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

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)|, depending on 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 > 0.¹ (ii) If no further assumptions are made on the sets of C, then we show that there is a positive integer r that depends only on s such that |R(C)| = O(n^2-1/r). (iii) If C consists of two collections C₁ and C₂ where C₁ is a collection of m 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 C₂ is a collection of polygons with a total of n sides, then |R(C)| = O(m^2/3n^2/3 + m + n), and this bound is tight in the worst case.