Predicates for line transversals to lines and line segments in three-dimensional space

When an observer is in a 3D scene, a topological change in the view arises when the line of sight is tangent to four objects. If we consider polyhedral scenes, the relevant lines of sight are transversals to some edges of the polyhedra. In this paper we investigate predicates about visibility events arising in this context. Namely, we consider the predicates for counting the number of line transversals to lines and segments in 3D and the predicate for determining whether a line of sight is intersected by a triangle. We also consider a predicate that order these visibility events in the rotating plane-sweep algorithm of Bronnimann et al. (2007) We present a new approach for solving these predicates and show that the degree of the resulting procedures are significantly smaller than the naive approach based on Plücker coordinates. All the degrees are considered here in the Cartesian coordinates of the points defining the lines and segments. In particular, we present a procedure of degree 12 (resp. 15) for determining the number of transversals to four (resp. five or more) segments. We present procedures of degree 15 for the occlusion predicate and of degree 36 for the ordering predicate. In comparison, the degree of the standard procedure based on the Plücker coordinates for solving these predicates range from 36 to 168 [Everett et al. 2006].