On Lines Missing
نویسنده
چکیده
We show some combinatorial and algorithmic results concerning sets of lines and polyhedral objects in 3-space. Our main results include: (1) An O(n 3 2 c p log n) upper bound on the worst case complexity of the set of lines missing a star-shaped compact polyhedron with n edges, where c is a suitable constant. (2) An O(n 3 2 c p log n) upper bound on the worst case complexity of the set of lines that can be moved to innnity without intersecting a set of n given lines, where c is a suitable constant. This bound is almost tight. (3) An O(n 1:5+) randomized expected time algorithm that tests whether a direction v exists along which a set of n red lines can be translated away from a set of n blue lines without collisions. (4) Computing the intersection of two polyhedral terrains in 3-space with n total edges in time O(n 4=3+ + k 1=3 n 1+ + k log 2 n), where k is the size of the output, and > 0 an arbitrary small but xed constant. This algorithm improves on the best previous result of Chazelle at al. 8]. The tools used to obtain these results include Pl ucker coordinates of lines, random sampling and polarity transformations in 3-space.
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