On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Combinatorial complexity bounds for arrangements of curves and spheres
Discrete & Computational Geometry - Special issue on the complexity of arrangements
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications
SIAM Journal on Computing
Transversal numbers for hypergraphs arising in geometry
Advances in Applied Mathematics
A Singly-Expenential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications
ICALP '89 Proceedings of the 16th International Colloquium on Automata, Languages and Programming
Topological graphs with no self-intersecting cycle of length 4
Proceedings of the nineteenth annual symposium on Computational geometry
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
On locally Delaunay geometric graphs
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
On levels in arrangements of surfaces in three dimensions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Intersection reverse sequences and geometric applications
Journal of Combinatorial Theory Series A
Similar simplices in a d-dimensional point set
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
On the bichromatic k-set problem
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
On s-intersecting curves and related problems
Proceedings of the twenty-fourth annual symposium on Computational geometry
On levels in arrangements of curves, iii: further improvements
Proceedings of the twenty-fourth annual symposium on Computational geometry
Proceedings of the twenty-fourth annual symposium on Computational geometry
Extremal problems on triangle areas in two and three dimensions
Proceedings of the twenty-fourth annual symposium on Computational geometry
On a question of bourgain about geometric incidences
Combinatorics, Probability and Computing
Extremal problems on triangle areas in two and three dimensions
Journal of Combinatorial Theory Series A
Arrangements in geometry: recent advances and challenges
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Tangencies between families of disjoint regions in the plane
Proceedings of the twenty-sixth annual symposium on Computational geometry
On the bichromatic k-set problem
ACM Transactions on Algorithms (TALG)
Tangencies between families of disjoint regions in the plane
Computational Geometry: Theory and Applications
Intersection reverse sequences and geometric applications
GD'04 Proceedings of the 12th international conference on Graph Drawing
Improved bounds for incidences between points and circles
Proceedings of the twenty-ninth annual symposium on Computational geometry
Exact algorithms and APX-hardness results for geometric packing and covering problems
Computational Geometry: Theory and Applications
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A collection of simple closed Jordan curves in the plane is called a family of pseudo-circles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct pseudo-circles is said to be an empty lens if the closed Jordan region that it bounds does not intersect any other member of the family. We establish a linear upper bound on the number of empty lenses in an arrangement of n pseudo-circles with the property that any two curves intersect precisely twice. We use this bound to show that any collection of n x-monotone pseudo-circles can be cut into O(n8/5) arcs so that any two intersect at most once; this improves a previous bound of O(n5/3) due to Tamaki and Tokuyama. If, in addition, the given collection admits an algebraic representation by three real parameters that satisfies some simple conditions, then the number of cuts can be further reduced to O(n3/2(log n)O(α(s(n))), where α(n) is the inverse Ackermann function, and s is a constant that depends on the the representation of the pseudo-circles. For arbitrary collections of pseudo-circles, any two of which intersect exactly twice, the number of necessary cuts reduces still further to O(n4/3). As applications, we obtain improved bounds for the number of incidences, the complexity of a single level, and the complexity of many faces in arrangements of circles, of pairwise intersecting pseudo-circles, of arbitrary x-monotone pseudo-circles, of parabolas, and of homothetic copies of any fixed simply shaped convex curve. We also obtain a variant of the Gallai--Sylvester theorem for arrangements of pairwise intersecting pseudo-circles, and a new lower bound on the number of distinct distances under any well-behaved norm.