I'm having trouble finding the proper keywords to search for this type of treatment so I apologize in advance if this is quite obvious.
I have a collection of lines in 3D space approximately centered about a central point. I then calculate the angle that each line forms with the central point (i.e. the angle formed by the center, the beginning of a line and the end of line) and plot a histogram showing how often each angle occurs in my system, from 0 to 180 degrees.
Now conceptually, I would imagine this distribution to be biased towards 90 degrees (and it is in my case) since there is precisely one orientation in which each line can form an angle of 0 or 180 degrees but many ways in which intermediate angles can be formed (there is exactly one way to have the line be exactly 0 degrees. There are many ways to be off by 1 degree, and more ways to be off by 2 degrees than by 1 degree).
So my question:
Is there a way to normalize this distribution of angles in order to take into account the number of configurations that can lead to each angle?
I suspect it will require a normalization factor similar to $1/\sin(\theta)$ but I can't find any references supporting this notion.
I believe that the question, as asked, cannot be answered. Let me rephrase it slightly:
"Suppose that $S$ is a random line segment with endpoints $A$ and $B$, and that $P$ is the origin. Let
$\theta(S) = \arccos( \frac{(A - P) \cdot (B - P)}{\|A-P\| ~ \|B - P \|})$
be the angle subtended by the segment $S$ at the origin. What's the probability density for the random variable $\theta$?"
The difficulty here is with the starting statement "$S$ is a random line segment". That might mean that $A$ and $B$ are chosen uniformly from points within the unit sphere. Or you might hope it means that they're chosen uniformly from all possible points in space. (That doesn't work, because there's no uniform distribution on $R^3$.) Or you might mean that $A$ and $B$ are chosen uniformly from within the unit cube. You might say "Hey, there's a rotational invariance here, so I'm going to pick $A = (1, 0, 0)$ for all time, and pick $B$ at random." Or maybe it's something else. The bad news is that each of these gives a different result.
I'm therefore going to suggest, by way of an answer, that you formulate the question as precisely as you possibly can. Say exactly how the line segments are generated, and then maybe we can help you determine the distribution of $\theta$.