Vertical decomposition of a single cell in a three-dimensional arrangement of surfaces

Abstract

Let Sigma be a collection of n algebraic surface patches in R-3 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 A(Sigma) is O(n(2+epsilon)), for any epsilon > 0, where the constant of proportionality depends on epsilon 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.