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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, sn) with radius r+bn. 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, sN) with radius r+bN.

Case one.

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.

Case two.

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.

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    If you say the center at $t=0$ is the origin, then a point $(x,y)$ is in one of the circle's if and only if there exists a $t\in[0,N]$ such that $$x^2+(y-st)^2\le(r+xt)^2.$$ This can be rewritten as $(\text{quadratic in }t)\ge0$. You can find the interval(s) on $\mathbb{R}$ where this inequality holds using standard techniques and then check if one of them intersects $[0,N]$. The test for intersection can itself be written as an inequality in $N,x,y,s,$ and $r$ determined by the roots of the quadratic.2011-08-22
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    Thanks, that looks promising. I will investigate after I get some sleep. In the question, I made a poor choice of variable name, using x for the increase (e.g. r + x*t) although it's unrelated to the x coordinate. I've renamed that to b. Am I correct that only the second x in your equation should be renamed to b, while the first x refers to the x coordinate?2011-08-22
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    I didn't even notice that slip up! Yes your understanding of the formula is correct.2011-08-22
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    Hmm. Producing the (quadratic in t) >= 0 is easy enough. But it seems like the roots of the quadratic depend on both x and y. Then, finding the intervals where the inequality holds seems hard because it's not a 2d curve with a set of zero crossing points, but a 3d surface whose zero crossings are themselves curves (ellipses?). Getting the correct set of zero crossings also seems dependent on t being bounded. I'm probably just stating the obvious here, but I seem to have gotten stuck. I'll keep thinking about it, though, and hope I have some insight... :-)2011-08-22
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    It's a quadratic *in t*, there are only at most two roots. Remember $x,y,s,b$ are fixed.2011-08-22

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