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On the multilinear restriction and Kakeya conjectures
 Acta Math
"... Abstract. We prove dlinear analogues of the classical restriction and Kakeya conjectures in Rd. 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 correspondin ..."
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Cited by 52 (12 self)
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Abstract. We prove dlinear analogues of the classical restriction and Kakeya conjectures in Rd. 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.
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 51 (15 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 46 (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 45 (14 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
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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 21 (6 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...
On Lines and Joints
, 2009
"... Let L be a set of n lines in R d, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplifica ..."
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Cited by 13 (2 self)
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Let L be a set of n lines in R d, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [9], and of the followup simpler proof of Elekes et al. [6]. Some extensions, e.g., to the case of joints of algebraic curves, are also presented.
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 12 (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