# Almost tight upper bounds for the single cell and zone problems in three dimensions

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DOI: 10.1007/BF02570714

- Cite this article as:
- Halperin, D. & Sharif, M. Discrete & Computational Geometry (1995) 14: 385. doi:10.1007/BF02570714

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## Abstract

We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement of*n* low-degree algebraic surface patches in 3-space. We show that this complexity is*O*(*n*^{2+ε}), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries. This extends several previous results, almost settles a 9-year-old open problem, and has applications to motion planning of general robot systems with three degrees of freedom. As a corollary of the above result, we show that the overall complexity of all the three-dimensional cells of an arrangement of*n* low-degree algebraic surface patches, intersected by an additional low-degree algebraic surface patch σ (the so-called*zone* of σ in the arrangement) is*O*(*n*^{2+ε}), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries.