80 research outputs found
3D realization of two triangulations of a onvex polygon
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 -orthogonal Latin squares
Two Latin squares of order are -orthogonal if, when superimposed,
there are exactly distinct ordered pairs. The spectrum of all values of
for Latin squares of order is known. A Latin square of order is
-self-orthogonal if and its transpose are -orthogonal. The spectrum
of all values of is known for all orders . We develop randomized
algorithms for computing pairs of -orthogonal Latin squares of order and
algorithms for computing -self-orthogonal Latin squares of order
Edge Routing with Ordered Bundles
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
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