M. Overmars. Point location in fat subdivisions. Information Processing Letters, 44:261-265, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....cells of equal volume, and at each instance the objects are assigned to one or more cells. Collisions are checked between all object pairs belonging to each cell. In fact, Overmars presented an e#cient algorithm based on hash table to e#ciently perform point location queries in fat subdivisions [Ove92] see also Chapter 34) This approach works well for sparse environments in which the objects are uniformly distributed through the space. Another approach operates directly on 4D volumes swept out by object motion over time [Cam90] E#cient algorithms for maintenance and self collision testing ....

M.H. Overmars. Point location in fat subdivisions. Inform. Proc. Lett., 44:261--265, 1992.

....oliveira cs.uiowa.edu Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA. E mail: dstewart math.uiowa.edu sets f S i ae R j i = 1; 2; N g, rather than a collection of points. Also note that this task is not the point location problem studied previously [9, 10], in that the sets S i are not usually disjoint. Our analysis is also different to that of [9, 10] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. x S S S S S S 1 2 3 4 5 7 Figure 1: Example of distribution of ....

....USA. E mail: dstewart math.uiowa.edu sets f S i ae R j i = 1; 2; N g, rather than a collection of points. Also note that this task is not the point location problem studied previously [9, 10] in that the sets S i are not usually disjoint. Our analysis is also different to that of [9, 10] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. x S S S S S S 1 2 3 4 5 7 Figure 1: Example of distribution of support sets S i For each x and S i , we assume that we can test if x 2 S i in Theta(1) time. The ....

M. H. Overmars. Point location in fat subdivisions. Infomation Proc. Letters, 44:261--265, 1992.

....of line segments with bounded length ratios, i.e. the ratio of the lengths of the longest and shortest segments is bounded by a constant. Their result for constructing BSP of linear size for sets of fat objects is based on a lemma about the fatness property obtained by van der Stappen et al. see [Ove92, vdSHO93] Although this is an interesting result from a theoretical point of view, their algorithm, with running time of O(n log n log log n) employs a rather complicated transformation and a segment arrangement construction which may not be useful for practical implementations. The authors ....

M. H. Overmars. Point location in fat subdivisions. Inform. Process. Lett., 44:261--265, 1992.

....the worst case. Recently, computational geometers have become interested in so called fat objects. Well known geometric problems can be reconsidered for cases where the given objects or subdivision satisfy a certain fatness condition, and more efficient, simpler algorithms can often be obtained [3, 6, 10, 12, 21, 25]. Fat objects are important in practice, since generally one does not deal with objects that are very thin. With respect to the contour size, Matouek et al. 19] observed that for triangles, a quadratic lower bound example can only be constructed if the triangles have sharp angles. They proved that ....

....the ratio of the radii of the maximum inscribed circle and the minimum enclosing circle is at least a constant [6] iii) for convex polygons, if the ratio of the width and diameter is at least a constant. A fourth definition, for fatness of simple polygons and not equivalent to ours, is given in [21, 25]. This paper deals with fatness according to Definition 1 above, and the results also hold for fatness according to the three equivalent definitions just mentioned. To avoid confusion, we use the term wide for our definition. Theorem I Let S be a set of J wide simple polygons with n vertices in ....

Overmars, M., Point location in fat subdivisions. Tech. Rep. RUU-CS-91-40, Dept. of Comp. Science, Utrecht University, 1991.

....the size of the smallest object, and returns the set of objects intersecting the range. This data structure is interesting because it implements range searching queries by performing many point location queries in the set of objects, which are themselves implemented using a structure by Overmars [4]. The number of point location queries depends on the shape of the objects van der Stappen could only prove bounds for convex and for polygonal fat objects , their size, the size of the query range, and on the fatness coefficient of the objects (see below for details) Even in simple cases, ....

....can be performed in time O(log n) We also give exact bounds that show the dependence on the size of the query range and on the low density parameter. 2 Fatness and low density environments As mentioned in the introduction, the concept of fatness has received quite some attention recently [1, 2, 3, 4, 9, 10]. For completeness, we review the definition of fatness used by Overmars and van der Stappen [5] They define an object E in R to be t fat, for a given parameter t 0, if for every axis parallel hypercube H with center in E that does not contain E fully, the volume of H E is at least 1=t of ....

