Stabbing triangulations by lines in 3D

Let S be a set of (possibly degenerate) triangles in R³ whose interiors are disjoint. A triangulation of R³ with respect to S, denoted by T(S), is a simplicial complex in which each face of T(S) is either disjoint from S or contained in a higher dimensional face of S. The line stabbing number of T(S) is the maximum number of tetrahedra of T(S) intersected by a segment that does not intersect any triangle of S. We investigate the line stabbing number of triangulations in several cases-when S is a set of points, when the triangles of 5 form the boundary of a convex or a nonconvex polyhedron, or when the triangles of S form the boundaries of k disjoint convex polyhedra. We prove almost tight worst-case upper and lower bounds on line stabbing numbers for these cases. We also estimate the number of tetrahedra necessary to guarantee low stabbing number.