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

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.