> ***Q***. Is there a linkage in the plane that traces out a circle $C$ in such a manner that the interior of the disk bounded by $C$ is never intersected by any link througout the motion? What I mean by "a linkage in the plane" is best illustrated by the famous [Peaucellier linkage][1], which traces out a straight line segment: <sub>(Wikipedia image.)</sub> <br /> ![WikipediaPeacellier][2] <br /> Some vertices are "pinned to the plane." All vertex joints are universal in the sense of allowing full $360^\circ$ motion. All links are rigid segments, which can pass over one another in a physical, layered model, e.g., <br /> ![HowRound][3] <br /> <sub>(Image from [this *How Round Is Your Circle* web page][4].)</sub> <hr /> Of course it is trivial without the restriction that the links not intersect the interior of $C$: One radial link pinned to the center of $C$ suffices. And the challenge in the Peaucellier linkage was to convert the natural circular motions of linkage components into straight-line motion. Here I am seeking a vertex of the linkage to follow a natural circular motion, but with the restriction to not intersect the interior of that circle $C$. Following a subarc of $C$—say, a semicircle—without intersecting the interior of $C$, would also be quite interesting. It may be that the 19th-century masters (Peaucellier, Lipkin, Watt, Chebyshev, et al.) did not investigate this question. But perhaps there is an easy construction I am not seeing...? [1]: http://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_linkage [2]: https://i.sstatic.net/BCSuT.gif [3]: https://i.sstatic.net/yBUea.jpg [4]: http://web.mat.bham.ac.uk/C.J.Sangwin/howroundcom/straightline/exact.html