Almost Tight Bound for the Union of Fat Tetrahedra in Three Dimensions

Abstract

We show that the combinatorial complexity of the. union of 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2+epsiv</sup> ),for any epsiv > 0: the bound is almost tight in the worst case, thus almost settling a conjecture of Pach el al. [24]. Our result extends, in a significant way, the result of Pach et al. [24] for the restricted case of nearly congruent cubes. The analysis uses cuttings, combined with the Dobkin-K'irkpatrick 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 Delta behave as fat dihedral wedges in Delta. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> having arbitrary side lengths, is O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2+epsiv</sup> ), for any epsiv > 0 again, significantly extending the result of [24]. Our analysis can easily he 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> . Finally, we show that a simple variant of our analysis implies a nearly-linear bound on the complexity of the union of fat triangles in the plane.