Almost tight upper bounds for the single cell and zone problems in three dimensions
- 40 Downloads
We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement ofn low-degree algebraic surface patches in 3-space. We show that this complexity isO(n2+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 ofn low-degree algebraic surface patches, intersected by an additional low-degree algebraic surface patch σ (the so-calledzone of σ in the arrangement) isO(n2+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.
Unable to display preview. Download preview PDF.
- 6.B. Aronov and M. Sharir, On translational motion planning in 3-space.Proc. 10th ACM Symp. on Computational Geometry, 1994, pp. 21–30.Google Scholar
- 10.M. de Berg, L. J. Guibas, and D. Halperin, Vertical decompositions for triangles in 3-space,Proc. 10th ACM Symp. on Computational Geometry, 1994, pp. 1–10.Google Scholar
- 15.D. Halperin, Algorithmic Motion Planning via Arrangements of Curves and of Surfaces, Ph.D. Dissertation, Computer Science Department, Tel Aviv University, July 1992.Google Scholar
- 17.D. Halperin and M. Sharir, Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment.Proc. 34th IEEE Symp. on Foundations of Computer Science, 1993, pp. 382–391.Google Scholar