Subject: Re: CCL: Smallest rectangular box

From: Herbert Homeier t4720 <[email-masked]>
Date: Thu, 13 Jun 96 11:40:18 +0200

> Dear CCL'ers,
> I've got the problem to find the smallest possible rectangular Box
> around a set of points (atoms) in space. Any suggestions or ideas
> are wellcome. I'll summerize the results.
> 
> Heiko Gerwens <[email-masked]> 

Hi,

here is my answer that I believe to avoid the drawbacks mentioned by
Friedrich Ackermann <[email-masked]>

Let the original points be \vec r_i = (x_i,y_i,z_i).
For a given rotation R(\alpha,\beta,\gamma) defined via 
Euler angles \alpha, \beta, \gamma the rotated points
are \vec r_i' = R(\alpha,\beta,\gamma) \vec r_i = (x_i',y_i',z_i').

Define 

F(\alpha,\beta,\gamma) = ( max(x_i') - min(x_i') )
( max(y_i') - min(y_i') )( max(z_i') - min(z_i') )

where max and min are meant as "for all i". This is the volume of the
smallest box parallel to the coordinate axes for the rotated points.

Then the volume of the smallest box should be

min_{\alpha,\beta,\gamma} F(\alpha,\beta,\gamma)

This comes close to an algorithm already... 

Or put as a sentence: Find that frame that the minimal box 
parallel to the coordinate axes in that frame is minimal!

Obviously, one could give also a "passive" interpretation (rotate
coordinate axes instead of points ...).

Maybe some mathematician will prove that the result really is THE
smallest box. But intuitively this method should be ok.

Best regards

Herbert Homeier
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Dr. Herbert H. H. Homeier
Institut fuer Physikalische und Theoretische Chemie
Universitaet Regensburg
D-93040 Regensburg, Germany
Phone: +49-941-943 4720                FAX  : +49-941-943 2305
email: [email-masked]
<A HREF="http://rchs1.uni-regensburg.de/%7Ec5008/">HOMEPAGE</A>
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