Ray-Shooting on Triangles in 3-Space1

We present a uniform approach to problems involving lines in 3-space. This approach is based on mapping lines in R 3 into points and hyperplanes in 5-dimensional projective space (Pl ucker space). We obtain new results on the following problems: 1. Preprocess n triangles so as to eeciently answer the query: \Given a ray, which is the rst triangle hit?" (Ray-shooting problem). We discuss the ray-shooting problem for both dis-joint and non-disjoint triangles. 2. Construct the intersection of two non-convex polyhedra in an output sensitive way with a subquadratic overhead term. 3. Construct the arrangement of n intersecting triangles in 3-space in an output-sensitive way, with a subquadratic overhead term. 4. EEciently detect the rst face hit by any ray in a set of axis-oriented polyhedra. 5. Preprocess n lines (segments) so as to eeciently answer the query \Given two lines, is it possible to move one into the other without crossing any of the initial lines (segments)?" (Isotopy problem). If the movement is possible produce an explicit representation of it.