 
Summary: On the Complexity of Many Faces in Arrangements of Circles #
Pankaj K. Agarwal + Boris Aronov # Micha Sharir §
Abstract
We obtain improved bounds on the complexity of m dis
tinct faces in an arrangement of n circles and in an arrange
ment of n unit circles. The bounds are worstcase 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.
1 Introduction
Problem statement and motivation. The arrangement
A(#) of a finite collection # of curves or surfaces in R d is
the decomposition of the space into relatively open con
nected cells of dimensions 0, . . . , d induced by #, where
each cell is a maximal connected set of points lying in the
intersection of a fixed subset of # and avoiding all other el
ements of #. The combinatorial complexity (or complexity
for short) of a cell # in A(#), denoted as #, is the number
of faces of A(#) of all dimensions that lie on the boundary
of #. Besides being interesting in their own right, due to
