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Separation and Approximation of Polyhedral Objects
, 1993
"... Given a family of disjoint polygons P1, P2, : ::, Pk in the plane, and an integer parameter m, it is NPcomplete to decide if the Pi's can be pairwise separated by a polygonal family with at most m edges, that is, if there exist polygons R1; R2; : ::; Rk with pairwisedisjoint boundaries such that P ..."
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Cited by 30 (3 self)
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Given a family of disjoint polygons P1, P2, : ::, Pk in the plane, and an integer parameter m, it is NPcomplete to decide if the Pi's can be pairwise separated by a polygonal family with at most m edges, that is, if there exist polygons R1; R2; : ::; Rk with pairwisedisjoint boundaries such that Pi Ri andP jRij m. In three dimensions, the problem is NPcomplete even for two nested convex polyhedra. Many other extensions and generalizations of the polyhedral separation problem, either to families of polyhedra or to higher dimensions, are also intractable. In this paper, we present e cient approximation algorithms for constructing separating families of nearoptimal size. Our main results are as follows. In two dimensions, we give an O(n log n) time algorithm for constructing a separating family whose size is within a constant factor of an optimal separating family; n is the number of edges in the input family of polygons. In three dimensions, we show how to separate a convex polyhedron from a nonconvex polyhedron with a polyhedral surface whose facetcomplexity is O(log n) times the optimal, where n = jPj+jQj is the complexity of the input polyhedra. Our algorithm runs in O(n4) time, but improves to O(n3) time if the two polyhedra are nested and convex. Our algorithm for separating a convex polyhedron from a nonconvex polyhedron extends to higher dimensions. In d dimensions, for d 4, the facetcomplexity of the approximation polyhedron is O(d log n) times the optimal, and the algorithm runs in O(nd+1) time. Finally, we also obtain results on separating sets of points, a family of convex polyhedra, and separation by nonpolyhedral surfaces, such as spherical patches.
Automatic Generation of Triangular Irregular Networks using Greedy Cuts
, 1995
"... We propose a new approach to the automatic generation of triangular irregular networks from dense terrain models. We have developed and implemented an algorithm based on the greedy principle used to compute minimumlink paths in polygons. Our algorithm works by taking greedy cuts ("bites") out of a ..."
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Cited by 22 (5 self)
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We propose a new approach to the automatic generation of triangular irregular networks from dense terrain models. We have developed and implemented an algorithm based on the greedy principle used to compute minimumlink paths in polygons. Our algorithm works by taking greedy cuts ("bites") out of a simple closed polygon that bounds the yettobe triangulated region. The algorithm starts with a large polygon, bounding the whole extent of the terrain to be triangulated, and works its way inward, performing at each step one of three basic operations: ear cutting, greedy biting, and edge splitting. We give experimental evidence that our method is competitive with current algorithms and has the potential to be faster and to generate many fewer triangles. Also, it is able to keep the structural terrain fidelity at almost no extra cost in running time and it requires very little memory beyond that for the input height array. 1 Introduction A terrain is the graph of a function of two variabl...
Approximation algorithms for geometric separation problems
 Department of
, 1993
"... In computer graphics and solid modeling, one is interested in representing complex geometric objects with combinatorially simpler ones. It turns out that via a “fattening ” transformation, one obtains a formulation of the approximation problem in terms of separation: Find a minimumcomplexity surface ..."
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Cited by 14 (4 self)
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In computer graphics and solid modeling, one is interested in representing complex geometric objects with combinatorially simpler ones. It turns out that via a “fattening ” transformation, one obtains a formulation of the approximation problem in terms of separation: Find a minimumcomplexity surface that separates two sets. In this paper, we provide approximation algorithms for several geometric separation problems, including: • Given a set of triangles T and a set S of points that lie within the union of the triangles, find a minimumcardinality set, T ′ , of pairwisedisjoint triangles, each contained within some triangle of T, that cover the point set S. • Given finite sets of “red ” and “blue ” points in the plane, determine a simple polygon of fewest edges that separates the red points from the blue points. More generally, given finite sets of points of many color classes, determine a planar “separating ” subdivision of minimum combinatorial complexity, which has the property that each face of the subdivision contains points of at most one color class; • Given two polyhedral terrains, P and Q, over a common support set (e.g., the unit square), with P lying above Q, compute a nested polyhedral terrain R that lies between P and Q such that R has a minimum number of facets. Exact solution of the above problems in polynomial time is highly unlikely: The decision versions of all three problems are known to be NPhard. We provide polynomialtime algorithms that are guaranteed to produce an answer within a logarithmic factor (O(log n), where n is the complexity of the input problem instance) of optimal. (The error factor is constant in the orthogonal case — coverage by disjoint aligned rectangles, or separation of orthohedral terrains.) We also discuss extensions to higher dimensions. 1
On the Complexity of Optimization Problems for 3Dimensional Convex Polyhedra and Decision Trees
 Comput. Geom. Theory Appl
, 1995
"... We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent i ..."
