**Particle tracking using IDL -- John C. Crocker and
Eric R. Weeks**

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# extra comments

I wrote this in response to a question I got, asking to clarify
my special tricks.

For a given particle and a given distance r, you are trying
to count all the particles that lie between r and r+dr
away. In 3D this is a spherical shell with thickness dr.
To correctly normalize g(r), you have to then account for
the volume contained within that shell. But, if your shell
lies outside your computation box, then you have to figure out
exactly how much volume lies within your box. Assuming dr is
small, then really it's like computing how much surface area
of the sphere of radius r lies within the box.

So, you're left with a nontrivial trigonometry problem,
especially if
you have a small cubical box. For most values of r,
you'd have several pieces of your sphere lying outside
your box, and computing the area of what's left inside would be
difficult.

In 2D the math is not too hard (for a circle and a
rectangular box). My trick there was to consider each quadrant
of the circle separately. At most, each quadrant of the circle
could intersect (or lie outside of) two edges of the rectangle.
Then I could determine how much of the quadrant was within those
two edges. Working it out took a few sheets of paper but the
resulting formulas were fairly simple (although at this point
embedded within my program and so I don't have them easily at
hand).

But in 3D it's a bit harder. I cheated by forcing the program to
stay away from the edges of the box in X and Y, so the only
intersections I have to consider are with the faces of the box in
Z. The math then wasn't too bad (to find the surface area of a
spherical cap lying outside the box).

Anyway, if you have a cubical box, I don't know if there is any
simple way to do it. Perhaps my trick in 2D with quadrants and
two edges of the rectangle can be extended to 3D with octants and
3 faces of the cube. But you'll have to see if you can do the
trigonometry and find a tractable way to compute the area
contained within the box.

## Contact us

- This page was written by
Eric Weeks:
weeks(at)physics.emory.edu