# Help indentifying unknown method of computing area by rolling a ball about the perimeter

Some time ago I saw a brief presentation about a newly discovered method used to compute the area of certain two-dimensional shapes by rolling a circle (of varying radius) about the perimeter; the area traced out by the circle was the area enclosed by the figure.

I believe the method only worked for convex shapes. I didn't catch the name of the discoverer of the method, but the name sounded French, and the method was discovered I think in $$2002$$ or thereabouts.

Any help identifying this method would be appreciated.

Edit: I'm not completely sure that convexity was required, but given that the perimeter of a non-complex shape can go to infinity without changing the area, and given that as the circle travels around the perimeter the area swept out is strictly increasing, it seems reasonable. There were only two or three examples, and all of them were convex. However the circle's radius also depended on...something, and it's possible the circle would have been much smaller for a non-convex shape with large perimeter and small area.

The only two examples I remember were some sort of simple polygon (a triangle or a rectangle or something like that) and some sort of trig function like $$\sin^2 x$$ from 0 to $$\pi$$. In the latter case the method was billed as being easier than integrating the curve directly.

Do you mean a planimeter? These have been around for several centuries, but I'm sure improvements are invented from time to time.

This mechanism seems to me to be no more than an embodiment of Green's theorem, as explained in the Wikipedia article. The OP's mention that the new method only works with convex areas is hard to assimilate into a Green's theorem story, however. So I'm puzzled, and would like to know what the OP saw originally.

• Maybe something like this? This seems to be a physical instrument, while the method I'm thinking about was purely mathematical. But the idea seems similar: trace the perimeter, use that to find the area. I should maybe add that this method was billed as, in some cases, a more convenient/computationally easier way of finding the area than integration would be. Jan 21, 2019 at 3:52
• @Galendo I have edited my non-answer. Could the convexity requirement be a hint that a naive triangulation recipe will work? Jan 21, 2019 at 4:16
• @kimichi lover: I've edited my question to hopefully clarify things. Jan 21, 2019 at 17:19
• H'm. Maybe you can reconstruct the context in which you heard about this new method, and work backwards from there? Like, where, when, from whom. Was there an application area? Was the presenter a pure mathematician? And so on. (If we knew it was all the rage among chemical engineers in London, say, we might be able to work from there.) Jan 21, 2019 at 17:24
• Unfortunately there isn't much to be said. This method was only a small part of a talk that was geared largely toward non-mathematicians, though a certain familiarity (probably high-school level) was assumed. If there were any practical applications of the method, other than being easier than integration in some cases, I don't remember them. Jan 21, 2019 at 17:59