TY - GEN

T1 - Lines avoiding balls in three dimensions revisited

AU - Rubin, Natan

PY - 2010/7/30

Y1 - 2010/7/30

N2 - Let ℬ be a collection of n arbitrary balls in ℝ3. We establish an almost-tight upper bound of O(n3+ε), for any ε > 0, on the complexity of the space ℱ(ℬ) of all the lines that avoid all the members of ℬ. In particular, we prove that the balls of ℬ admit O(n3+ε) free isolated tangents, for any ε > 0. This generalizes the result of Agarwal et al. [1], who established this bound only for congruent balls, and solves an open problem posed in that paper. Our bound almost meets the recent lower bound of Ω(n3) of Glisse and Lazard [6]. Our approach is constructive and yields an algorithm that computes a discrete representation of the boundary of ℱ(ℬ) in O(n3+ε) time, for any ε > 0.

AB - Let ℬ be a collection of n arbitrary balls in ℝ3. We establish an almost-tight upper bound of O(n3+ε), for any ε > 0, on the complexity of the space ℱ(ℬ) of all the lines that avoid all the members of ℬ. In particular, we prove that the balls of ℬ admit O(n3+ε) free isolated tangents, for any ε > 0. This generalizes the result of Agarwal et al. [1], who established this bound only for congruent balls, and solves an open problem posed in that paper. Our bound almost meets the recent lower bound of Ω(n3) of Glisse and Lazard [6]. Our approach is constructive and yields an algorithm that computes a discrete representation of the boundary of ℱ(ℬ) in O(n3+ε) time, for any ε > 0.

KW - Arrangements

KW - Combinatorial complexity

KW - Free lines

KW - Lines in space

KW - Tangency surfaces

UR - http://www.scopus.com/inward/record.url?scp=77954897308&partnerID=8YFLogxK

U2 - 10.1145/1810959.1810970

DO - 10.1145/1810959.1810970

M3 - Conference contribution

AN - SCOPUS:77954897308

SN - 9781450300162

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 58

EP - 67

BT - Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10

T2 - 26th Annual Symposium on Computational Geometry, SoCG 2010

Y2 - 13 June 2010 through 16 June 2010

ER -