Abstract
Cycloids, hypocycloids and epicycloids have an often forgotten common property: they are homothetic to their evolutes. But what if we use convex symmetric polygons as unit balls, can we define evolutes and cycloids which are genuinely discrete? Indeed, we can! We define discrete cycloids as eigenvectors of a discrete double evolute transform which can be seen as a linear operator on a vector space we call curvature radius space. We are also able to classify such cycloids according to the eigenvalues of that transform, and show that the number of cusps of each cycloid is well determined by the ordering of those eigenvalues. As an elegant application, we easily establish a version of the four-vertex theorem for closed convex polygons. The whole theory is developed using only linear algebra, and concrete examples are given.
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Notes
Choosing a convex symmetric unit ball is the same as choosing a norm in the plane, so we are in the realms of Minkowski Geometry [7].
Ok, there is no physical hearing in this context, but we wanted to cite [4].
In fact, all the theory could be done if P were any locally convex polygon that goes around the origin m times (not necessarily repeating itself at each turn), but then the geometric interpretation of P as a unit ball is somewhat diminished. Three articles for the price of one!
References
Arnold, M., Fuchs, D., Izmestiev, I., Tabachnikov, S., Tsukerman, E.: Iterating evolutes and involutes. Discrete Comput. Geom. 58(1), 80–143 (2017)
Craizer, M., Martini, H.: Involutes of polygons of constant width in Minkowski planes. Ars Math. Contemp. 11(1), 107–125 (2016)
Craizer, M., Teixeira, R., Balestro V.: Closed cycloids in a normed plane. Monatsh. Math. https://doi.org/10.1007/s00605-017-1030-5
Kac, M.: Can one hear the shape of a drum? Am. Math. Mon. 73(4), 1–23 (1966)
Schneider, R.: The middle hedgehog of a planar convex body. Beitr. Algebra Geom. 58(2), 235–245 (2017)
Tabachnikov, S.: A four vertex theorem for polygons. Am. Math. Mon. 107(9), 830–833 (2000)
Thompson, A.C.: Minkowski Geometry. Encyclopedia of Mathematics and Its Applications, vol. 63. Cambridge University Press, Cambridge (1996)
Wang, Y., Shi, Y.: Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions. J. Math. Anal. Appl. 309(1), 56–69 (2005)
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Craizer, M., Teixeira, R. & Balestro, V. Discrete Cycloids from Convex Symmetric Polygons. Discrete Comput Geom 60, 859–884 (2018). https://doi.org/10.1007/s00454-017-9955-y
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DOI: https://doi.org/10.1007/s00454-017-9955-y