Euler Transformation of Polyhedral Complexes
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We propose an Euler transformation that transforms a given d-dimensional cell complex K for d=2,3 into a new d-complex K̂ in which every vertex is part of a uniform even number of edges. Hence every vertex in the graph Ĝ that is the 1-skeleton of K̂ has an even degree, which makes Ĝ Eulerian, i.e., it is guaranteed to contain an Eulerian tour. Meshes whose edges admit Eulerian tours are crucial in coverage problems arising in several applications including 3D printing and robotics. For 2-complexes in R^2 (d=2) under mild assumptions (that no two adjacent edges of a 2-cell in K are boundary edges), we show that the Euler transformed 2-complex K̂ has a geometric realization in R^2, and that each vertex in its 1-skeleton has degree 4. We bound the numbers of vertices, edges, and 2-cells in K̂ as small scalar multiples of the corresponding numbers in K. We prove corresponding results for 3-complexes in R^3 under an additional assumption that the degree of a vertex in each 3-cell containing it is 3. In this setting, every vertex in Ĝ is shown to have a degree of 6. We also presents bounds on parameters measuring geometric quality (aspect ratios, minimum edge length, and maximum angle) of K̂ in terms of the corresponding parameters of K (for d=2, 3). Finally, we illustrate a direct application of the proposed Euler transformation in additive manufacturing.
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