Iteration of Involutes of Constant Width Curves in the Minkowski Plane

TL;DR

This paper explores properties of convex curves in the Minkowski plane, focusing on evolutes and involutes, and demonstrates convergence of iterative involutes to symmetric curves with a common center.

Contribution

It introduces a Minkowski framework where constant width curves are analyzed via evolutes and involutes, proving convergence to symmetric curves with a shared center.

Findings

AE is contained within the CSS region and has smaller signed area

Iterative involutes generate sequences of constant width curves

Sequences converge to symmetric curves with the same center

Abstract

In this paper we study properties of the area evolute (AE) and the center symmetry set (CSS) of a convex planar curve $\gamma$. The main tool is to define a Minkowski plane where $\gamma$ becomes a constant width curve. In this Minkowski plane, the CSS is the evolute of $\gamma$ and the AE is an involute of the CSS. We prove that the AE is contained in the region bounded by the CSS and has smaller signed area. The iteration of involutes generate a pair of sequences of constant width curves with respect to the Minkowski metric and its dual, respectively. We prove that these sequences are converging to symmetric curves with the same center, which can be regarded as a central point of the curve $\gamma$.