Stabbing and ray shooting in 3 dimensional space

Abstract

In this paper we consider the following problems: given a set T of triangles in 3-space, with |T| = n,answer the query “given a line l, does l stab the set of triangles?” (query problem). find whether a stabbing line exists for the set of triangles (existence problem). Given a ray &rgr;, which is the first triangle in T hit by &rgr;? The following results are shown.There is an &OHgr;(n3) lower bound on the descriptive complexity of the set of all stabbers for a set of triangles. The existence problem for triangles on a set of planes with g different plane inclinations can be solved in &Ogr;(g2n2 log n) time (Theorem 2). The query problem is solvable in quasiquadratic &Ogr;(n2+ε) preprocessing and storage and logarithmic &Ogr;(log n) query time (Theorem 4). If we are given m rays we can answer ray shooting queries in &Ogr;(m5/6-δ n5/6+5δ log2 n + m log2 n + n log n log m) randomized expected time and &Ogr;(m + n) space (Theorem 5). In time &Ogr;((n+m)5/3+4δ) it is possible to decide whether two non convex polyhedra of complexity m and n intersect (Corollary 1). Given m rays and n axis-oriented boxes we can answer ray shooting queries in randomized expected time &Ogr;(m3/4-δ n3/4+3δ log4 n + m log4 n + n log n log m) and &Ogr;(m + n) space (Theorem 6).