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We show that the number of incidences between m distinct points and n distinct lines in R4 is O(2{c sqrt{logm}} (m{2/5}n{4/5}+m) + m{1/2}n{1/2}q{1/4} +m{2/3}n{1/3}s{1/3} + n), for a suitable absolute constantc, provided that no 2-plane contains more than s input lines, and no hyper plane or quadric contains more than q lines. The bound holds without the extra factor 2{c sqrt{log m}} when mlen{6/7}...
We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz (2010) [9], to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum possible number of incidences between n lines in R3 and m of their joints (points incident to at least three non-coplanar lines) is Θ(m1/3n) for m⩾n, and Θ(m2/3n2/3+m+n)...
This paper investigates the computational complexity of planning the motion of a body B in 2-D or 3-D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational complexity has not previously been investigated. We provide evidence that the 3-D dynamic movement problem is...
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