Computational geometry: an introduction
Computational geometry: an introduction
Characterizing and efficiently computing quadrangulations of planar point sets
Computer Aided Geometric Design
Converting triangulations to quadrangulations
Computational Geometry: Theory and Applications
Quadrangulations of Planar Sets
WADS '95 Proceedings of the 4th International Workshop on Algorithms and Data Structures
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Let Pn be a set of n points on the plane in general position, n ≥ 4. A convex quadrangulation of Pn is a partitioning of the convex hull Conv(Pn) of Pn into a set of quadrilaterals such that their vertices are elements of Pn, and no element of Pn lies in the interior of any quadrilateral. It is straightforward to see that if P admits a quadrilaterization, its convex hull must have an even number of vertices. In [6] it was proved that if the convex hull of Pn has an even number of points, then by adding at most •n/• Steiner points in the interior of its convex hull, we can always obtain a point set that admits a convex quadrangulation. The authors also show that n/• Steiner points are sometimes necessary. In this paper we show how to improve the upper and lower bounds of [6] to •n/• +2 and to n/• respectively. In fact, in this paper we prove an upper bound of n, and with a long and unenlightening case analysis (over fifty cases!) we can improve the upper bound to •n/• + 2, for details see [9].