Iteration of involutes of constant width curves in the Minkowski plane
Let \(\gamma\) be a smooth closed convex curve in the plane. A chord connecting two points of \(\gamma\) with parallel tangents is called a \textit{diameter}. The set \(M\) of midpoints of all diameters is called the \textit{area evolute} (AE), and the envelope of the diameters is called the \textit{centre symmetry set} (CSS) of \(\gamma\). The leading idea of this paper is to view the plane as a Minkowski plane, introducing a norm via the symmetrized curve \(\gamma\). In this Minkowski plane, \(\gamma\) becomes a curve of constant width, the CSS becomes the evolute of \(\gamma\), and the AE becomes an involute of the CSS. Further introduced is a closed convex curve \(N\), which is the AE of the involutes of \(M\) with respect to the dual Minkowski norm. Many relations between \(M\) and \(N\), for example with regard to Minkowski curvatures or signed areas, are presented. Among various results, it is proved that the AE is contained in the region bounded by the CSS. By iteration of involutes, a pair of sequences of curves of constant width, in the Minkowski norm or its dual, is constructed, which converge to symmetric curves with the same center.
scientific article; zbMATH DE number 6362511
scientific article