma about the rotated z axis (z').
Of course it may well be that Phil's equation is based on an older
version of Jorge's analysis perhaps using a different convention in his
Acta Cryst. (1994), A50, 157-163 paper, but unfortunately I don't have
online access to AC(A) to check it out, maybe someone who has access
could do so.
In fact it's quite obvious looking at the individual matrices Rz(a) &
Ry(b) at the bottom of page 1358 in Phil's paper that they must apply to
fixed not rotating axes, because if say the Ry(b) matrix were for
rotation about the rotated y axis, it would have to be a function of
gamma: applying the Rz(g) matrix as given in the paper first to the
y-axis vector (0,1,0) gives the rotated y-axis vector
(-sin(g),cos(g),0). Similarly if the Rz(a) matrix represented rotation
about the rotated z axis it would have to be a function of both beta &
gamma and plainly it's not.
This all goes to show that a) even the experts sometimes get it wrong
particularly where matrix algebra is concerned, and b) you should avoid
the concept of rotating about rotated axes like the plague!
-- Ian
> -----Original Message-----
> From: [log in to unmask]
> [mailto:[log in to unmask]] On Behalf Of Bernhard Rupp
> Sent: 12 August 2007 20:37
> To: [log in to unmask]
> Subject: CCP4 rotation convention
>
> Dear programmers -
>
> Phil Evans writes in acta D57 1355 (2001) on p 1358 section 5.2:
>
> "....the convention used in AMoRe (Navaza, 1994)
> and other CCP4 programs (Collaborative Computational
> Project, Number 4, 1994) is to rotate by gamma around z, then by >
beta
> around the new y, then by alpha around the new z again,
> R = Rz(al)Ry(be)Rz(ga)"
>
> This seems correct, as the first rotation is applied first to
> vector x, then the second to the new one, etc, thus
> x' = (Rz(al)(Ry(be)(Rz(ga)x)))
>
> In J.Appl.Cryst. 30 402-410 (1977) in the convrot description,
> Sascha Uzhumtsev lists in table one for (Navaza 1994):
>
> alpha about Z, beta about Y and gamma about new Z
> and gives the *same* resulting rotation
> Rz(al)Ry(be)Rz(ga)
>
> This seems to be a contradiction I cannot resolve?
>
> Thx, br
> -----------------------------------------------------------------
> Bernhard Rupp
> 001 (925) 209-7429
> +43 (676) 571-0536
> [log in to unmask]
> [log in to unmask]
> http://www.ruppweb.org/
> -----------------------------------------------------------------
> People can be divided in three classes:
> The few who make things happen
> The many who watch things happen
> And the overwhelming majority
> who have no idea what is happening.
> -----------------------------------------------------------------
>
>
Disclaimer
This communication is confidential and may contain privileged information
intended solely for the named addressee(s). It may not be used or disclosed
except for the purpose for which it has been sent. If you are not the
intended recipient you must not review, use, disclose, copy, distribute or
take any action in reliance upon it. If you have received this communication
in error, please notify Astex Therapeutics Ltd by emailing
[log in to unmask] and destroy all copies of the message and
any attached documents.
Astex Therapeutics Ltd monitors, controls and protects all its messaging
traffic in compliance with its corporate email policy. The Company accepts
no liability or responsibility for any onward transmission or use of emails
and attachments having left the Astex Therapeutics domain. Unless expressly
stated, opinions in this message are those of the individual sender and not
of Astex Therapeutics Ltd. The recipient should check this email and any
attachments for the presence of computer viruses. Astex Therapeutics Ltd
accepts no liability for damage caused by any virus transmitted by this
email. E-mail is susceptible to data corruption, interception, unauthorized
amendment, and tampering, Astex Therapeutics Ltd only send and receive
e-mails on the basis that the Company is not liable for any such alteration
or any consequences thereof.
Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science
Park, Cambridge CB4 0QA under number 3751674
|
