. We present new algorithms for creating and rendering visual hulls in real-time. Unlike voxel or sampled approaches, we compute an exact polyhedral representation for the visual hull directly from the silhouettes. This representation has a number of advantages: 1) it is a view-independent representation, 2) it is well-suited to rendering with graphics hardware, and 3) it can be computed very quickly. We render these visual hulls with a view-dependent texturing strategy, which takes into account visibility information that is computed during the creation of the visual hull. We demonstrate these algorithms in a system that asynchronously renders dynamically created visual hulls in real-time. Our system outperforms similar systems of comparable computational power. 1

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...

A visibility ordering of a set of objects, from a given viewpoint, is a total order on the objects such that if object a obstructs object b,thenb precedes a in the ordering. Such orderings are extremely useful for rendering volumetric data. We present an algorithm that generates a visibility ordering of the cells of an unstructured mesh, provided that the cells are convex polyhedra and nonintersecting, and that the visibility ordering graph does not contain cycles. The overall mesh may be nonconvex and it may have disconnected components. Our technique employs the sweep paradigm to determine an ordering between pairs of exterior (mesh boundary) cells which can obstruct one another. It then builds on Williams' MPVO algorithm [33] which exploits the ordering implied by adjacencies within the mesh. The partial ordering of the exterior cells found by sweeping is used to augment the DAG created in Phase II of the MPVO algorithm. Our method thus removes the assumption of the MPVO algorithm t...

In this paper, we reexamine in the framework of robust computation the Bentley-Ottmann algorithm for reporting intersecting pairs of segments in the plane. This algorithm has been reported as being very sensitive to numerical errors. Indeed, a simple analysis reveals that it involves predicates of degree 5, presumably never evaluated exactly in most implementation. Within the exact-computation paradigm we introduce two models of computation aimed at replacing the conventional model of real-number arithmetic. The first model (predicate arithmetic) assumes the exact evaluation of the signs of algebraic expressions of some degree, and the second model (exact arithmetic) assumes the exact computation of the value of...

Given a polygonal object (simple polygon, geometric graph, wire-frame, skeleton or more generally a set of line segments) in three dimensional Euclidean space, we consider the problem of computing a variety of "nice" parallel (orthographic) projections of the object. We show that given a general polygonal object consisting of n line segments in space, deciding whether it admits a crossing-free projection can be done in O(n 2 log n+k) time and O(n 2 +k) space, where k is the number of edge intersections of forbidden quadrilaterals (i.e. set of directions that admits a crossing) and varies from zero to O(n 4 ). This implies for example that given a simple polygon in 3-space we can determine if there exists a plane on which the projection is a simple polygon, within the same complexity. Furthermore, if such a projection does not exist, a minimum-crossing projection can be found in O(n 4 ) time and space. We show that an object always admits a regular projection (of interest to k...

We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, replacing randomized algorithms by deterministic ones with as small decrease of efficiency as possible. Related to the problem of derandomization is the question of reducing the amount of random bits needed by a randomized algorithm while retaining its efficiency; the derandomization can be viewed as an ultimate case. Randomized algorithms are also related to probabilistic proofs and constructions in combinatorics (which came first historically), whose development has similarly been accompanied by the effort to replace them by explicit, non-random constructions whenever possible. Derandomization of algorithms can be seen as a part of an effort to map the power of randomness and explain its role. ...

For a set S of n line segments in the plane, we give the first work-optimal deterministic parallel algorithm for constructing their arrangement. It runs in O(log 2 n) time using O(n log n + k) work in the EREW PRAM model, where k is the number of intersecting line segment pairs, and provides a fairly simple divide-and-conquer alternative to the optimal sequential “plane-sweep ” algorithm of Chazelle and Edelsbrunner. Moreover, our method can be used to output all k intersecting pairs while using only O(n) working space, which solves an open problem posed by Chazelle and Edelsbrunner. We also describe a sequential algorithm for computing a single face in an arrangement of n line segments that runs in O(n 2 (n) log n) time, which improves on a previous O(n log 2 n) time algorithm. For collections of simplices in IR d, we give methods for constructing a set of m = O(n d,1 log c n+k) cells of constant descriptive complexity that covers their arrangement, where c> 1 is a constant and k is the number of faces in the arrangement. The construction is performed sequentially in O(m) time, or in O(log n) time using O(m) work in the EREW PRAM model. The covering can be augmented to answer point location queries in O(log n) time. In addition to supplying the first parallel methods for these problems, we improve on the previous best sequential methods by reducing the query times (from O(log 2 n) in IR 3 and O(log 3 n) in IR d, d>3), and also the size and construction cost of the covering (from O(n d,1+ + k)). 1

We consider whether restricted sets of geometric predicates support efficient algorithms to solve line and curve segment intersection problems in the plane. Our restrictions are based on the notion of algebraic degree, proposed by Preparata and others as a way to guide the search for efficient algorithms that can be implemented in more realistic computational models than the Real RAM.

A variety of techniques for performing a spatial join are reviewed. Instead of just summarizing the literature and presenting each technique in its entirety, distinct components of the different techniques are described and each is decomposed into an overall framework for performing a spatial join. A typical spatial join technique consists of the following components: partitioning the data, performing internal-memory spatial joins on subsets of the data, and checking if the full polygons intersect. Each technique is decomposed into these components and each component addressed in a separate section so as to compare and contrast similar aspects of each technique. The goal of this survey is to describe the algorithms within each component in detail, comparing and contrasting competing methods, thereby enabling further analysis and experimentation with each component and allowing the best algorithms for a particular situation to be built piecemeal, or, even better, enabling an optimizer to choose which algorithms to use. Categories and Subject Descriptors: H.2.4 [Database Management]: Systems—Query processing; H.2.8 [Database Management]: Database Applications—Spatial databases and GIS

Introduction Computational geometry evolves from the classical discipline of design and analysis of algorithms, and has received a great deal of attention in the last two decades since its inception in 1975 by M. Shamos[108]. It is concerned with the computational complexity of geometric problems that arise in various disciplines such as pattern recognition, computer graphics, computer vision, robotics, VLSI layout, operations research, statistics, etc. In contrast with the classical approach to proving mathematical theorems about geometry-related problems, this discipline emphasizes the computational aspect of these problems and attempts to exploit the underlying geometric properties possible, e.g., the metric space, to derive efficient algorithmic solutions. The classical theorem, for instance, that a set S is convex if and only if for any 0 ff 1 the convex combination ffp + (1 \Gamma<F