Small Strictly Convex Quadrilateral Meshes of Point Sets

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.