A note on ovals and their evolutoides (Q839705)

The author considers special curves related to ovals. These curves can uniformely be generated by families of secants with respect to an arbitrary given oval. These secants envelop a curve and this curve gives informations about the oval. If the secants connect the points of contact of parallel tangents of the given oval the envelope is called midenvelope. Using such lines of a one-parameter set of oriented lines which intersect the oval isogonally with the constant angle \(\alpha\) the corresponding envelope \(e_{\alpha}\) is called evolutoide. With the help of these envelopes several properties are discussed. In particular the author demonstrates that to any evolutoide of the given oval a global property can be shown which is invariant under regular affine transformations. This is a generalization of a corresponding result about the evolute of a oval given by \textit{O. Giering} [Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 198, No.~1--3, 45--66 (1989; Zbl 0661.52002)].