The two wheels of a bicycle typically make separate tracks. One exception is a bike traveling in a straight line. The convoluted path shown in the graphic is an example of a nonstraight track along which a bike travels so that the rear wheel follows exactly in the track made by the front wheel. If is the parametrized curve, then , and the white unit tangent vector at ends at . This vector represents the bicycle.

The curve is defined by starting with the red path and then extending rightward by adding the unit tangent vector. This construction yields an ambiguous bicycle-unicycle path for all . The initial curve from to that gets the process started is the singular function , which shows a single small bump. Snapshot 1 shows how that single bump becomes a double bump in the next segment. All derivatives of this function at or are , except the first derivative, which is . This property is preserved at the end points of all the other unit-time-interval segments going forward. This construction is due to David Finn [1].

Reference

[1] D. Finn, "Can a Bicycle Create a Unicycle Track?," The College Mathematics Journal, 33(4), 2002 pp. 283–292. doi:10.2307/1559048.