On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
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 κ-curved objects
Proceedings of the fourteenth annual symposium on Computational geometry
The complexity of the union of (&agr;, &bgr;)-covered objects
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
On the boundary complexity of the union of fat triangles
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
ACM SIGACT News
On the Complexity of the Union of Geometric Objects
JCDCG '00 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
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A {\em dihedral (trihedral) wedge} is the intersection of two (resp. t hree) half-spaces in $\reals^3$. It is called {\em $\alpha$-fat} if the angle (resp., solid angle) determined by these half-spaces is at least $\alpha0$. If, in addition, the sum of the three face angles of a trihedral wedge is at least $\gamma 4\pi/3$, then it is called {\em $(\gamma,\alpha)$-substantially fat}. We prove that, for any fixed $\gamma4\pi/3, \alpha0$, the combinatorial complexity of the union of $n$ (a) $\alpha$-fat dihedral wedges, (b) $(\gamma,\alpha)$-substantially fat trihedral wedges is at most $O(n^{2+\eps})$, for any $\eps0$, where the constants of proportionality depend on $\eps$, $\alpha$ (and $\gamma$).We obtain as a corollary that the same upper bound holds for the combinatorial complexity of the union of $n$ (nearly) congruent cubes in $\reals^3$. These bounds are not far from being optimal.