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M. H. Overmars. Point location in fat subdivisions. Inform. Process. Lett., 44:261--265, 1992.

....heap updated. Only the objects whose bounding boxes for their swept volumes during the time step intersect with other boxes are selected and included in the heap. The intersections between these n boxes can be done in O(n(1 log R) R being the ratio of largest to smallest box size, as shown in [64]. A similar idea is followed in [51,24,67,54] where temporal coherence is exploited not only to speed up pairwise intersection detection (as mentioned in Section 4.2.3) but also to perform less of these pairwise tests. If time steps are small enough, there will be little change in the positions ....

Overmars, M. Point location in fat subdivisions. Information Processing letters 44 (1992), 261--265. 28

....collision detection are based on spatial decomposition techniques. These algorithms partition the space into uniform or adaptive grids (i.e. volumetric approaches) octrees [32] k D trees or BSP s [26] To overcome the problem of large memory requirements for volumetric approaches, some authors [28] have proposed the use of hash tables. Utilizing Frame to Frame Coherence: In many simulations, the objects move only a little between successive frames. Many efficient algorithms that utilize frame to frame coherence have been proposed for convex polytopes [22, 8, 3] Cohen et al. 9] have used ....

M. H. Overmars. Point location in fat subdivisions. Inform. Proc. Lett., 44:261--265, 1992.

....Institute of Mathematical Sciences, New York University. sharir math.tau.ac.il 1 Introduction 2 Figure 1: Two (fi; ffi) covered objects properties, which were used by many authors to obtain more efficient solutions to a variety of algorithmic problems, when the underlying objects are fat. See [4, 6, 9, 17, 18, 21, 23, 25, 26] for a sample of these results. In a recent paper [17] Katz has designed a data structure of nearly linear size for sets of convex ff fat objects in the plane. By augmenting the data structure in various ways he obtains efficient and simple solutions to several query type problems, including the ....

M.H. Overmars, Point location in fat subdivisions, Information Processing Letters 44 (1992), 261--265.

....between theory and practice lead de Berg et al. 5] to study BSPs for scenes consisting of fat objects. Fat objects are objects that do not have long and skinny 2 parts; a formal definition is given in Section 2.3. Recently, fat objects have attracted a lot of attention in computational geometry [1, 9, 28, 13, 15, 26, 27]. De Berg et al. proved that scenes of fat objects always admit a BSP of linear size. Their algorithm for constructing a BSP runs in O(n log n log log n) time, where n is the number of objects. Unfortunately, their method only works in the plane, so it is not very useful in computer graphics ....

.... an extra search structure on top of our BSP tree, which guarantees that point location queries can be performed efficiently: if the scene is uncluttered then the query time is O(log n) and the amount of storage for the data structure is O(n) This improves and generalizes a result of Overmars [15], who showed that point location queries in a set of n disjoint fat objects can be done in O(log d Gamma1 n) time with a structure using O(n log d Gamma1 n) storage. If the scene consists of disjoint fat objects (or, more generally, is a so called low density scene) then our point location ....

M. H. Overmars. Point location in fat subdivisions. Inform. Process. Lett., 44:261--265, 1992.

....this operation for each of the rectangles in a given set requires a total time of O(n log n K) where K is the total number of intersecting rectangles. 6.3. 4 Uniform Spatial Subdivision We can divide the space into unit cells (or volumes) and place each object (or bounding box) in some cell(s) Ove92] To check for collisions, we have to examine the cell(s) occupied by each box to verify if the cell(s) is(are) shared by other objects. The memory required is O(p 3 ) when using a p Thetap Thetap grid. It is difficult to set a near optimal size for each cell, and if the size of the cell is not ....

M. H. Overmars. Point location in fat subdivisions. Inform. Proc. Lett., 44:261--265, 1992.

....as tries, such as are used for string matching [2] However, the task studied in this paper involves preprocessing a collection of sets S i # R d i = 1, 2, N , rather than a collection of points. Also note that this task is not the point location problem previously studied [11, 12], in that here the sets S i are usually not disjoint. Our analysis is also di erent to that of [11, 12] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. While the data structures described here are those of Samet [15] Samet ....