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Cited by 14 (0 self)
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We show that several wellknown optimization problems involving 3dimensional convex polyhedra and decision trees are NPhard or NPcomplete. One of the techniques we employ is a lineartime method for realizing a planar 3connected triangulation as a convex polyhedron, which may be of independent interest. Key words: Convex polyhedra, approximation, Steinitz's theorem, planar graphs, art gallery theorems, decision trees. 1 Introduction Convex polyhedra are fundamental geometric structures (e.g., see [20]). They are the product of convex hull algorithms, and are key components for problems in robot motion planning and computeraided geometric design. Moreover, due to a beautiful theorem of Steinitz [20, 38], they provide a strong link between computational geometry and graph theory, for Steinitz shows that a graph forms the edge structure of a convex polyhedra if and only if it is planar and 3connected. Unfortunately, algorithmic problems dealing with 3dimensional convex polyhedra ...
Convex Approximation by Spherical Patches
, 2006
"... 3 Partially supported by the IST Programme of the EU as a Sharedcost RTD (FET Open) Project under Contract No IST006413 (ACS – Algorithms for Complex Shapes) Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such t ..."
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Cited by 1 (1 self)
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3 Partially supported by the IST Programme of the EU as a Sharedcost RTD (FET Open) Project under Contract No IST006413 (ACS – Algorithms for Complex Shapes) Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such that all points are within some specified tolerance bound ε of the approximating surface. We describe a greedy algorithm which constructs an approximating surface whose spherical patches are associated to the faces of an inscribed polytope formed from a subset of the input points. We show that deciding whether an approximation with not more than a given number of spherical patches exists is NPhard by a reduction from planar 3SAT. 1
Greedy Cuts: An AdvancingFront Terrain Triangulation Algorithm
"... We propose advancingfront techniques for the problem of simplification of dense digitized terrain models. While most simplification algorithms have been based on either incremental refinement or decimation techniques, our GreedyCuts algorithms use a simple triangulationgrowth procedure. They work ..."
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We propose advancingfront techniques for the problem of simplification of dense digitized terrain models. While most simplification algorithms have been based on either incremental refinement or decimation techniques, our GreedyCuts algorithms use a simple triangulationgrowth procedure. They work by taking greedy cuts (“bites”) out of a simple closed polygon that bounds a connected component of the yettobe triangulated region. The method begins with a large polygon, bounding the whole extent of the terrain to be triangulated, and works its way inward, performing at each step one of three basic operations: ear cutting, greedy biting, and edge splitting. In this paper, we present both the basic GreedyCuts framework (which has been introduced in our earlier paper) and a new enhancement of the GreedyCuts method that improves the quality of the resulting triangulation. This improvement is made possible through the maintenance of two “fronts”, a real front and a virtual front, that bound between them a region of the terrain that has only a tentative triangulation. By allowing simple local operations (edge collapses and edge flips) in the tentative triangulation, we are able to avoid many of the artifacts of the basic GreedyCuts advancingfront technique, while not significantly affecting memory usage or running time. Our implementation of GreedyCuts, as well as its multifront enhancement is publicly available in the GcTin system. We give experimental evidence of the effectiveness of the multifront enhancement to the GreedyCuts method and show that our method is competitive with current algorithms in terms of running time. One of the major advantages of our implementation is that it requires very little memory beyond that for the input height array. 2 1
Octreebased Simplifications of Polyhedral Solids
, 1999
"... Automatic simplification of polyhedral objects is a major topic in many computer graphics applications. In this work, simplification algorithms for the generation of a multiresolution family of solid representations from an initial polyhedral solid are discussed. We introduce the Discretized Polyhed ..."
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Automatic simplification of polyhedral objects is a major topic in many computer graphics applications. In this work, simplification algorithms for the generation of a multiresolution family of solid representations from an initial polyhedral solid are discussed. We introduce the Discretized Polyhedra Simplification (DPS), a framework for polyhedra simplification using space decomposition models. DPS is based on a new error distance and provides a sound scheme for errorbounded, geometry and topology simplification while preserving the validity of the model. A method following this framework, trihedral DPS, is presented and discussed. Trihedral DPS uses an octree for topology simplification and error control, and generates valid solid representations. Our method is also able to generate approximations that do not interpenetrate the original model, either being completely contained in the input solid or bounding it. Unlike most of the current methods, restricted to triangle meshes, our ...
MINIMUM VERTEX HULLS FOR POLYHEDRAL DOMAINS 1
"... In this paper we investigate several variations of the following problem: Given a collection of pairwise disjoint polygons and their spatial positions in the plane, cover each with a polygonal hull such that (i) the hulls are pairwise disjoint, and ..."
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In this paper we investigate several variations of the following problem: Given a collection of pairwise disjoint polygons and their spatial positions in the plane, cover each with a polygonal hull such that (i) the hulls are pairwise disjoint, and
Spherical approximation of convex shapes
"... Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such that all points are within some specified tolerance bound ε of the approximating surface. We describe a greedy algorithm which constructs an approximating surfa ..."
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Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such that all points are within some specified tolerance bound ε of the approximating surface. We describe a greedy algorithm which constructs an approximating surface whose spherical patches are associated to the faces of an inscribed polytope formed from a subset of the input points. We show that deciding whether an approximation with not more than a given number of spherical patches exists is NPhard by a reduction from planar 3SAT.