On the union of κ-round objects
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Vertical ray shooting for fat objects
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Small-size ε-nets for axis-parallel rectangles and boxes
Proceedings of the forty-first annual ACM symposium on Theory of computing
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On the union of fat tetrahedra in three dimensions
Journal of the ACM (JACM)
Computing the visibility map of fat objects
Computational Geometry: Theory and Applications
Better bounds on the union complexity of locally fat objects
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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
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
SIAM Journal on Computing
Improved bound for the union of fat triangles
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Improved bounds on the union complexity of fat objects
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
Computing the visibility map of fat objects
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
The union of colorful simplices spanned by a colored point set
Computational Geometry: Theory and Applications
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A triangle is said to be {\it $\delta$-fat\/} if its smallest angle is at least $\delta0$. A connected component of the complement of the union of a family of triangles is called a {\it hole}. It is shown that any family of n $\delta$-fat triangles in the plane determines at most $O\left(\frac{n}{\delta}\log\frac{2}{\delta}\right)$ holes. This improves on some earlier bounds of Efrat, Rote, Sharir, and Matousek, et al. Solving a problem of Agarwal and Bern, we also give a general upper bound for the number of holes determined by n triangles in the plane with given angles. As a corollary, we obtain improved upper bounds for the boundary complexity of the union of fat polygons in the plane, which, in turn, leads to better upper bounds for the running times of some known algorithms for motion planning, for finding a separator line for a set of segments, etc.