I'm investigating the following problem.
Imagine a circle at the origin with radius r. The circle moves upward with speed s and its radius simultaneously increases by a length of b per time unit. That is, at time t=0, the circle is at the origin with radius r. At time t=n, the circle is at (0, s*n) with radius r+b*n. As the circle moves and expands, it sweeps a shape. What is the formula for the shape swept by the circle in a given amount of time N?
It occurs to me that there are two cases. First, when s <= b, the motion of the circle is overshadowed by the expansion, and the final shape ends up being a circle at (0, s*N) with radius r+b*N.
But when s > b, the expansion can't keep up with the motion, and it sweeps a shape through space. It appears to me that the exact shape is that of both circles connected by line segments tangent to both, like two pulleys with a belt.
This related question says that it's called the convex hull of two circles. The questioner wants to know how to determine whether a point is within the convex hull. I could also like to know how, but I'd like to have a formula rather than an algorithm. I imagine there must be a formula that describes this shape, but unfortunately, I've been unable to derive one myself. :-/ It would be great if one formula handled both cases, but I'd be happy to have just the formula for case two.
I'd also be satisfied with a formula to determine the distance from the origin to the outer edge of the shape at a given angle. In fact, that is the main thing I would hope to calculate from the formula, in addition to using the formula to plot it.
Does anybody know? Thank you.