The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

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We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with Ω(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have Ω(n bd=2c ) facets, the previously best lower bound for these problems is only Ω(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is ⌈d/2⌉-hard, in the in the sense of Gajentaan and Overmars.

Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the complexity of the Delaunay triangulation of points in may be quadratic in the worst-case, we show in this paper that it is only linear when the points are distributed on a fixed number of well-sampled facets of (e.g. the facets of a polyhedron). Our bound is deterministic and the constants are explicitly given. Categories and Subject Descriptors I.3.5 [Computing Methodologies]: Computational Geometry and

Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearest-neighbor searching, clustering, finite-element mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since three-dimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worst-case running time \Omega (n2). However, this behavior is almost never observed in practice except for highly-contrived inputs. For all practical purposes, three-dimensional Delaunay triangulations appear to have linear complexity. This frustrating

A space-efficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. We describe four space-efficient algorithms for computing the convex hull of a planar point set.

Using a divide, prune, and conquer approach based on geometric partitioning, we obtain: (1) An output sensitive algorithm for computing a weighted Voronoi diagram in R 4 (the projection of certain polyhedra in R 5) that runs in time O((n+f) log 3 f) where n is the number of sites and f is the number of output cells; and (2) a deterministic parallel algorithm in the EREW PRAM model for computing an algebraic planar Voronoi diagram (in which bisectors between sites are simple curves consisting of a constant number of algebraic pieces of constant degree) that runs in time O(log 2 n) using optimal O(n log n) work. The first result implies an algorithm for the problems of computing the convex hull of a point set and the intersection of a set of halfspaces in R 5, and computing the Euclidean Voronoi diagram in R 4. The second result implies both sequential and parallel work-optimal deterministic algorithms for a number of Voronoi diagram problems (including line segments in the plane), and other non-Voronoi diagram problems that can fit in the framework (including the intersection of equal radius balls in R 3 and some lower envelope problems in R 3). 1

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.

Randomized incremental constructions are widely used in computational geometry, but they perform very badly on large data because of their inherently random memory access patterns. We define a biased randomized insertion order which removes enough randomness to significantly improve performance, but leaves enough randomness so that the algorithms remain theoretically optimal.

We consider the problems of computing 2- and 3-d Voronoi diagrams for large data sets efficiently. We describe a cache-oblivious distribution data structure (bu#er tree) that is the basis for the cache oblivious implementation of a random incremental construction for geometric problems. We then apply this to the construction of 2- and 3-d Voronoi diagrams. We also describe a very simple variant of the standard random incremental construction based on history dag, which has optimal running time and is likely to be I/O-efficient because the pattern of insertions is also local (but we don't have theoretical bounds). Finally, we describe a practical variant that has been implemeted and present some experimental results.