### On the union of fat tetrahedra in three dimensionsOn the union of fat tetrahedra in three dimensions

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 Author Ezra, Esther ♦ Sharir, Micha Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2009 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword (1/r)-cuttings ♦ Union of simply-shaped bodies ♦ Curve-sensitive cuttings ♦ Hierarchical decomposition of convex polytopes Abstract We show that the combinatorial complexity of the union of $\textit{n}$ “fat” tetrahedra in 3-space (i.e., tetrahedra all of whose solid angles are at least some fixed constant) of arbitrary sizes, is $O(n^{2+ϵ}),$ for any ϵ > 0;the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et al. [2003]. Our result extends, in a significant way, the result of Pach et al. [2003] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combined with the Dobkin-Kirkpatrick hierarchical decomposition of convex polytopes, in order to partition space into subcells, so that, on average, the overwhelming majority of the tetrahedra intersecting a subcell Δ behave as fat $\textit{dihedral}$ wedges in Δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of $\textit{n}$ cubes in $R^{3},$ having arbitrary side lengths, is $O(n^{2+ϵ}),$ for any ϵ > 0 (again, significantly extending the result of Pach et al. [2003]). Finally, our analysis can easily be extended to yield a nearly quadratic bound on the complexity of the union of arbitrarily oriented fat triangular prisms (whose cross-sections have arbitrary sizes) in $R^{3}.$ ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2009-11-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 57 Issue Number 1 Page Count 23 Starting Page 1 Ending Page 23

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Source: ACM Digital Library