Abstract
We study closed smooth convex plane curves Λ enjoying the following property: a pair of pointsx, y can traverse Λ so that the distances betweenx andy along the curve and in the ambient plane do not change; such curves are calledbicycle curves. Motivation for this study comes from the problem how to determine the direction of the bicycle motion by the tire tracks of the bicycle wheels; bicycle curves arise in the (rare) situation when one cannot determine which way the bicycle went.
We discuss existence and non-existence of bicycle curves, other than circles; in particular, we obtain restrictions on bicycle curves in terms of the ratio of the length of the arcxy to the perimeter, length of Λ, the number and location of their vertices, etc. We also study polygonal analogs of bicycle curves, convex equilateraln-gonsP whosek-diagonals all have equal lengths. For some values ofn andk we prove the rigidity result thatP is a regular polygon, and for some we construct flexible bicycle polygons.
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References
A. Abrams, J. Cantarella, J. Fu, M. Ghomi and R. Howard,Circles minimize most knot energies, Topology42 (2003), 381–394.
V. Arnold,The geometry of spherical curves and quanternion algebra, Russian Mathematical Surveys50 (1995), 1–68.
S. Dunbar, R. Bosman and S. Nooij,The track of a bicycle back tire, Mathematics Magazine74 (2001), 273–287.
D. Finn,Can a bicycle create a unicycle track?, College Mathematical Journal, September, 2002.
D. Finn's, at www.rose-hulman.edu/finn/research/bicycle/tracks.html
H. Guggenheimer,Differential Geometry, Dover, New York, 1977.
E. Gutkin,Billiard tables of constant width and dynamical characterization of the circle, Penn. State Workshop Proc., Oct. 1993.
J. Konhauser, D. Velleman and S. Wagon,Which way did the bicycle go?…, and other intriguing mathematical mysteries, Mathematical Association of America, Washington DC, 1996.
M. Kovalev,A characteristic property of a disc, Proceedings of the Steklov Institute of Mathematics152 (1980), 124–137.
S. Tabachnikov,Billiards, Société Mathématique de France, Montrouge, 1995.
H. Auerbach,Sur un problem de M. Ulam concernant l'equilibre des corps flottants, Studia Mathematica7 (1938), 121–142.
J. Bracho, L. Montejano and D. Oliveros,A classification theorem for Zindler carrousels, Journal of Dynamical and Control Systems7 (2001), 367–384.
J. Bracho, L. Montejano and D. Oliveros,Carousels, Zindler curves and floating body problem, preprint.
R. Mauldin,The Scottish Book, Mathematics from the Scottish Café, Birkhäuser, Basel, 1981.
E. Salkowski,Eine kennzeichnende Eigenschaft des Kreises, Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse (1934), 57–62.
K. Zindler,Uber konvexe Gebilde II, Monatshefte für Mathematik und Physik31 (1921), 25–57.
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Partially supported by an NSF grant.
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Tabachnikov, S. Tire track geometry: Variations on a theme. Isr. J. Math. 151, 1–28 (2006). https://doi.org/10.1007/BF02777353
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DOI: https://doi.org/10.1007/BF02777353