The construction of Voronoi diagrams and Delaunay triangulations finds wide application in many branches of science. Delaunay triangulations have properties which make them more desirable than other triangulations for the same node set. Delaunay has characterized his triangulations by the empty circle property. The partitioning and flipping methods which have been developed for digital construction of Voronoi diagrams and Delaunay triangulations only make indirect use of this property. A novel method of construction is proposed, which is based directly on the empty circle property of Delaunay. The geometry of the steps of the algorithm is simple and can be grasped intuitively. The method can be applied to constrained triangulations, in which a triangulation domain and some of the edges are prescribed. A data structure for triangulations of concave and multiply-connected domains is presented which permits convenient specification of the constraints and the triangulation. The method is readily implemented, efficient and robust.Keywords: Delaunay, Voronoi, empty circle, shortest diagonal, triangulation theorem, partitioning, flipping, constrained, half edge, dihedral cycle Voronoi Diagrams and Delaunay TriangulationsOver centuries, outstanding scientists have considered the following problem: Given a set of sites in a plane, partition the plane into regions in such a way that all points of a region are at least as near to a particular site as to any other site of the set. This problem has arisen in many applications [1] [2] [3]. Johannes Keppler (1571-1639) encountered Voronoi diagrams when he considered the densest packing of spheres, René Descartes (1596-1650) when he investigated the distribution of matter relative to fixed stars, Johann Dirichlet (1805-1859) when he constructed integer lattices, John Snow (1813-1853) when he related the outbreak of cholera in London to the location of water pumps, Georges Voronoi (1868Voronoi ( -1908 when he extended the Dirichlet tessellations to higher dimensions and Boris Delaunay when he generalized Voronoi diagrams and their duals to irregularly spaced sites in ddimensional space. Many additional examples of Voronoi diagrams have been reported, for example in gold mining, crystallography, metallurgy and meteorology. Today diagrams which relate sets of nearest points to sites in a plane are known as Voronoi diagrams, sometimes also as Dirichlet tessellations. Extensions of the nearness concept to weighted tessellations

The effective buckling length of a column in a steel frame depends on the sidesway of the frame. The classification sidesway-no sidesway of a frame depends on all members of the frame and is made on an empirical basis. A change of class leads to large changes in the effective column length, and thus affects the buckling load and the economy of the column design. In order to avoid the uncertainties of the empirical classification, it is proposed to determine the buckling load of the complete frame with a nonlinear analysis. The method is illustrated with an unbraced and a braced frame, which are analyzed for hinged as well as fixed columns at ground floor level. The forces in the columns at buckling of the frames are compared to the buckling loads of the single columns. The design of high-rise steel frames against buckling by sidesway-no sidesway categorization has been compared to the buckling analysis of the frames as a whole with nonlinear models. The results confirm the large differences between the buckling loads of braced and unbraced high-rise frames, which are well known from analytical solutions for simple portal frames.

Stability analysis in civil engineering is traditionally centred on the stability of individual components of a structure, rather than on the stability of the assemblage of structural components. This may be explained by the lack of adequate tools for the stability analysis of complete structures in the past. Recently, the necessity of the development of general rational methods of stability analysis with a model of the complex structure is widely recognized. These methods should reliably predict the overall stability of the structure, the interaction between the components of the structure in providing restraint against instability of individual members, and the local stability of each individual member. Development of such theories and corresponding algorithms require a thorough investigation. The aim of this paper is to investigate the instability of single columns without large deflections by means of the second order structural theory and to study the influence of imperfections on the behaviour of such structural elements.

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