Towards exact geometric computation
Computational Geometry: Theory and Applications - Special issue: computational geometry, theory and applications
Root comparison techniques applied to computing the additively weighted Voronoi diagram
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Transversals to Line Segments in Three-Dimensional Space
Discrete & Computational Geometry
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Algebraic methods and arithmetic filtering for exact predicates on circle arcs
Computational Geometry: Theory and Applications
Lines through segments in 3d space
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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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 Brönnimann 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].