Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3-space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.

This paper presents how the space of spheres and shelling may be used to delete a point from a d-dimensional triangulation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O(k log k), but we notice that this number only applies to low cost operations, while time consuming computations are only done a linear number of times. This algorithm may be viewed as a variation of Heller’s algorithm,[1, 2] which is popular in the geographic information system community. Unfortunately, Heller algorithm is false, as explained in this paper.

This dissertation explores some techniques for automatic approximation of geometric objects. My thesis is that using and extending concepts from computational geometry can help us in devising efficient and parallelizable algorithms for automatically constructing useful detail hierarchies for geometric objects. We have demonstrated this by developing new algorithms for two kinds of geometric approximation problems that have been motivated by a single driving problem --- the efficient computation and display of smooth solvent-accessible molecular surfaces. The applications of these detail hierarchies are in biochemistry and computer graphics. The smooth solvent-accessible surface of a molecule is useful in studying the structure and interactions of proteins, in particular for attacking the protein-substrate docking problem. We have developed a parallel linear-time algorithm for computing molecular surfaces. Molecular surfaces are equivalent to the weighted ff-hulls. Thus our work is pot...

Treemaps are a well-known method for the visualization of attributed hierarchical data. Previously proposed Treemap layout algorithms are limited to rectangular shapes, which causes problems with the aspect ratio of the rectangles as well as with identifying the visualized hierarchical structure. The approach of Voronoi Treemaps presented in this paper eliminates these problems through enabling subdivisions of and in polygons. Additionally, this allows for creating Treemap visualizations within areas of arbitrary shape, such as triangles and circles, thereby enabling a more flexible adaptation of Treemaps for a wider range of applications.

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We show that the combinatorial complexity of the Euclidean Voronoi diagram of n lines in R 3 that have at most c distinct orientations is O(c 3 n 2+ε), for any ε> 0. This result is a step towards proving the long-standing conjecture that the Euclidean Voronoi diagram of lines in three dimensions has near-quadratic complexity. It provides the first natural instance in which this conjecture is shown to hold. In a broader context, our result adds a natural instance to the (rather small) pool of instances of general 3-dimensional Voronoi diagrams for which near-quadratic complexity bounds are known. 1

Understanding the process of categorization is a primary research goal in artificial intelligence. The conceptual space framework provides a flexible approach to modeling context-sensitive categorization via a geometrical representation designed for modeling and managing concepts. In this paper we show how algorithms developed in computational geometry, and the Region Connection Calculus can be used to model important aspects of categorization in conceptual spaces. In particular, we demonstrate the feasibility of using existing geometric algorithms to build and manage categories in conceptual spaces, and we show how the Region Connection Calculus can be used to reason about categories and other conceptual regions. 1

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

An algorithm for rapid computation of Richards's smooth molecular surface is described. The entire surface is computed analytically, triangulated, and displayed at interactive rates. The faster speeds for our program have been achieved by algorithmic improvements, parallelizing the computations, and by taking advantage of the special geometrical properties of such surfaces. Our algorithm is easily parallelizable and it has a time complexityofO#klog k#over n processors, where n is the number of atoms of the molecule and k is the average number of neighbors per atom. 1 Introduction The smooth molecular surface of a molecule is de#ned as the surface which an exterior probe-sphere touches as it is rolled over the spherical atoms of that molecule. This de#nition of a molecular surface was #rst proposed by Richards #16#. This surface is useful in studying the structure and interactions of proteins, in particular for attacking the protein-substrate docking problem. An example of suc...

Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have “nicely shaped” triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions. The main contributions are rigorous, theory-tested definitions of constrained Delaunay triangulations and piecewise linear complexes (geometric domains that incorporate nonconvex faces with “internal ” boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.