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Proceedings of GO 2005, pp. 263 – 267.Optimal Triangulation: Old and New Problems ∗Yinfeng Xu 1 and Wenqiang Dai 21 School of Management, Xi’an Jiaotong University, Xi’an, 710049, P. R. China,The State Key Lab for Manufacturing Systems Engineering, P. R. China, yfxu@mail.xjtu.edu.cn2 School of Management, Xi’an Jiaotong University, Xi’an, 710049, P. R. China, wqdai@mail.xjtu.edu.cnAbstractKeywords:The optimal triangulation is an important topic in computational geometry and arises in many differentfields. This paper surveys several unsolved problems in optimal triangulations and especiallyfocus on some new problems which need further research.minimum weight triangulation, Delaunay triangulation, k−optimality, β−skeleton, computationalcomplexity1. IntroductionA triangulation of a given points set S in the plane is a maximal set of non-crossing line segments(called edges) which have both endpoints in S. In two dimensions, a triangulation partitionsthe interior of the convex hull of the given point set into triangles and these trianglesinterest only at shared edges and vertices. There are many areas of engineering and scientificapplications for triangulation, such as finite element, numerical computation, computer aideddesign(CAD), computational geometry, etc [4, 7].From the view of application, it is important to confine geometric constraints on the shapeof triangle in obtained triangulation. Several measures of triangle quality have been proposed,which are based on edges length, angles, areas and other elements of the individual trianglesin a triangulation. An optimal triangulation is the one that is best according to some criterionof these measures.More recently, there have already some surveys on the optimal triangulation, such as [5, 5,8, 20, 22]. This paper also deals with this topic, but our focus is the set of planar points withthe Euclidean metric in E 2 , which is the most important case for applications. The emphasisof our work is on problems of optimal triangulation. The goal is to survey some unsolvedproblems and propose some new problems which need further research.2. Optimal Triangulation2.1 Delaunay TriangulationThe Delaunay triangulation of a point set S, DT (S), is the planar dual of the Voronoi diagram,V (S). The Voronoi diagram is a partition of the plane into polygonal cells, one for each inputpoint, so that the cell for input point s consists of the region of the plane closer to s thanto any other input point. See [4, 13, 18] for extensive discussion and surveys. Various global∗ This research is supported by NSF of China under Grants No. 10371094 and 70471035.