80 research outputs found

    3D realization of two triangulations of a onvex polygon

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    We study the problem of construction of a convex 3-polytope whose (i) shadow boundary has n vertices and (ii) two hulls, upper and lower, are isomorphic to two given triangulations of a convex n-gon. Barnette [℄ D. W. Barnette. Projections of 3-polytopes. Israel J. Math., 8:304{308, 1970] proved the existence of a convex 3-polytope in general case. We show that, in our case, a polytope can be constructed using an operation of edge creation

    Computing random rr-orthogonal Latin squares

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    Two Latin squares of order nn are rr-orthogonal if, when superimposed, there are exactly rr distinct ordered pairs. The spectrum of all values of rr for Latin squares of order nn is known. A Latin square AA of order nn is rr-self-orthogonal if AA and its transpose are rr-orthogonal. The spectrum of all values of rr is known for all orders n14n\ne 14. We develop randomized algorithms for computing pairs of rr-orthogonal Latin squares of order nn and algorithms for computing rr-self-orthogonal Latin squares of order nn

    Edge Routing with Ordered Bundles

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    Edge bundling reduces the visual clutter in a drawing of a graph by uniting the edges into bundles. We propose a method of edge bundling drawing each edge of a bundle separately as in metro-maps and call our method ordered bundles. To produce aesthetically looking edge routes it minimizes a cost function on the edges. The cost function depends on the ink, required to draw the edges, the edge lengths, widths and separations. The cost also penalizes for too many edges passing through narrow channels by using the constrained Delaunay triangulation. The method avoids unnecessary edge-node and edge-edge crossings. To draw edges with the minimal number of crossings and separately within the same bundle we develop an efficient algorithm solving a variant of the metro-line crossing minimization problem. In general, the method creates clear and smooth edge routes giving an overview of the global graph structure, while still drawing each edge separately and thus enabling local analysis