TY - JOUR

T1 - Almost tight upper bounds for the single cell and zone problems in three dimensions

AU - Halperin, D.

AU - Sharif, M.

PY - 1995/12

Y1 - 1995/12

N2 - 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(n 2+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 of n low-degree algebraic surface patches, intersected by an additional low-degree algebraic surface patch σ (the so-called zone of σ in the arrangement) is O(n 2+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.

AB - 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(n 2+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 of n low-degree algebraic surface patches, intersected by an additional low-degree algebraic surface patch σ (the so-called zone of σ in the arrangement) is O(n 2+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.

UR - http://www.scopus.com/inward/record.url?scp=21844493276&partnerID=8YFLogxK

U2 - 10.1007/BF02570714

DO - 10.1007/BF02570714

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AN - SCOPUS:21844493276

SN - 0179-5376

VL - 14

SP - 385

EP - 410

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 1

ER -