Involutes of polygons of constant width in Minkowski planes

Abstract

Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and −P . Then the polygon U , as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of constant U -width, and we prove that many properties of the smooth case, which is already completely studied, are preserved. The iteration of involutes generates a pair of sequences of polygons of constant width with respect to the Minkowski norm and its dual norm, respectively. We prove that these sequences are converging to symmetric polygons with the same center, which can be regarded as a central point of the polygon P .

Concepts of curvatures in normed planes

“…In [10], Craizer defines the circular curvature inspired by contact geometry (see also [11] for the discrete framework). The idea is to consider a Minkowski circle with second order contact in each point of the curve, and to define the curvature to be the inverse of the curvature radius.…”

“…Among the papers in which these and other related concepts were studied we may refer to [1], [4], [10], [12], [35], and [39]. For the discrete framework, see [11] and [41].…”

“…Guggenheimer [20] works with an osculating anti-circle, obtaining, as we shall see, the isoperimetric curvature. A related discrete framework appears in [11] and [41].…”

Discrete Cycloids from Convex Symmetric Polygons

“…Our goal in this section is to pair up algebraic properties of r ∈ R 2n with geometric properties of M (compare this to the similar analysis done in [3]). We start defining a suitable inner product in L P : Definition 2 Given two radii vectors r and s in L P , we define their P -inner product as r, s P = 2n i=1 r i s i β i Some interesting subspaces of L P are listed below:…”

“…In this paper, we present a discrete evolute transform and then define discrete cycloids as polygonal lines which are homothetic to their double evolutes. Our evolute construction is an alternative to [1] -we follow instead the approach in [3]. By using a suitable representation, we can represent our polygonal lines as vectors in a space we call L P .…”