Lines in Space: Combinatorics and Algorithms

Raising Roofs, Crashing Cycles, and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions

“…The relative orientation of two oriented lines λ and is defined to be the orientation of any simplex abcd, where λ passes through the points a and b, and through c and d, in that order. Equivalently, the relative orientation is given by the inner product of the Plücker coordinates of the two lines [22], [59].…”

“…Finally, for the last level, we only need to know whether a query line 4 lies entirely above a set of lines. Chazelle et al [22] describe a data structure that supports such queries in time O(n 1+ε /s 1/2 ).…”

An Improved Bound for Joints in Arrangements of Lines in Space

“…Clearly, this count is a (probably gross) overestimate of the number of joints under consideration. 5 We construct a (1/r )-cutting τ of the cross section within of the hyperplanes π , for ∈ L τ , using, as above, a generic triangulation of the arrangement A(R τ ), for an appropriate sample R τ of O(r ) of these hyperplanes. As above, the size of τ is O(r 4 log r ), and we may assume that each of its cells τ contains at most (n/r )/(r 4 log r ) = n/(r 5 log r ) blue Plücker points p , for ∈ L τ , and is crossed by at most (n/(r 4 log r ))/r = n/(r 5 log r ) red Plücker hyperplanes π , for ∈ L τ .…”

“…Two lines , ∈ L meet if and only if p lies on π (and p lies on π ). See [5] for more details on this transformation.…”

“…The proof proceeds by mapping the lines of L into points and/or hyperplanes in projective 5-space, using Plücker coordinates [5]. We then apply a two-stage decomposition process, which partitions the problem into subproblems, using cuttings of arrangements of appropriate subsets of the Plücker hyperplanes.…”