Curves, points, incidences and covering

TL;DR

This paper investigates the minimum number of various types of curves needed to cover a grid of points and explores the inverse problem of point configurations covered by a limited number of lines.

Contribution

It provides bounds on the number of curves required for coverage and analyzes the structure of point sets covered by few lines, advancing understanding of geometric covering problems.

Findings

Established bounds for covering grid points with different curves

Analyzed configurations of points covered by few lines

Connected covering problems with point set structure

Abstract

Given a point set, mostly a grid in our case, we seek upper and lower bounds on the number of curves that are needed to cover the point set. We say a curve covers a point if the curve passes through the point. We consider such coverings by monotonic curves, lines, orthoconvex curves, circles, etc. We also study a problem that is converse of the covering problem -- if a set of $n^2$ points in the plane is covered by $n$ lines then can we say something about the configuration of the points?