Is there an algorithm similar to the Delaunay triangulation which can organize a set of points into a set of four sided polygons instead of triangles?
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I don't know of any quadrangulation having similar properties to a Delaunay triangulation, but you can always convert a triangulation to a quadrangulation by adding additional points, called Steiner points. See the first section of http://www.cs.mcgill.ca/~cs507/projects/1998/rachelp/ for instance.