On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Cutting hyperplane arrangements
Discrete & Computational Geometry
Fat triangles determine linearly many holes
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
On the union of fat wedges and separating a collection of segments by a line
Computational Geometry: Theory and Applications
Fat Triangles Determine Linearly Many Holes
SIAM Journal on Computing
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
The Union of Convex Polyhedra in Three Dimensions
SIAM Journal on Computing
On Translational Motion Planning of a Convex Polyhedron in 3-Space
SIAM Journal on Computing
On the union of k-curved objects
Computational Geometry: Theory and Applications
On the Boundary Complexity of the Union of Fat Triangles
SIAM Journal on Computing
Determining the Separation of Preprocessed Polyhedra - A Unified Approach
ICALP '90 Proceedings of the 17th International Colloquium on Automata, Languages and Programming
On the Complexity of the Union of Geometric Objects
JCDCG '00 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
The Complexity of the Union of $(\alpha,\beta)$-Covered Objects
SIAM Journal on Computing
SIAM Journal on Computing
On the Union of κ-Round Objects in Three and Four Dimensions
Discrete & Computational Geometry
Improved Bounds on the Union Complexity of Fat Objects
Discrete & Computational Geometry
On the Union of Cylinders in Three Dimensions
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Counting and representing intersections among triangles in three dimensions
Computational Geometry: Theory and Applications
Improved bound for the union of fat triangles
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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 O(n2+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 dihedral wedges in Δ. As an immediate corollary, we obtain that the combinatorial complexity of the union of n cubes in R3, having arbitrary side lengths, is O(n2+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 R3.