We give a complete description of the Voronoi diagram, in R³, of three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. In particular, we show that the topology of the Voronoi diagram is invariant for three such lines. The trisector consists of four unbounded branches of either a non-singular quartic or of a non-singular cubic and a line that do not intersect in real space. Each cell of dimension two consists of two connected components on a hyperbolic paraboloid that are bounded, respectively, by three and one of the branches of the trisector. We introduce a proof technique, which relies heavily upon modern tools of computer algebra, and is of interest in its own right. This characterization yields some fundamental properties of the Voronoi diagram of three lines. In particular, we present linear semi-algebraic tests for separating the two connected components of each two-dimensional Voronoi cell and for separating the four connected components of the trisector. This enables us to answer queries of the form, given a point, determine in which connected component of which cell it lies. We also show that the arcs of the trisector are monotonic in some direction. These properties imply that points on the trisector of three lines can be sorted along each branch using only linear semi-algebraic tests.

We show that the complexity of the Voronoi diagram of a collection of disjoint polyhedra in general position in 3-space that have n vertices overall, under a convex distance function induced by a polyhedron with O(1) facets, is O(n 2+ε), for any ε> 0. We also show that when the sites are n segments in 3-space, this complexity is O(n 2 α(n) log n). This generalizes previous results by Chew et al. [10] and by Aronov and Sharir [4], and solves an open problem put forward by Agarwal and Sharir [2]. Specific distance functions for which our results hold are the L1 and the L ∞ metrics. These results imply that we can preprocess a collection of polyhedra as above into a near-quadratic data structure that can answer δapproximate Euclidean nearest-neighbor queries amidst the polyhedra in time O(log(n/δ)), for an arbitrarily small δ> 0.

Abstract. This paper presents a dynamic algorithm for the construction of the Euclidean Voronoi diagram of a set of convex objects in the plane. We consider first the Voronoi diagram of smooth convex objects forming pseudo-circles set. A pseudo-circles set is a set of bounded objects such that the boundaries of any two objects intersect at most twice. Our algorithm is a randomized dynamic algorithm. It does not use a conflict graph or any sophisticated data structure to perform conflict detection. This feature allows us to handle deletions in a relatively easy way. In the case where objects do not intersect, the randomized complexity of an insertion or deletion can be shown to be respectively O(log 2 n) and O(log 3 n). Our algorithm can easily be adapted to the case of pseudo-circles sets formed by piecewise smooth convex objects. Finally, given any set of convex objects in the plane, we show how to compute the restriction of the Voronoi diagram in the complement of the objects ’ union. 1

In a typical range emptiness searching (resp., reporting) problem, we are given a set P of n points in R d, and wish to preprocess it into a data structure that supports efficient range emptiness (resp., reporting) queries, in which we specify a range σ, which, in general, is a semialgebraic set in R d of constant description complexity, and wish to determine whether P ∩σ = ∅, or to report all the points in P ∩ σ. Range emptiness searching and reporting arise in many applications, and have been treated by Matouˇsek [33] in the special case where the ranges are halfspaces bounded by hyperplanes. As shown in [33], the two problems are closely related, and have solutions (for the case of halfspaces) with similar performance bounds. In this paper we extend the analysis to general semi-algebraic ranges, and show how to adapt Matouˇsek’s technique, without the need to linearize the ranges into a higher-dimensional space. This yields more efficient solutions to several useful problems, and we demonstrate the new technique in four applications, with the following results: (i) An algorithm for ray shooting amid balls in R 3, which uses O(n) storage and O ∗ (n) preprocessing, 1 and answers a query in O ∗ (n 2/3) time, improving the previous bound of O ∗ (n 3/4).

Let C be a set of geometric objects in R d. The combinatorial complexity of the union U(C) of C is the total number of faces of all dimensions, of the arrangement of the boundaries of the objects, which lie on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These problems play a central role in the design and analysis of many geometric algorithms arising in robotics, molecular modeling, solid modeling, and shape matching, and the techniques used for their solutions are interesting in their own right.

Abstract. Given a set S of line segments in the plane, we introduce a new family of partitions of the convex hull of S called segment triangulations of S. The set of faces of such a triangulation is a maximal set of disjoint triangles that cut S at, and only at, their vertices. Surprisingly, several properties of point set triangulations extend to segment triangulations. Thus, the number of their faces is an invariant of S. In thesameway,ifS is in general position, there exists a unique segment triangulation of S whose faces are inscribable in circles whose interiors do not intersect S. This triangulation, called segment Delaunay triangulation, is dual to the segment Voronoi diagram. The main result of this paper is that the local optimality which characterizes point set Delaunay triangulations [10] extends to segment Delaunay triangulations. A similar result holds for segment triangulations with same topology as the Delaunay one. 1

This paper presents a dynamic algorithm for the construction of the Euclidean Voronoi diagram of a set of convex objects in the plane. We consider first the Voronoi diagram of smooth convex objects forming pseudocircles set. A pseudo-circles set is a set of bounded objects such that the boundaries of any two objects intersect at most twice. Our algorithm is a randomized dynamic algorithm. It does not use a conflict graph or any sophisticated data structure to perform conflict detection. This feature allows us to handle deletions in a relatively easy way. In the case where objects do not intersect, the randomized complexity of an insertion or deletion can be shown to be respectively O(log 2 n) and O(log 3 n). Our algorithm can easily be adapted to the case of pseudo-circles sets formed by piecewise smooth convex objects. Finally, given any set of convex objects in the plane, we show how to compute the restriction of the Voronoi diagram in the complement of the objects ’ union. 1

Abstract: This paper presents a dynamic algorithm for the construction of the Euclidean Voronoi diagram of a set of convex objects in the plane. We consider first the Voronoi diagram of smooth convex objects forming pseudo-circles set. A pseudo-circles set is a set of bounded objects such that the boundaries of any two objects intersect at most twice. Our algorithm is a randomized dynamic algorithm. It does not use a conflict graph or any sophisticated data structure to perform conflict detection. This feature allows us to handle deletions in a relatively easy way. In the case where objects do not intersect, the randomized complexity of an insertion or deletion can be shown to be respectively and. Our algorithm can easily be adapted to the case of pseudocircles sets formed by piecewise smooth convex objects. Finally, given any set of convex objects in the plane, we show how to compute the restriction of the Voronoi diagram in the complement of the objects ’ union. Key-words: Vorono diagram; Delaunay triangulation; Euclidean distance; abstract Voronoi diagram;