A Note on Ovals and their Evolutoides

Marco Hamann

Abstract: In this paper we consider curves related to ovals, which can uniformly be generated by certain families of secants with respect to an arbitrary given oval. If e.\,g. the generating secants connect the points of contact of parallel tangents of the given oval then they envelop a curve, which contains information about the oval analogously to its evolute for example. Those curves are treated in different context in [G] and \cite{midpoint} too. In this article several properties can be treated into detail and extended respectively. In particular with help of this curve a global property to any evolutoide of the given oval can be shown, which is invariant under (regular) affine transformations. It generalizes a corresponding result about the evolute of the oval, see [G].

[G] Giering, O.: \textit{Über Eilinien und mit ihnen verknüpfte Mittelpunktkurven}. Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. \textbf{198} (1--3) (1989), 45--66.

Keywords: evolute, evolutoide, kinematics, minimal curve, oval, certain secants

Classification (MSC2000): 53A04, 51M04, 51N20

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