🚀 go-pugleaf

In article <Pine.SGI.3.91.951127153848.8482A-100000@helios.physics> Daniel Cote <dcote@Physics.UToronto.CA> writes:
>From: Daniel Cote <dcote@Physics.UToronto.CA>
>Subject: Cut-projection method
>Date: Mon, 27 Nov 1995 20:44:02 GMT


>This is about the cut-projection method (and some other related stuff).

>I know this is kind of basic, but I don't understand how an intersection=20
>in real space becomes a projection in Fourier space.  I looked at many,=20
>many books, and they all state "We know that...".   But I don't.

>So my question is extremely simple:

>How do you show (mathematically) that an intersection in real space=20
>becomes a projection in fourier space?

>I really need to understand the basic idea behind that technique, and I=20
>can't see it.

>Thank you,

>-------------
>Daniel C=F4t=E9                       "If it moves, it's Biology.
>U of Toronto Physics Dept.             If it stinks, it's Chemistry.
>dcote@physics.utoronto.ca                  If it doesn't work, it's Physics=
>."

   A single section of electron density can be represented as the product of 
the full 3-d map and a "plane".  The plane would be zero everywhere except at 
the location of the section where it would have a value of one.  By the 
convolution thereom the Fourier transform of the section would be the 
convolution of the transform of the 3-d map and the transform of the plane.  
The transform of the map is your structure factors.

   The Fourier transform of the plane is more difficult to justify.  A simple 
way to explore the function is to look at its variability in different 
directions.  Assume the plane is z=5.  For any x and y, as you vary z to cross 
the plane you will find a delta function.  The Fourier transform of a delta 
function is a constant for all resolutions.  For any z,  as you vary x or y 
you will never have any variation in the function.  The Fourier transform of a 
constant function is zero for all resolutions (except for the 000 term).  
While this is not a mathematical proof it indicates that the Fourier transform 
of a plane in a line, perpendicular to the plane and passing through the 
origin.

   Now all you need to do is to convolute the structure factors with the line 
to get the Fourier transform of the section.  You should be able to work out 
that this is a projection operator.

   Of course this argument works both directions.  This is why a precession 
photo gives you a projection in real space.

Dale Tronrud
University of Oregon