....preprocessing a collection of sets S i # R d i = 1, 2, N , rather than a collection of points. Also note that this task is not the point location problem previously studied [11, 12] in that here the sets S i are usually not disjoint. Our analysis is also di erent to that of [11, 12] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. While the data structures described here are those of Samet [15] Samet considers their use only for the intersection problem: nd all pairs (i, j) where S i #S j #= #. Here ....

M. H. Overmars. Point location in fat subdivisions. Infomation Proc. Letters, 44:261265, 1992.

....fat objects possess several desirable properties. This has led many authors to reconsider various known problems in computational and combinatorial geometry under the assumption that the input objects are fat, which resulted in an array of more efficient solutions than in the general case. See [2, 4, 9, 11, 18, 22, 25, 26, 30, 31, 32, 34] for a small sample of these results. In practice, the fatness assumption often holds, thus the search for more efficient solutions for collections of fat objects is definitely justified. We use the following definition of fatness (also used in [2] An object c in the plane is ff fat, for some ....

M.H. Overmars, Point location in fat subdivisions, Inf. Proc. Letters 44 (1992), 261-- 265.

....collision detection are based on spatial decomposition techniques. These algorithms partition the space into uniform or adaptive grids (i.e. volumetric approaches) octrees [38] k D trees or BSP s [32] To overcome the problem of large memory requirements for volumetric approaches, some authors [34] have proposed the use of hash tables. Utilizing Frame to Frame Coherence: In many simulations, objects move only a little between successive frames. Many efficient algorithms that utilize frame to frame coherence have been proposed for convex polytopes [3,8,27] Cohen et al. 9] have used ....

.... Assumptions 1 and 2, the expected collision query time using the hybrid data structures for possible intersections between P and a line segment L is O(1) The basic dictionary operations using hash tables require only O(1) time on average [12] and this is extended for the point location problem [34]. There are typically less than two such queries, since S 1 given Assumptions 1 and 4. So, we can query the hash table as to which OBBTrees need to be checked in expected O(1) time. According to Lemma 5, querying an OBBTree for intersection with L takes O(KH) time, where H is the maximum height ....

M. H. Overmars. Point location in fat subdivisions. Inform. Proc. Lett., 44:261-- 265, 1992.

....Mathematics, The University of Iowa, Iowa City, IA 52242, USA. E mail: dstewart math.uiowa.edu 1 2 X. Han, S. Oliveira and D.E. Stewart sets f S i ae R d j i = 1; 2; N g, rather than a collection of points. Also note that this task is not the point location problem studied previously [9, 10], in that the sets S i are not usually disjoint. Our analysis is also different to that of [9, 10] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. x S S S S 6 S S S 1 2 3 4 5 7 Figure 1: Example of distribution of support sets ....

....2 X. Han, S. Oliveira and D.E. Stewart sets f S i ae R d j i = 1; 2; N g, rather than a collection of points. Also note that this task is not the point location problem studied previously [9, 10] in that the sets S i are not usually disjoint. Our analysis is also different to that of [9, 10] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. x S S S S 6 S S S 1 2 3 4 5 7 Figure 1: Example of distribution of support sets S i For each x and S i , we assume that we can test if x 2 S i in Theta(1) time. The point about ....

M. H. Overmars. Point location in fat subdivisions. Infomation Proc. Letters, 44:261--265, 1992.

....with intersecting bounding volumes for pairwise collision detection. SiLVIA provides two alternative methods. The rst one is based on space partitioning and tries to nd possible intersections among boxes in space using a hierarchical hashing table. The basic technique was rst presented in [Ove92] and then modi ed in [Mir96] to exploit coherence between moving objects. The second method uses coordinate sorting to nd all intersections between the boxes. 2.2 Hierarchical Bounding Volumes On top of the ACIS r kernel, we built a data structure that allows collision detection and ....

