Abstract.
Let Σ be a collection of n algebraic surface patches in \({\Bbb R}^3\) 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 \({\cal A}(\Sigma)\) 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.
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Received May 30, 1996, and in revised form February 18, 1997.
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Schwarzkopf, O., Sharir, M. Vertical Decomposition of a Single Cell in a Three-Dimensional Arrangement of Surfaces . Discrete Comput Geom 18, 269–288 (1997). https://doi.org/10.1007/PL00009319
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DOI: https://doi.org/10.1007/PL00009319