In this paper we study various predicates concerning line transversals to lines and segments in 3D. We compute the degrees of standard methods of evaluating these predicates. The degrees of some of these methods are surprisingly high, which may explain why computing line transversals with finite precision is prone to error. Our results suggest the need to explore alternatives to the standard methods of computing these quantities. is occluded by a triangle. Finally, we study the predi- cate for ordering planes through two fixed points, each containing a third rational point or a line transversal to four segments or lines. This predicate arises in the rotational plane sweep algorithm of Goaoc (9) that com- putes the maximal free segments tangent to four among k convex polyhedra in 3D. This algorithm performs n rotational sweeps of a plane, one about each edge in the scene, which we call the reference edge. All transversals to the reference edge and three other edges are com- puted in one sweep. The events of the sweep correspond to planes that contain a vertex not on the reference line (i.e., the line containing the reference edge) or that con- tain a line transversal to the reference line and three other segments. This algorithm is the asymptotically fastest known for this problem. Our study shows that standard procedures for solv- ing these predicates have high degree. In particular, we show that determining whether a minimal segment transversal to four line segments is occluded by a tri- angle can be evaluated by a degree 90 predicate Also, the predicate for comparing, in a rotational sweep, two planes, each defined by a line transversal, can be evalu- ated by a degree 168 procedure. These very high degrees may help explain why fixed-precision implementations for solving 3D visibility problems are prone to errors when given real-world data.
['Hazel Everett', 'Sylvain Lazard', 'William Lenhart', 'Jeremy Redburn', 'Linqiao Zhang']
Predicates for Line Transversals in 3D