Summary: On Lines Avoiding Unit Balls in Three Dimensions #
Pankaj K. Agarwal + Boris Aronov # Vladlen Koltun § Micha Sharir ¶
December 10, 2003
Let B be a set of n unit balls in R 3 . We show that the combinatorial complexity of the space of lines in R 3 that
miss all the balls of B is O(n 3+e ), for any e > 0. This result has connections to problems in visibility, ray shooting,
motion planning, and geometric optimization.
# Work on this paper has been supported by a joint grant from the U.S.--Israel Binational Science Foundation. Work by Pankaj Agarwal was also
supported by NSF under grants CCR0086013 EIA9870724, EIA9972879, EIA0131905, and CCR0204118. Work by Boris Aronov was
also supported by NSF Grants CCR9972568 and ITR CCR0081964. Work by Vladlen Koltun was also supported by NSF Grant CCR0121555.
Work by Micha Sharir was also supported by NSF Grants CCR9732101 and CCR0098246, by a grant from the Israeli Academy of Sciences for
a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski--MINERVA Center for Geometry at Tel
+ Department of Computer Science, Duke University, Durham, NC 277080129, USA; firstname.lastname@example.org.
# Department of Computer and Information Science, Polytechnic University, Brooklyn, NY 112013840, USA; email@example.com.
§ Computer Science Division, University of California, Berkeley, CA 947201776, USA; firstname.lastname@example.org.
¶ School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and Courant Institute of Mathematical Sciences, New York University,
New York, NY 10012, USA; email@example.com.