Cutting Circles into Pseudo-Segments and Improved Bounds for Incidences% and Complexity of Many Faces
نویسندگان
چکیده
We show that n arbitrary circles in the plane can be cut into O(n) arcs, for any ε > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree.
منابع مشابه
On the Complexity of Many Faces in Arrangements of Pseudo-Segments and of Circles
We obtain improved bounds on the complexity of m distinct faces in an arrangement of n pseudo-segments, n circles, or n unit circles. The bounds are worst-case optimal for unit circles; they are also worst-case optimal for the case of pseudo-segments, except when the number of faces is very small, in which case our upper bound is a polylogarithmic factor from the best-known lower bound. For gen...
متن کاملCutting Circles into Pseudo-segments and Improved Bounds for Incidences
We show that n arbitrary circles in the plane can be cut into O(n) arcs, for any ε > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant m...
متن کاملImproved Bounds for Incidences and Complexity of Many Faces in Arrangements of Circles and of Polynomial Arcs *
We derive improved upper bounds for the number of incidences between m points and n circles in the plane, and for the complexity of m distinct faces in an arrangement of circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree.
متن کاملOn the Complexity of Many Faces in Arrangements of Circles
We obtain improved bounds on the complexity of m distinct faces in an arrangement of n circles and in an arrangement of n unit circles. The bounds are worst-case tight for unit circles, and, for general circles, they nearly coincide with the best known bounds for the number of incidences between m points and n circles.
متن کاملCutting algebraic curves into pseudo-segments and applications
We show that a set of n algebraic plane curves of constant maximum degree can be cut into O(n3/2 polylog n) Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments. This extends a similar (and slightly better) bound for pseudo-circles due to Marcus and Tardos. Our result is based on a technique of Ellenberg, Solymosi and Zahl that transform...
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عنوان ژورنال:
- Discrete & Computational Geometry
دوره 28 شماره
صفحات -
تاریخ انتشار 2002