TY - JOUR

T1 - On the union of fat tetrahedra in three dimensions

AU - Ezra, Esther

AU - Sharir, Micha

PY - 2009/11/1

Y1 - 2009/11/1

N2 - We showthat the combinatorial complexity of the union of n "efat"e 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 behave as fat dihedral wedges in δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of 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.

AB - We showthat the combinatorial complexity of the union of n "efat"e 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 behave as fat dihedral wedges in δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of 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.

KW - (1/r )-cuttings

KW - Curve-sensitive cuttings

KW - Hierarchical decomposition of convex polytopes

KW - Union of simply-shaped bodies

UR - http://www.scopus.com/inward/record.url?scp=71449101504&partnerID=8YFLogxK

U2 - 10.1145/1613676.1613678

DO - 10.1145/1613676.1613678

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AN - SCOPUS:71449101504

SN - 0004-5411

VL - 57

JO - Journal of the ACM

JF - Journal of the ACM

IS - 1

M1 - 2

ER -