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80
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 484 (122 self)
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An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
New Bounds for Lower Envelopes in Three Dimensions, with Applications to Visibility in Terrains
 Geom
, 1997
"... We consider the problem of bounding the complexity of the lower envelope of n surface patches in 3space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect i ..."
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Cited by 64 (24 self)
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We consider the problem of bounding the complexity of the lower envelope of n surface patches in 3space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect in at most two points. We show that the number of vertices on the lower envelope of n such surface patches is O(n 2 \Delta 2 c p log n ), for some constant c depending on the shape and degree of the surface patches. We apply this result to obtain an upper bound on the combinatorial complexity of the `lower envelope' of the space of all rays in 3space that lie above a given polyhedral terrain K with n edges. This envelope consists of all rays that touch the terrain (but otherwise lie above it). We show that the combinatorial complexity of this rayenvelope is O(n 3 \Delta 2 c p log n ) for some constant c; in particular, there are at most that many rays that pass above th...
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 39 (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 33 (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
On Fat Partitioning, Fat Covering and the Union Size of Polygons
, 1993
"... The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has unio ..."
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Cited by 33 (3 self)
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The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity is O(nloglogn), which is a generalization of a similar result for fat triangles [19]. Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, etc. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(1) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any coveting of a fat simple polygon by a linear number of fat triangles.
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 32 (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.
Sparse Arrangements and the Number of Views of Polyhedral Scenes
, 1992
"... Given a collection of n lowdegree algebraic surface patches in 3space with the property that any vertical line stabs at most k of them, we wish to determine the maximum combinatorial complexity, D(n, k), of the entire arrangement that they induce. We show that D(n, k) = O(n2k). We extend this r ..."
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Cited by 30 (11 self)
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Given a collection of n lowdegree algebraic surface patches in 3space with the property that any vertical line stabs at most k of them, we wish to determine the maximum combinatorial complexity, D(n, k), of the entire arrangement that they induce. We show that D(n, k) = O(n2k). We extend this result to collections of hypersurfaces in 4space and to collections of (d 1)simplices in &space. We apply these results to obtain upper bounds on the maximum number of views of a polyhedral terrain consisting of n edges and vertices. Our bounds are O(n4A4(n)) for views from infinity and O(n7A4(n)) for perspective views, where A4(rl) is a nearlinear function related to DavenportSchinzel sequences. Furthermore, we show that these bounds are almost tight in the worst case. In the
Efficient Algorithms for Petersen's Matching Theorem
, 1999
"... Petersen's theorem is a classic result in matching theory from 1891, stating that every 3regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general gra ..."
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Cited by 29 (3 self)
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Petersen's theorem is a classic result in matching theory from 1891, stating that every 3regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n^3/2) time for 3regular graphs. We have developed an O(n log^4 n)time algorithm for perfect matching in a 3regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.
On Incremental Rendering of Silhouette Maps of a Polyhedral Scene
 SODA
, 2000
"... We consider the problem of incrementally rendering a polyhedral scene while the viewpoint is moving. In practical situations the number of geometric primitives to be rendered can be very large  hundreds of thousands or millions, but these may come from only a moderate number of objects that happe ..."
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Cited by 28 (4 self)
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We consider the problem of incrementally rendering a polyhedral scene while the viewpoint is moving. In practical situations the number of geometric primitives to be rendered can be very large  hundreds of thousands or millions, but these may come from only a moderate number of objects that happen to have been finely tessellated. It is sometimes advantageous to render only the silhouettes of the objects, rather than the objects themselves, and then exploit coherence or other methods to optimize the rendering of singleobject regions with uniform reflectance properties. Such an approach is also regularly used in the domain of nonphotorealistic rendering, where the rendering of silhouette edges plays a key role. The hard part in e#ciently implementing a kinetic approach to this problem is to realize when the rendered silhouette undergoes a combinatorial change. In this paper, we obtain bounds on a number of combinatorial problems involving the complexity of these events for a collec...