Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Sharp upper and lower bounds on the length of general Davenport-Schinzel Sequences
Journal of Combinatorial Theory Series A
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
On the general motion-planning problem with two degrees of freedom
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
On the two-dimensional davenport-schinzel problem
Journal of Symbolic Computation
On the sum of squares of cell complexities in hyperplane arrangements
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Arrangements of curves in the plane—topology, combinatorics, and algorithms
Theoretical Computer Science
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
On the zone theorem for hyperplane arrangements
SIAM Journal on Computing
New bounds for lower envelopes in three dimensions, with applications to visibility in terrains
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
On the zone of a surface in a hyperplane arrangement
Discrete & Computational Geometry
Computing a face in an arrangement of line segments and related problems
SIAM Journal on Computing
Computational geometry: a retrospective
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
ACM Computing Surveys (CSUR)
ACM SIGACT News
Aspect-ratio Voronoi diagram and its complexity bounds
Information Processing Letters
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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(n2+&egr;), for any &egr;0, where the constant of proportionality depends on &egr; and on the maximum degree of the given surfaces and of their boundaries. This extends several previous results, almost settles a 7-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 &sgr; (the so-called zone of &sgr; in the arrangement) is O(n2+&egr;), for any &egr;0, where the constant of proportionality depends on &egr; and on the maximum degree of the given surfaces and of their boundaries.