Drilling a Square Hole
One way of getting a curve of constant width is to start with a right isosceles triangle and draw arcs of circles centered at the vertices as indicated and an additional smaller-radius arc at the top. Then if the resulting curve is rotated so that it stays inside the square that surrounds it, the locus of the apex is an exact square. By placing a cutting tool at the apex (red) this device can be used to build a drill that drills perfect square holes. One would start with a circular hole in the target material and then use the rotor to remove material so that the hole is an exact square.
The idea of this drill goes back to a 1939 article in Mechanical World. A full description can be found in the book How Round Is Your Circle? Color plate 21 shows a working drill using this method that the authors constructed.
J. Bryant and C. Sangwin, How Round Is Your Circle?, Princeton, NJ: Princeton University Press, 2007.