Computational geometry column 8

Shortest diagonal [CG4] [CG6] I was incorrect in stating that the shortest diagonal can be foun d as an edge of the constrained Delaunay triangulation . Warren Smith (the originator of th e problem) provided the following counterexample . Let p(8) be the point on the unit circl e centered on the origin at angle O . Then form a convex quadrilateral from the points p(180°) , p(—l°), p(O°), and p(l°) . Now push p(0) towards the left along the x-axis by c; call this ne w point p'(0) . Clearly (p(—1), p(l)) is the shortest diagonal of the resulting convex quadrilateral , but it is not a Delaunay edge, since the circumcircle of the triangle (p(180°), p(—1°), p(1°) ) includes a visible vertex p'(0) . However, both Lingas and Smith have shown that the shortest diagonal is either an edge of th e constrained Delaunay triangulation, or a diagonal that satisfies a simple property . Smith' s property is that the diagonal cuts off a single polygon vertex . Clearly the above exampl e satisfies this condition . Lingas considered the more general case of "Voronoi diagrams wit h barriers" [L], and his property is correspondingly more complex . But in either case, the characterizations of shortest diagonals lead to (not without some complications) 0(n log n) algorithms . Achieving o(n log n) remains open .