As a means of curiosity, I am working on a problem involving the rotation of various shapes. The thing that I am interested in is the cross-sectional length of the shapes as they cross the x axis. So imagine that you have a circle with a pivot point at its bottom. You then rotate it about the origin. I'm interested in the length of the x axis that is enclosed in the circle at a time t. This turns out to be easy as it can be expressed in polar coordinates as r=2(radius)sin(ωt). That's not too bad, but what if the circle is rotating about a point that is h units above the origin? Clearly where h≥d, it never touches, but what if h≤d? How can we think about this? Further, what if we want to consider different shapes?