Drawing the double circle on a grid of minimum size

TL;DR

This paper extends Jarník's grid size bounds from convex polygons to double circles, providing an optimal construction method and an efficient algorithm for drawing these configurations on a minimal grid.

Contribution

It proves that double circles can be drawn within the same minimal grid size as convex polygons and introduces an O(n)-time construction algorithm.

Findings

Double circles can be embedded in a grid of size proportional to n^{3/2}.

An O(n)-time algorithm constructs such point sets efficiently.

The grid size is proven to be optimal up to a constant factor.

Abstract

In 1926, Jarn\'ik introduced the problem of drawing a convex $n$-gon with vertices having integer coordinates. He constructed such a drawing in the grid $[1,c\cdot n^{3/2}]^2$ for some constant $c>0$, and showed that this grid size is optimal up to a constant factor. We consider the analogous problem for drawing the double circle, and prove that it can be done within the same grid size. Moreover, we give an O(n)-time algorithm to construct such a point set.