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Title:
Breaking depth-order cycles and other adventures in 3D
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Abstract:
Given a collection of non-vertical lines in general position in 3D, there is a natural above/below relation defined on the lines. One line is “above” another if the unique vertical line that meets both meets the former at a point above the point where it meets the latter. One can similarly define the (partial) above/below relation on any set of reasonably well-behaved pairwise disjoint objects; a pair of objects is not related at all, if no vertical line meets both. Motivated by a computer graphics problem, the following question was asked more than 35 years ago: What is the minimum number of pieces one must cut N lines into, in the worst case, to make sure that the resulting pieces have no cycles in their above/below relation? An N2 upper bound is easy, but is the answer sub-quadratic? An approximately N3/2 lower bound was known, but there were no non-trivial upper bounds, in the general case. Restricted versions of the problem have been studied and will be briefly discussed. We present a near-optimal near-N3/2 upper bound. We also sketch how to extend this to the original motivating question for computer graphics, which until now was unreachable: How many pieces does one have to cut N triangles into, to eliminate all cycles in the above/below relationship, as above? We obtain a near-N3/2 bound in this case as well, though slightly weaker. Joint work with Micha Sharir (Tel Aviv U), and also with Edward Y. Miller (NYU), for the extension to triangles.
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