Arrangements in geometry: recent advances and challenges
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Better bounds on the union complexity of locally fat objects
Proceedings of the twenty-sixth annual symposium on Computational geometry
Tangencies between families of disjoint regions in the plane
Proceedings of the twenty-sixth annual symposium on Computational geometry
The union of colorful simplices spanned by a colored point set
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part I
Tangencies between families of disjoint regions in the plane
Computational Geometry: Theory and Applications
The union of colorful simplices spanned by a colored point set
Computational Geometry: Theory and Applications
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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 {\rm O}(n^{2 + \varepsilon } ) for any \varepsilon\le 0; the bound is almost tight in the worst case, thus almost settling a conjecture of Pach et 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-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 \vartriangle behave as fat dihedral wedges in \vartriangle. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in \mathbb{R}^3, having arbitrary side lengths, is {\rm O}(n^{2 + \varepsilon } ), for any \varepsilon\le 0 (again, significantly extending the result of [24]). 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 \mathbb{R}^3. 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.