Bicycle or Unicycle Tracks?

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.


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