M. H. Overmars. Point location in fat subdivisions. Inform. Process. Lett., 44:261-265, 1992.

....phases. The broad phase culls most of the O(n 2 ) pairs of bodies from further consideration. It uses simple bounding boxes around the objects to quickly determine which pairs are in no danger of colliding. The bounding boxes are stored in a multi resolution spatial hash table as described in [Overmars, 1992]. This data structure can efficiently report which bounding boxes overlap a region of three dimensional space. 1 www.merl.com people mirtich berkeleyHtml massProps.html MERL TR 98 11 January 4 The pairs which the broad phase can not eliminate are passed to the slower but more precise narrow ....

Overmars, M. (1992). Point location in fat subdivisions. Information Processing Letters, 44:261--265. MERL-TR-98-11 January 15

....collision detection are based on spatial decomposition techniques. These algorithms partition the space into uniform or adaptive grids (i.e. volumetric approaches) octrees [33] k D trees or BSP s [27] To overcome the problem of large memory requirements for volumetric approaches, some authors [29] have proposed the use of hash tables. Utilizing Frame to Frame Coherence: In many simulations, the objects move only a little between successive frames. Many efficient algorithms that utilize frame to frame coherence have been proposed for convex polytopes [23, 8, 3] Cohen et al. 9] have used ....

M. H. Overmars. Point location in fat subdivisions. Inform. Proc. Lett., 44:261--265, 1992.

....time for each primitive and our time bounds. Our algorithm can easily be generalized to report the set of points lying within the range, and the running time increases to include the time to output the points. Approximate range searching is probably interesting only for fat ranges. Overmars [14] defines an object Q to be k fat if for any point p in Q, and any ball B with p as center that does not fully contain Q in its interior, the portion of B covered by Q is at least 1=k. For ranges that are not k fat, the diameter of the range may be arbitrarily large compared to the thickness of the ....

M. H. Overmars. Point location in fat subdivisions. Inform. Process. Lett., 44:261--265, 1992.

....the size of the fixed cube to be large enough to contain the object at any orientation. We define this axis aligned cube by a center and a radius. Fixed cubes are easy to recompute as objects move, making them well suited to dynamic environments. If an object is nearly spherical, or fat [Ove92] the fixed cube fits it well. As preprocessing steps we calculate the center and radius of the fixed cube. At each time step as the object moves, we recompute the cube as follows: 1. Transform the center using one vector matrix multiplication. 2. Compute the minimum and maximum x, y, and ....

....by other objects. But, it is difficult to set a near optimal size for each cell and it requires a tremendous amount of allocated memory. If the size of the cell is not properly chosen, the computation can be rather expensive. For an environment where almost all objects are of uniform size and fat [Ove92] like a vibrating parts feeder bowl, or molecular modeling application [Lev66, Tur89] this algorithm is ideal, especially for execution on a parallel machine. Alonso et al. ASF94] have an algorithm that combines space partitioning with hierarchical bounding boxes. The running time of their ....

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M. H. Overmars. Point location in fat subdivisions. Inform. Proc. Lett., 44:261-- 265, 1992.

....Consider an environment where most of objects are elongated and only a few objects (probably just the robot manipulators in most situations) are moving, then rectangular bounding boxes are preferable. In a more dynamic environment like a vibrating parts feeder where all objects are rather fat [68] and bouncing around, then spherical bounding boxes are more desirable. If the objects are concave or articulated, then a subpart hierarchical bounding box representation (similar to subpart hierarchical tree representation, with each node storing a bounding box) should be employed. The reasons ....

....can easily dominate the run time, especially when a collision occurs. The tree structures also cannot capture the temporal and spatial coherence well. 5.3. 2 Uniform Spatial Subdivision We can divide the space into unit cells (or volumes) and place each object (or bounding box) in some cell(s) [68]. To check for collisions, we have to examine the cell(s) occupied by each box to verify if the cell(s) is(are) shared by other objects. But, it is difficult to set a near optimal size for each cell and it requires tremendous amount of allocated memory. If the size of the cell is not properly ....

