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15
Efficient Hidden Surface Removal for Objects with Small Union Size
, 1991
"... Let S be a set of n nonintersecting objects in space for which we want to determine the portions visible from some viewing point. We assume that the objects are ordered by depth from the viewing point (e.g., they are all horizontal and are viewed from infinity from above). In this paper we give ..."
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Cited by 48 (17 self)
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Let S be a set of n nonintersecting objects in space for which we want to determine the portions visible from some viewing point. We assume that the objects are ordered by depth from the viewing point (e.g., they are all horizontal and are viewed from infinity from above). In this paper we give an algorithm that computes the visible portions in time O((U(n)+ k)log 2 n), where U(n ) is a superadditive bound on the maximal complexity of the union of (the projections on a viewing plane of) any n objects from the family under consideration, and k is the complexity of the resulting visibility map. The algorithm uses O(U(n)logn) working storage. The algorithm is useful when the objects are "fat" in the sense that the union of the projection of any subset of them has small (i.e., subquadratic) complexity. We present three applications of this general technique: (i) For disks (or balls in space) we have U(n) = O(n), thus the visibility map can be computed in time O((n + k) log 2 n). (ii) For 'fat' triangles (where each internal angle is at least some fixed 0 degrees) we have U(n) = O(nloglogn) and the algorithm runs in time O((n log log n + k)log 2 n). (iii) The method also applies to computing the visibility map for a polyhedral terrain viewed from a fixed point, and yields an O((na(n)+ k)logn) algorithm.
Fast Rendering of Irregular Grids
, 2007
"... We propose a fast algorithm for rendering general irregular grids. Our method uses a sweepplane approach to accelerate ray casting, and can handle disconnected and nonconvex (even with holes) unstructured irregular grids with a rendering cost that decreases as the “disconnectedness” decreases. The ..."
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Cited by 43 (11 self)
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We propose a fast algorithm for rendering general irregular grids. Our method uses a sweepplane approach to accelerate ray casting, and can handle disconnected and nonconvex (even with holes) unstructured irregular grids with a rendering cost that decreases as the “disconnectedness” decreases. The algorithm is carefully tailored to exploit spatial coherence even if the image resolution differs substantially from the object space resolution. In this paper, we establish the practicality of our method through experimental results based on our implementation, and we also provide theoretical results, both lower and upper bounds, on the complexity of ray casting of irregular grids.
An Exact Interactive Time Visibility Ordering Algorithm for Polyhedral Cell Complexes
, 1998
"... 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 orderin ..."
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Cited by 39 (13 self)
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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...
The Lazy Sweep Ray Casting Algorithm for Rendering Irregular Grids
 IEEE Transactions on Visualization and Computer Graphics
, 1997
"... Lazy Sweep Ray Casting is a fast algorithm for rendering general irregular grids. It is based on the sweepplane paradigm, and it is able to accelerate ray casting for rendering irregular grids, including disconnected and nonconvex (even with holes) unstructured irregular grids with a rendering cost ..."
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Cited by 38 (8 self)
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Lazy Sweep Ray Casting is a fast algorithm for rendering general irregular grids. It is based on the sweepplane paradigm, and it is able to accelerate ray casting for rendering irregular grids, including disconnected and nonconvex (even with holes) unstructured irregular grids with a rendering cost that decreases as the "disconnectedness" decreases. The algorithm is carefully tailored to exploit spatial coherence even if the image resolution differs substantially from the object space resolution. Lazy Sweep Ray Casting has several desirable properties, including its generality, (depthsorting) accuracy, low memory consumption, speed, simplicity of implementation and portability (e.g., no hardware dependencies). We establish the practicality of our method through experimental results based on our implementation, which is shown to be substantially faster (by up to two orders of magnitude) than other algorithms implemented in software. We also provide theoretical results, both lower and ...
Complexity Measures for Assembly Sequences
 In Proc. IEEE Int. Conf. on Robotics and Automation
, 1996
"... Our work examines various complexity measures for twohanded assembly sequences. For many products there exists an exponentially large set of valid sequences, and a natural goal is to use automated systems to select wisely from the choices. Since assembly sequencing is a preprocessing phase for a lo ..."
