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19
RAY SHOOTING AND PARAMETRIC SEARCH
, 1993
"... Efficient algorithms for the ray shooting problem are presented: Given a collection F of objects in d, build a data structure so that, for a query ray, the first object of F hit by the ray can be quickly determined. Using the parametric search technique, this problem is reduced to the segment emptin ..."
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Cited by 127 (25 self)
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Efficient algorithms for the ray shooting problem are presented: Given a collection F of objects in d, build a data structure so that, for a query ray, the first object of F hit by the ray can be quickly determined. Using the parametric search technique, this problem is reduced to the segment emptiness problem. For various ray shooting problems, space/querytime tradeoffs of the following type are achieved: For some integer b and a parameter m (n _< m < n b) the queries are answered in time O((n/m /b) log <) n), with O(m!+) space and preprocessing time (t> 0 is arbitrarily small but fixed constant), b Ld/2J is obtained for ray shooting in a convex dpolytope defined as an intersection of n half spaces, b d for an arrangement of n hyperplanes in d, and b 3 for an arrangement of n half planes in 3. This approach also yields fast procedures for finding the first k objects hit by a query ray, for searching nearest and farthest neighbors, and for the hidden surface removal. All the data structures can be maintained dynamically in amortized time O (m + / n) per insert/delete operation.
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.
An efficient outputsensitive hiddensurface removal algorithm for polyhedral terrains
, 1994
"... In this paper, we present an algorithm for hidden surface removal for a class of polyhedral surfaces which have a property that they can be ordered relatively quickly. For example, our results apply directly to terrain maps. A distinguishing feature of our algorithm is that its running time is sen ..."
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Cited by 36 (1 self)
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In this paper, we present an algorithm for hidden surface removal for a class of polyhedral surfaces which have a property that they can be ordered relatively quickly. For example, our results apply directly to terrain maps. A distinguishing feature of our algorithm is that its running time is sensitive to the actual size of the visible image, rather than the total number of intersections in the image plaue which can be much larger than the visible image. The time complexity of this algorithm is O((k + n) log ’ n) where n and /c are, respectively, the input and the output sizes. Thus, in a significant number of situations this will be faster than the worst case optimal algorithms which have running time of n(n²) irrespective of the output size.
Efficient Ray Shooting and Hidden Surface Removal
 ALGORITHMICA
, 1991
"... In this paper we study the ray shooting problem for three special classes of polyhedral objects in space: axisparallel polyhedra, curtains (unbounded polygons with three edges, two of which are parallel to the zaxis and extend downward to minus infinity) and fat horizontal triangles (triangles ..."
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Cited by 30 (5 self)
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In this paper we study the ray shooting problem for three special classes of polyhedral objects in space: axisparallel polyhedra, curtains (unbounded polygons with three edges, two of which are parallel to the zaxis and extend downward to minus infinity) and fat horizontal triangles (triangles parallel to the yplane whose angles are greater than some fixed constant). For all three problems structures are presented using O(n 2+) preprocessing, for any fixed e > 0, with O(log n) query time. We also study the general ray shooting problem in an arbitrary set of (possibly intersecting) triangles. Here we present a structure that uses O(n 4+e) preprocessing and has a query time of O(log n). As an
Visibility with a moving point of view
 Algorithmica
, 1994
"... We investigate 3d visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering rayshooting queries for rays with origin on the ..."
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Cited by 28 (1 self)
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We investigate 3d visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering rayshooting queries for rays with origin on the flightpath. Three progressively more specialized problems are considered: general scenes, terrains, and terrains with vertical flightpaths. 1.
Dynamic Maintenance of Kinematic Structures
, 1996
"... We consider the following dynamic data structure problem. Given a collection of rigid bodies moving in 3dimensional space and hinged together in a kinematic structure, our goal is to efficiently maintain a data structure that allows us to quickly answer range queries as the bodies move. This kinema ..."
