### Approximating uniform triangular meshes in polygons

Franz Aurenhammer, Naoki Katoh, Hiromichi Kojima, Makoto Ohsaki, Yinfeng Xu
2002 Theoretical Computer Science
24 Franz Aurenhammer et al. large. Our experiments reveal a much better behaviour, concerning the quality as well as the runtime. With minor modifications, our method works for arbitrary polygons (with possible holes), and yields the same approximation result for Problem 1. Concerning Problems 2 and 3, the approximation factors above can be guaranteed for a restricted class of non-convex polygons. We first develop a heuristic we called canonical Voronoi insertion which approximately solves a
more » ... tain extreme packing problem for point sets within P . The method is similar to the one used in Gonzalez [9] and Feder and Greene [8] developed for clustering problems. We then show how to modify the heuristic, in order to produce a set of n points whose Delaunay triangulation within P constitutes a constant approximation for the problems stated above. Note that the solution we construct is neccessarily a triangulation of constant vertex degree. Generating triangular meshes is one of the fundamental problems in computational geometry, and has been extensively studied; see e.g. the survey article by Bern and Eppstein [3] . Main fields of applications are finite element methods and computer aided design. In finite element methods, for example, it is desirable to generate triangulations that do not have too large or too small angles. Along this direction, various algorithms have been reported [4, 11, 6, 2, 5, 15] . Restricting angles means bounding the edge length ratio for the individual triangles, but not necessarily for a triangulation in global, which might be desirable in some applications. That is, the produced triangulation need not be uniform concerning, e.g., the edge length ratio of its triangles. Chew [6] and Melisseratos and Souvaine [11] construct uniform triangular meshes in the weaker sense that only upper bounds on the triangle size are required. To the knowledge of the authors, the problems dealt with in this paper have not been studied in the field of computational geometry. The mesh refinement algorithms in Chew [6] and in Ruppert [15] are similar in spirit to our Voronoi insertion method, but do not proceed in a canonical way and aim at different optimality criteria. A particular application of length-uniform triangulation arises in designing structures such as plane trusses with triangular units, where it is required to determine the shape from aesthetic points of view under the constraints concerning stress and nodal displacement. The plane truss can be viewed as a triangulation of points in the plane by regarding truss members and nodes as edges and points, respectively. When focusing on the shape, edge lengths should be as equal as possible from the viewpoint of design, mechanics and manufacturing; see [12, 13] . In such applications, the locations of the points are usually not fixed, but can be viewed as decision variables. In view of this application field, it is quite natural to consider Problems 1, 2, and 3. The following notation will be used throughout. For two points x and y in the plane, let l(x, y) denote their Euclidean distance. The minimum (non-zero) distance between two point sets X and Y is defined as l(X, Y ) = min{l(x, y) | x ∈ X, y ∈ Y, x = y}. When X is a singleton set {x} we simply write l(X, Y ) as l(x, Y ). Note that l(X, X) defines the minimum interpoint distance among the point set X.