Data structures and algorithms 3: multi-dimensional searching and computational geometry
Data structures and algorithms 3: multi-dimensional searching and computational geometry
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
The complexity and construction of many faces in arrangements of lines and of segments
Discrete & Computational Geometry - Special issue on the complexity of arrangements
Partitioning arrangements of lines, part II: applications
Discrete & Computational Geometry
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Cutting hyperplanes for divide-and-conquer
Discrete & Computational Geometry
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Cutting cylces of rods in space
Proceedings of the fourteenth annual symposium on Computational geometry
Handbook of discrete and computational geometry
Constructing Planar Cuttings in Theory and Practice
SIAM Journal on Computing
On levels in arrangements of curves
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
On the Complexity of Many Faces in Arrangements of Circles
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
On s-intersecting curves and related problems
Proceedings of the twenty-fourth annual symposium on Computational geometry
Proceedings of the twenty-fourth annual symposium on Computational geometry
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A collection L of n x-monotone unbounded Jordan curves in the plane is called a family of pseudo-lines if every pair of curves intersect in at most one point, and the two curves cross each other there. Let P be a set of m points in R2. We define a duality transform that maps L to a set L* of points in R2 and P to a set P* of pseudo-lines in R2, so that the incidence and the "above-below" relationships between the points and pseudo-lines are preserved. We present an efficient algorithm for computing the dual arrangement A (P* under an appropriate model of computation. We also propose a dynamic data structure for reporting, in O(mε + k) time, all k points of P that lie below a query arc, which is either a circular arc or a portion of the graph of a polynomial of fixed degree. This result is needed for computing the dual arrangement for certain classes of pseudo-lines arising in our applications, but is also interesting in its own right. We present a few applications of our dual arrangement algorithm, such as computing incidences between points and pseudo-lines and computing a subset of faces in a pseudo-line arrangement.Next, we present an efficient algorithm for cutting a set of circles into arcs so that every pair of arcs intersect in at most one point, i.e., the resulting arcs constitute a collection of pseudo-segments. By combining this algorithm with our algorithm for computing the dual arrangement of pseudo-lines, we obtain efficient algorithms for a number of problems involving arrangements of circles or circular arcs, such as detecting, counting, or reporting incidences between points and circles.