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Cited by 22 (3 self)
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Our work examines various complexity measures for twohanded assembly sequences. For many products there exists an exponentially large set of valid sequences, and a natural goal is to use automated systems to select wisely from the choices. Since assembly sequencing is a preprocessing phase for a long and expensive manufacturing process, any work towards ndinga\better" assembly sequence isofgreat value when it comes time to assemble the physical product in mass quantities. We take a step in this direction by introducing a formal framework for studying the optimization of several complexity measures. This framework focuses on the combinatorial aspect of the family of valid assembly sequences, while temporarily separating out the speci c geometric assumptions inherent to the problem. With an exponential number of possibilities, nding the true optimal cost solution is nontrivial. In fact in the most general case, our results show that even nding an approximate solution is hard. Furthermore, we can show several hardness results, even in simple geometric settings. Future work is directed towards using this model to study how the original geometric assumptions can be reintroduced toprove stronger approximation results. 1
On the multilinear restriction and Kakeya conjectures
 Acta Math
"... Abstract. We prove dlinear analogues of the classical restriction and Kakeya conjectures in R d. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the correspondi ..."
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Cited by 18 (6 self)
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Abstract. We prove dlinear analogues of the classical restriction and Kakeya conjectures in R d. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variablecoefficient problems and the socalled “joints ” problem, as well as presenting some nlinear analogues for n < d. 1.
On the Relative Complexities of Some Geometric Problems
 In Proc. 7th Canad. Conf. Comput. Geom
, 1995
"... We consider the relative complexities of a large number of computational geometry problems whose complexities are believed to be roughly \Theta(n 4=3 ). For certain pairs of problems, we show that the complexity of one problem is asymptotically bounded by the complexity of the other. Almost all of ..."
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Cited by 17 (7 self)
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We consider the relative complexities of a large number of computational geometry problems whose complexities are believed to be roughly \Theta(n 4=3 ). For certain pairs of problems, we show that the complexity of one problem is asymptotically bounded by the complexity of the other. Almost all of the problems we consider can be solved in time O(n 4=3+ffi ) or better, and there are\Omega\Gamma n 4=3 ) lower bounds for a few of them in specialized models of computation. However, the best known lower bound in any general model of computation is only\Omega\Gamma n log n). The paper is naturally divided into two parts. In the first part, we consider a large number of problems that are harder than Hopcroft's problem. These problems include various ray shooting problems, sorting line segments in IR 3 , collision detection in IR 3 , and halfspace emptiness checking in IR 5 . In the second, we survey known reductions among problems involving lines in threespace, and among highe...
Online Point Location in Planar Arrangements and Its Applications
 GEOM
, 2002
"... Recently, HarPeled [HP99b] presented a new randomized technique for online construction of the zone of a curve in a planar arrangement of arcs. In this paper, we present several applications of this technique, which yield improved solutions to a variety of problems. These applications include: ( ..."
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Cited by 10 (6 self)
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Recently, HarPeled [HP99b] presented a new randomized technique for online construction of the zone of a curve in a planar arrangement of arcs. In this paper, we present several applications of this technique, which yield improved solutions to a variety of problems. These applications include: (i) an efficient mechanism for performing online point location queries in an arrangement of arcs; (ii) an efficient algorithm for computing an approximation to the minimumweight Steinertree of a set of points, where the weight is the number of intersections between the tree edges and a given collection of arcs; (iii) a subquadratic algorithm for cutting a set of pseudoparabolas into pseudosegments; (iv) an algorithm for cutting a set of line segments (`rods') in 3space so as to eliminate all cycles in the vertical depth order; and (v) a nearoptimal algorithm for reporting all bichromatic intersections between a set R of red arcs and a set B of blue arcs, where the unions of the arcs in each set are both connected.
An improved bound for joints in arrangements of lines in space
, 2003
"... Let L be a set of n lines in space. A joint of L is a point in R 3 where at least three noncoplanar lines meet. We show that the number of joints of L is O(n ..."
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Cited by 9 (3 self)
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Let L be a set of n lines in space. A joint of L is a point in R 3 where at least three noncoplanar lines meet. We show that the number of joints of L is O(n