 
Summary: ALGORITHMS FOR LINE TRANSVERSALS IN SPACE
Preliminary Report
David Avis
Rephael Wenger
School of Computer Science
McGill University
805 Sherbrooke St. W.
Montreal, Canada, H3A 2K6
ABSTRACT
Algorithms are developed for determining if a set of polyhedral
objects in R 3 can be intersected by a common transversal (stabbing) line.
It can be determined in O(n) time if a set of n lines in space has a line
transversal, and such a transversal can be found in the same time bound.
For a set of n line segments, the complexity of finding such a transversal
becomes O(nlogn). Finally, for a set of polyhedra with a total of n ver
tices, we give a O(n 5 ) algorithm for determining the existence of, and
computing, a line transversal. Hellytype theorems for lines and segments
are also given. In particular, it is shown that if every six of a set of lines in
space are intersected by a common transversal, then the entire set has a
common transversal.
