Small Strictly Convex Quadrilateral Meshes of Point Sets

TL;DR

This paper investigates the minimum and maximum number of Steiner points needed to create strictly convex quadrilateral meshes for planar point sets, providing bounds and specific constructions.

Contribution

It establishes tight bounds on Steiner points required for convex quadrilateral meshes of planar point sets, including both sufficiency and necessity results.

Findings

3*floor(n/2) Steiner points always suffice

There exist point sets requiring at least ceil((n-3)/2)-1 Steiner points

Bounds are tight for the number of Steiner points needed

Abstract

In this paper, we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that $3{\lfloor\frac{n}{2}\rfloor}$ internal Steiner points are always sufficient for a convex quadrilateral mesh of $n$ points in the plane. Furthermore, for any given $n\geq 4$, there are point sets for which $\lceil\frac{n-3}{2}\rceil-1$ Steiner points are necessary for a convex quadrilateral mesh.