Results 1 
4 of
4
Local polyhedra and geometric graphs
 In Proc. 14th ACMSIAM Sympos. on Discrete Algorithms
, 2003
"... We introduce a new realistic input model for geometric graphs and nonconvex polyhedra. A geometric graph G is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor and (2) the lengths of the longest and shortest ed ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
We introduce a new realistic input model for geometric graphs and nonconvex polyhedra. A geometric graph G is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor and (2) the lengths of the longest and shortest edges differ by at most a polynomial factor. A polyhedron is local if all its faces are simplices and its edges form a local geometric graph. We show that any boolean combination of any two local polyhedra in IR d each with n vertices, can be computed in O(n log n) time, using a standard hierarchy of axisaligned bounding boxes. Using results of de Berg, we also show that any local polyhedron in IR d has a binary space partition tree of size O(n log d1 n). Finally, we describe efficient algorithms for computing Minkowski sums of local polyhedra in two and three dimensions.
Guarding Scenes against Invasive Hypercubes
, 1998
"... A set of points G is a guarding set for a set of objects O, if any hypercube not containing a point from G in its interior intersects at most objects of O. This definition underlies a new input model, that is both more general than de Berg's unclutteredness, and retains its main property: a dd ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
A set of points G is a guarding set for a set of objects O, if any hypercube not containing a point from G in its interior intersects at most objects of O. This definition underlies a new input model, that is both more general than de Berg's unclutteredness, and retains its main property: a ddimensional scene satisfying the new model's requirements is known to have a linearsize binary space partition. We propose several algorithms for computing guarding sets, and evaluate them experimentally. One of them appears to be quite practical. 1. Introduction Recently de Berg et al. [4] brought together several of the realistic input models that have been proposed in the literature, namely fatness, low density, unclutteredness, and small simplecover 1 Supported by the ESPRIT IV LTR Project No. 21957 (CGAL). 2 Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. 3 Supported by the Netherlands' Organization for Scientific Research...
Walking Around Fat Obstacles
, 1999
"... We prove that if an object O is convex and fat then, for any two points a and b on its boundary, there exists a path along the boundary, from a to b, whose length is bounded by the length of the line segment ab times some constant fi. This constant is a function of the fatnessconstant and the di ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We prove that if an object O is convex and fat then, for any two points a and b on its boundary, there exists a path along the boundary, from a to b, whose length is bounded by the length of the line segment ab times some constant fi. This constant is a function of the fatnessconstant and the dimension d. We prove bounds for fi and show how to efficiently find paths on the boundary of O whose lengths are within these bounds. As an application of this result, we present a method for computing short paths among convex, fat obstacles in R d by applying de Berg's method for producing a linearsize subdivision of the space. Given a start site and a destination site in the free space, a standard obstacleavoiding "straightline" path that is at most some multiplicative constant factor longer than the length of the segment between the sites can be computed efficiently. 1 Introduction An object O is fffat if for any hyperball B that does not entirely contain O and whose center ...
Linear Binary Space Partitions and Hierarchy of Object Classes
, 2003
"... We consider the problem of constructing binary space partitions for the set P of ddimensional objects in ddimensional space. There are several classes of objects defined for such settings, which support design of effective algorithms. We extend the existing the de Berg hierarchy of classes [8] by ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We consider the problem of constructing binary space partitions for the set P of ddimensional objects in ddimensional space. There are several classes of objects defined for such settings, which support design of effective algorithms. We extend the existing the de Berg hierarchy of classes [8] by the definition of new classes derived from that one and we show desirability of such an extension. Moreover we propose a new algorithm, which works on generalized λlow density scenes [20] (defined in this paper) and provides BSP tree of linear size. The tree can be constructed in O(n log 2 n) time and space, where n is the number of objects. Moreover, we can tradeoff between size and balance of the BSP tree fairly simply.