Vertex-Rounding a Three-Dimensional Polyhedral Subdivision

Abstract. Let P be a polyhedral subdivision in R3 with a total of n faces. We show that there is an embedding σ of the vertices, edges, and facets of P into a subdivision Q , where every vertex coordinate of Q is an integral multiple of $2^{- \lceil \log_2 n + 2 \rceil}$ . For each face f of P , the Hausdorff distance in the L∈fty metric between f and σ(f) is at most 3/2 . The embedding σ preserves or collapses vertical order on faces of P . The subdivision Q has O(n4) vertices in the worst case, and can be computed in the same time.

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