On the union of κ-round objects

Authors:
B. Aronov;A. Efrat;V. Koltun;Micha Sharir
Affiliations:
Polytechnic University, Brooklyn, NY;University of Arizona, Tucson, AZ;University of California, Berkeley, CA;Tel Aviv University and Courant Institute of Mathematical Sciences, Tel Aviv, Israel and New York University, New York, NY
Venue:
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Year:
2004

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Abstract

A compact body c in ℝd is κ-round if for every point p∈ ∂c there exists a closed ball that contains p, is contained in c, and has radius κ diam c. We show that, for any fixed κ0, the combinatorial complexity of the union of n κ-round, not necessarily convex objects in ℝ3 (resp., in ℝ4) of constant description complexity is O(n2+ε) (resp., O(n3+ε)) for any ε0, where the constant of proportionality depends on ε, κ, and the algebraic complexity of the objects. The bound is almost tight.