View Full Version : Rotated Ellipses

07-28-2010, 09:49 AM
Hi all, I've been thinking about this problem for a while, and I haven't been able to find a good starting point. This may be more of a geometry/calculus problem than a probability one...but here's the setup:

Take a horizontal ellipse with semi-major axis = a, semi-minor axis = b, and center c at (x + a, y). Now, rotate that ellipse from theta = [0 - 2pi) about the point (x, y), making a full circle. The ellipse should draw out a circle, but there will be a radial distribution of where the ellipse overlapped itself the most. i.e., the inside of the circle will have the highest probability due to overlapping, while the outside will have the lowest. I've attached a picture using discrete steps to clarify, but the goal is to determine a distribution based on the rotation being continuous.


So the question is, how would I even start going about trying to quantify this distribution?