Almost Tight Upper Bounds for the Single Cell and Zone Problems in Three Dimensions

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+"), for any " > 0, where the constant of proportionality depends on " 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 (the so-called zone of in the arrangement) is O(n2+" ), for any " > 0, where the constant of proportionality depends on " and on the maximum degree of the given surfaces and of their boundaries.

• Type: Conference
• Year: 1994
• Where: COMPGEOM
• Authors: Dan Halperin, Micha Sharir
• Keywords: Algebraic Surface Patches, COMPGEOM 1994, Discrete Geometry, Low-degree Algebraic Surface, Maximum Degree