Details

The existence of such a device was stated as an open question in [1]. To turn this geometrical construction into a working drill with a driving end that follows plain circular motion, you could use an Oldham coupling, as shown in the Demonstration "Square-Hole Drill in Three Dimensions".

Snapshot 1: shows the case ; then the roller becomes the intersection of two copies of a vesica pisces (a vesica pisces is the intersection of two congruent circles, where the center of each circle lies on the boundary of the other one—see the MathWorld entry for "Lens" (Wolfram MathWorld)); in this case the locus of the cusp is identical to the ambient pentagon

Snapshot 2: shows the case where is large, and the locus is a rather small pentagon

Snapshot 3: shows the case of a small value of

The construction is based on an envelope construction similar to that in [1] to determine the proper radius of the four large circular arcs. Once is known, the various junction points and angles that define the arcs are easily computed. It is critical that, for each orientation of the roller, its position within the large pentagon is determined. The red tangency points show that this rigidity holds, since there are always enough points of contact to prevent either up-down or left-right motion.

Reference:

[1] B. Cox and S. Wagon, "Mechanical Circle-Squaring," College Mathematics Journal, 40, 2009 pp. 238–247.