On the Union of -Curved Objects

A (not necessarily convex) object C in the plane is -curved for some constant , < 1, if it has constant description complexity, and for each point p on the boundary of C, one can place a disk B whose boundary passes through p, its radius is diam(C) and it is contained in C. We prove that the combinatorial complexity of the boundary of the union of a set C of n -curved objects (e.g., fat ellipses or rounded hearts) is O( s (n) logn), for some constant s. We also describe an e cient dynamic data structure for point location queries for C.