[Article contains additional citation context not shown here]

M. H. Overmars. Point location in fat subdivisions. Inform. Proc. Lett., 44:261-- 265, 1992.

....of intervals. Therefore, reporting intersection among n rectangles can be done in O(nlogn K) where K is the total number of intersecting rectangles. 5. 4 Uniform spatial subdivision We can divide the space into unit cells (or volumes) and place each object (or bounding box) in some cell(s) [Ove92]. To check for collisions, we have to examine the cell(s) occupied by each box to verify if the cell(s) is(are) shared by other objects. But, it is difficult to set a near optimal size for each cell and it requires tremendous amount of allocated memory. If the size of the cell is not properly ....

....be expensive. For an environment where objects are of uniform size [Tur89] this is a rather ideal algorithm and especially suitable for parallelization. Overmars has shown that using a hash table to look up an entry and O(n) storage space we can perform the point location queries in constant time [Ove92]. 6 Public Domain Software Packages Most of public domain systems are applicable to polygonal models and some are also applicable to large environments composed of multiple moving objects. It is nearly impossible to compare different algorithms and systems fairly, since their performance varies, ....

M. H. Overmars. Point location in fat subdivisions. Inform. Proc. Lett., 44:261--265, 1992.

....n) preprocessing. The data structure and query algorithm are rather simple. 1 Introduction Fatness turns out to be an interesting phenomenon in computational geometry. Several papers present surprising combinatorial complexity reductions [3, 15, 22, 26, 32] and efficiency gains for algorithms [1, 4, 19, 28, 33] if the objects under consideration have a certain fatness. Fat objects are compact to some extent, rather than long and thin. Fatness is a realistic assumption, since in many practical instances of geometric problems the considered objects are fat. The aim of studying fatness is to find new fast ....

.... planning amidst fat obstacles [32] and efficient algorithms for computing depth orders on certain fat objects [1] binary space partitions for scenes of non intersecting fat objects in the plane [4] hidden surface removal for fat horizontal triangles [19] point location in fat subdivisions [28], and motion planning in certain realistic settings of fat obstacles in twoand three dimensional workspaces [33] In this paper we study two fundamental problems in computational geometry in a context of fat objects: point location and range searching. The point location problem aims at ....

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M.H. Overmars, Point location in fat subdivisions, Information Processing Letters 44 (1992), pp. 261-265.

....a molecule. Our work is closely related to the study of fatness. It has been shown that certain problems in computational geometry can be solved much more efficiently when the 2 Figure 2: Test molecules crambin (327 atoms) and felix (613 atoms) objects involved have no long and thin parts (see [18, 20, 23]) Clearly, spheres are fat in that sense. In 3 space, though, fatness is not enough to guarantee efficient algorithms when the objects are allowed to intersect. Fortunately, the extra properties of the hard sphere model provide sufficient additional constraints on the objects to allow for ....

M. H. Overmars, Point location in fat subdivisions, Inform. Proc. Lett., 44 (1992), pp. 261--265.

....results will be the linear free space complexity for problems in this class. 2.2 Fatness Fatness has turned out to be an interesting phenomenon in computational geometry. Several papers present surprising combinatorial complexity reductions [2, 10, 16, 19, 33] and efficiency gains for algorithms [1, 6, 14, 22, 23] if the objects under consideration have a certain fatness. Fat objects are compact to some extent, rather than long and thin. Fatness is a realistic assumption, since in many practical instances of geometric problems the considered objects are fat. The aim of studying fatness is to find new fast ....

.... fat obstacles [33] and efficient algorithms for computing depth orders on certain fat objects [1] binary space partitions for scenes of nonintersecting fat objects in the plane [6] hidden surface removal for fat horizontal triangles [14] and range searching and point location among fat objects [22, 23]. Contrary to many other definitions of fatness in literature [1, 2, 10, 14, 16, 19] the notion introduced in [33] and recaptured below, applies to general objects in arbitrary dimension d. The definition involves a parameter k, supplying a qualitative measure of the fatness of an object: the ....

M.H. Overmars, Point location in fat subdivisions, Information Processing Letters 44 (1992), pp. 261-265.

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M. Overmars. Point location in fat subdivisions. Information Processing Letters, 44:261-265, 1992.

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