On the number of regular vertices of the union of Jordan regions

Abstract

Let C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in C cross exactly twice, then their intersection points are called regular vertices of the arrangement A(C). Let R(C) denote the set of regular vertices on the boundary of the union of C. We present several bounds on ¦R(C)¦, determined by the type of the sets of C. (i) If each set of C is convex, then ¦R(C)¦ = O(n ^1.5+ε) for any ε > 0. (ii) If C consists of two collections C ₁ and C ₂ where C ₁ is a collection of n convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and C ₂ is a collection of polygons with a total of n sides, then ¦R(C)¦=O(n ^4/3), and this bound is tight in the worst case. (iii) If no further assumptions are made on the sets of C, then we show that there is a positive integer t that depends only on s such that ¦R(C)¦=O(n ^2−1/t).

The first and the fourth authors have been supported by a grant from the U.S.-Israeli Binational Science Foundation. Boris Aronov has also been supported by a Sloan Research Fellowship. Micha Sharir has also been supported by NSF Grants CCR-94-24398 and CCR-93-11127, and by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Dan Halperin has been supported by an Alon Fellowship, by ESPRIT IV LTR Project No. 21957 (CGAL), by the USA-Israel Binational Science Foundation, and by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Dan Halperin and Micha Sharir have also been supported by the Hermann Minkowski — Minerva Center for Geometry at Tel Aviv University. Part of the work on this paper has been done when Boris Aronov was visiting Tel Aviv University in March 1997.