 
Summary: Cutting Triangular Cycles of Lines in Space #
Boris Aronov + Vladlen Koltun # Micha Sharir §
March 22, 2004
Abstract
We show that n lines in 3space can be cut into O(n 21/69 log 16/69 n) pieces, such
that all depth cycles defined by triples of lines are eliminated. This partially resolves
a longstanding open problem in computational geometry, motivated by hiddensurface
removal in computer graphics.
1 Introduction
The problem. Let L be a collection of n lines in R 3 in general position. In particular, we
assume that no two lines in L intersect and that the xyprojections of no two of the lines
are parallel. For any pair #, # # of lines in L, we say that # passes above # # (equivalently, # #
passes below #) if the unique vertical line # that meets both # and # # intersects # at a point
that lies higher than its intersection with # # . We denote this relation as # # # #. The relation
# is total, but in general it need not be transitive and it can contain cycles of the form
# 1 # # 2 # · · · # # k # # 1 . We refer to k as the length of the cycle. Cycles of length three are
called triangular. See Figure 1(a).
If we cut the lines of L at a finite number of points, we obtain a collection of lines,
segments, and rays. We can extend the definition of the relation # to the new collection
in the obvious manner, except that it is now only a partial relation. Our goal is to cut the