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Cited by 13 (9 self)
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We consider the following dynamic data structure problem. Given a collection of rigid bodies moving in 3dimensional space and hinged together in a kinematic structure, our goal is to efficiently maintain a data structure that allows us to quickly answer range queries as the bodies move. This kinematic data structure problem arises in a variety of applications such as conformational search in molecular biology, simulation of hyperredundant robots, collision detection, and computer animation. We study several models for dynamic maintenance of such structures and devise algorithms under these models. We obtain tight results on the worstcase, amortized, and randomized complexity of the data structure problem. For the offline version of the problem, we establish NPhardness and provide efficient approximation algorithms.
The Area Bisectors of a Polygon and Force Equilibria in Programmable Vector Fields
, 1997
"... We consider the family of area bisectors of a polygon #possibly with holes# in the plane. Wesay that two bisectors of a polygon P are combinatorially distinct if they induce di#erent partitionings of the vertices of P.We show that there are simple polygons with n vertices that have# #n 2 # combi ..."
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Cited by 12 (7 self)
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We consider the family of area bisectors of a polygon #possibly with holes# in the plane. Wesay that two bisectors of a polygon P are combinatorially distinct if they induce di#erent partitionings of the vertices of P.We show that there are simple polygons with n vertices that have# #n 2 # combinatorially distinct area bisectors #matching the obvious upper bound#, and we present an outputsensitive algorithm for computing an explicit representation of all the bisectors of a given polygon. Our study is motivated by the development of novel, #exible feeding devices for parts positioning and orienting. The question of determining all the bisectors of polygonal parts arises in connection with the development of e#cient part positioning strategies when using these devices. 1 Introduction Let P be a polygon in the plane, possibly with holes, and having n vertices in total. We denote by V the set of vertices of P.For a directed line # in the plane, we denote by h l ### #resp. hr #### the...
Stabbing and Ray Shooting in 3 Dimensional Space
, 1994
"... In this paper we consider the following problems: given a set T of triangles in 3space, with jT j = n, a) answer the query " given a line l, does l stab the set of triangles?" (query problem). b) find whether a stabbing line exists for the set of triangles (existence problem). c) Given a ray ae, ..."
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Cited by 11 (3 self)
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In this paper we consider the following problems: given a set T of triangles in 3space, with jT j = n, a) answer the query " given a line l, does l stab the set of triangles?" (query problem). b) find whether a stabbing line exists for the set of triangles (existence problem). c) Given a ray ae, which is the first triangle in T hit by ae? The following results are shown. 1. There is an \Omega\Gamma n 3 ) lower bound on the descriptive complexity of the set of all stabbers for a set of triangles. 2. The existence problem for triangles on a set of planes with g different plane inclinations can be solved in O(g 2 n 2 log n) time (Theorem 2). 3. The query problem is solvable in quasiquadratic O(n 2+ffl ) preprocessing and storage and logarithmic O(log n) query time (Theorem 4). 4. All stabbing results for triangles extend, with the same asymptotic bounds, to sets of convex polyhedra with total complexity n. 5. Using O(n 3+ffl ) preprocessing time and storage we can det...
Parallel ObjectSpace Hidden Surface Removal
, 1990
"... A parallel objectspace hidden surface removal algorithm for polyhedral scenes is presented. The uniform grid technique is used to achieve parallelism for the hidden line removal. A conflictdetection and backoff strategy is then used to obtain parallelism for the visible region reconstruction from ..."
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Cited by 9 (3 self)
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A parallel objectspace hidden surface removal algorithm for polyhedral scenes is presented. The uniform grid technique is used to achieve parallelism for the hidden line removal. A conflictdetection and backoff strategy is then used to obtain parallelism for the visible region reconstruction from the visible segments. The algorithm has been implemented on a Sequent Balance 21000 sharedmemory parallel computer. An average speedup of 10 has been obtained using 15 processors.