On lines missing polyhedral sets in 3-space

We <?Pub Caret>study some combinatorial and algorithmic problems onsets of lines and polyhedral objects in 3-space. Our main resultsinclude: <list><item>(1) An <inline-equation><f>O<fen lp="par">n³2^{c<rad><rcd><rf>log</rf>n</rcd></rad>}<rp post="par"></fen></f></inline-equation> upper bound on the worst case complexity of the setof lines missing a star-shaped compact polyhedron with<?Pub Fmt italic>n<?Pub Fmt /italic> edges. </item><item>(2) Given a star-shaped compact polyhedron P with<?Pub Fmt italic>n<?Pub Fmt /italic> edges we can compute on-line the<?Pub Fmt italic>shadow<?Pub Fmt /italic> of P from a query direction<?Pub Fmt italic>v<?Pub Fmt /italic> in almost-optimal output-sensitivetime <?Pub Fmt italic>O(k log<supscrpt>4</supscrpt>n)<?Pub Fmt /italic>, where <?Pub Fmt italic>k<?Pub Fmt /italic> is thesize of the shadow. The storage used by the data structure is<?Pub Fmt italic>O(n<supscrpt>3+ε</supscrpt><?Pub Fmt /italic>. </item><item>(3) An <inline-equation><f>O<fen lp="par">n³2^{c<rad><rcd><rf>log</rf>n</rcd></rad>}<rp post="par"></fen></f></inline-equation> upper bound on the worst case complexity of the setof lines that can be moved to infinity without intersecting a set of<?Pub Fmt italic>n<?Pub Fmt /italic> given lines. This bound is almosttight. </item><item>(4) An<?Pub Fmt italic>O(n<supscrpt>1.5+ε</supscrpt>)<?Pub Fmt /italic>randomized expected time algorithm that tests the separation property:there exists a direction <?Pub Fmt italic>v<?Pub Fmt /italic> alongwhich a set of <?Pub Fmt italic>n<?Pub Fmt /italic> red lines can betranslated away from a set of <?Pub Fmt italic>n<?Pub Fmt /italic> bluelines without collisions? </item><item>(5) Computing the intersection of two polyhedral terrains in3-space with <?Pub Fmt italic>n<?Pub Fmt /italic> edges in time<?Pub Fmt italic>O(n<supscrpt>4/3+ε</supscrpt> +k<supscrpt>1/3</supscrpt>n<supscrpt>1+ε</supscrpt> + klog<supscrpt>2</supscrpt> n)<?Pub Fmt /italic>, where <?Pub Fmt italic>k<?Pub Fmt /italic> is the size of the output, andε > 0 an arbitrary small but fixed constant. This algorithmimproves on the best previous result of Chazelle et al. [7] </item></list>The tools used to obtain these results include Plu¨ckercoordinates of lines, random sampling and polarity transformations in3-space.