Arrangements of translates of a curve

TL;DR

This paper classifies five types of planar curves whose translates form arrangements equivalent to line arrangements, and explores their applications in defining norms with many unit distances among points.

Contribution

It identifies five specific curve types with arrangements equivalent to lines and links these to constructing norms with numerous unit distances.

Findings

Five types of curves have arrangements equivalent to lines.

These curves can be used to define norms with many unit distances.

Applications in geometric constructions and distance problems.

Abstract

We show that there are five types of planar curves such that arrangements of its translates are combinatorially equivalent to an arrangement of lines. These curves can be used to define norms giving constructions with many unit distances among points in the plane.