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 by the red curves and the blue curve 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 (the red triangle) this device can be used to build a drill that drills perfect square holes.
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.
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