Universal convex covering problems under translation and discrete rotations

TL;DR

This paper determines minimal-area convex coverings for planar objects of perimeter 2 under translation and discrete rotations, providing exact solutions for specific rotation sets and conjecturing minimality in some cases.

Contribution

It introduces new geometric solutions for universal coverings under discrete rotations, including exact minimal coverings for certain rotation multiples and conjectures for broader cases.

Findings

The minimal covering for translation and rotation of π is an equilateral triangle of height 1.

Convex coverings for rotations of multiples of π/2 and 2π/3 are provided.

Minimality is proven for the case of rotations of π/2, with conjectures for other cases.

Abstract

We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently closed curves of length 2) allowing translation and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translation and discrete rotation of $\pi$ are allowed. Our proof is purely geometric and elementary. We also give convex coverings of closed curves of length 2 under translation and discrete rotations of multiples of $\pi/2$ and $2\pi/3$. We show a minimality of the covering for discrete rotation of multiples of $\pi/2$, which is an equilateral triangle of height smaller than 1, and conjecture that the covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translation and discrete rotations $2\pi/k$ for all integers $k\ge 3$.