Vertical Decomposition of a Single Cell in a Three-Dimensional Arrangement of Surfaces

Let Σ be a collection of n algebraic surface patches in of constant maximum degree b, such that the boundary of each surface consists of a constant number of algebraic arcs, each of degree at most b as well. We show that the combinatorial complexity of the vertical decomposition of a single cell in the arrangement is O(n^{2+ɛ}), for any ɛ > 0, where the constant of proportionality depends on ɛ and on the maximum degree of the surfaces and of their boundaries. As an application, we obtain a near-quadratic motion-planning algorithm for general systems with three degrees of freedom. Received May 30, 1996, and in revised form February 18, 1997.