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39
Accurate and fast proximity queries between polyhedra using convex surface decomposition
, 2001
"... The need to perform fast and accurate proximity queries arises frequently in physicallybased modeling, simulation, animation, realtime interaction within a virtual environment, and game dynamics. The set of proximity queries include intersection detection, tolerance verification, exact and approxi ..."
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Cited by 97 (14 self)
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The need to perform fast and accurate proximity queries arises frequently in physicallybased modeling, simulation, animation, realtime interaction within a virtual environment, and game dynamics. The set of proximity queries include intersection detection, tolerance verification, exact and approximate minimum distance computation, and (disjoint) contact determination. Specialized data structures and algorithms have often been designed to perform each type of query separately. We present a unified approach to perform any of these queries seamlessly for general, rigid polyhedral objects with boundary representations which are orientable 2manifolds. The proposed method involves a hierarchical data structure built upon a surface decomposition of the models. Furthermore, the incremental query algorithm takes advantage of coherence between successive frames. It has been applied to complex benchmarks and compares very favorably with earlier algorithms and systems. 1.
Strategies for Polyhedral Surface Decomposition: An Experimental Study
, 1995
"... This paper addresses the problem of decomposing a complex polyhedral surface into a small number of "convex" patches (ie, boundary parts of convex polyhedra). The corresponding optimization problem is shown to be NPcomplete and an experimental search for good heuristics is undertaken. 1 Introductio ..."
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Cited by 50 (5 self)
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This paper addresses the problem of decomposing a complex polyhedral surface into a small number of "convex" patches (ie, boundary parts of convex polyhedra). The corresponding optimization problem is shown to be NPcomplete and an experimental search for good heuristics is undertaken. 1 Introduction Convex shapes are easiest to represent, manipulate, and render. Even though they form the building blocks of bottomup solid modelers, it is more often the case that the convex structure of a geometric shape is lost in its representation. We are then presented, not with the solidmodeling problem of putting together primitive convex objects, but with the reverse problem of extracting convexity out of a complex shape. The classical example is that of cutting up a 3polyhedron into convex pieces. This is often a useful, sometimes a required, preprocessing step in graphics, manufacturing, and mesh generation. The problem has been exhaustively researched in the last few years [2][18]. Despi...
Meshing Piecewise Linear Complexes by Constrained Delaunay Tetrahedralizations
 In Proceedings of the 14th International Meshing Roundtable
, 2005
"... Summary. We present a method to decompose an arbitrary 3D piecewise linear complex (PLC) into a constrained Delaunay tetrahedralization (CDT). It successfully resolves the problem of nonexistence of a CDT by updating the input PLC into another PLC which is topologically and geometrically equivalent ..."
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Cited by 41 (2 self)
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Summary. We present a method to decompose an arbitrary 3D piecewise linear complex (PLC) into a constrained Delaunay tetrahedralization (CDT). It successfully resolves the problem of nonexistence of a CDT by updating the input PLC into another PLC which is topologically and geometrically equivalent to the original one and does have a CDT. Based on a strong CDT existence condition, the redefinition is done by a segment splitting and vertex perturbation. Once the CDT exists, a practically fast cavity retetrahedralization algorithm recovers the missing facets. This method has been implemented and tested through various examples. In practice, it behaves rather robust and efficient for relatively complicated 3D domains. 1
A Condition Guaranteeing the Existence of HigherDimensional Constrained Delaunay Triangulations
 Proceedings of the Fourteenth Annual Symposium on Computational Geometry
, 1998
"... Let X be a complex of vertices and piecewise linear constraining facets embedded in E d . Say that a simplex is strongly Delaunay if its vertices are in X and there exists a sphere that passes through its vertices but passes through and encloses no other vertex. Then X has a ddimensional constra ..."
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Cited by 38 (3 self)
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Let X be a complex of vertices and piecewise linear constraining facets embedded in E d . Say that a simplex is strongly Delaunay if its vertices are in X and there exists a sphere that passes through its vertices but passes through and encloses no other vertex. Then X has a ddimensional constrained Delaunay triangulation if each kdimensional constraining facet in X with k d \Gamma 2 is a union of strongly Delaunay ksimplices. This theorem is especially useful in E 3 for forming tetrahedralizations that respect specified planar facets. If the bounding segments of these facets are subdivided so that the subsegments are strongly Delaunay, then a constrained tetrahedralization exists. Hence, fewer vertices are needed than in the most common practice in the literature, wherein additional vertices are inserted in the relative interiors of facets to form a conforming (but unconstrained) Delaunay tetrahedralization. 1 Introduction Many applications can benefit from triangulations...
Polyhedral surface decomposition with applications
 Computers and Graphics
"... This paper addresses the problem of decomposing a polyhedral surface into “meaningful ” patches. We describe two decomposition algorithms – flooding convex decomposition and watershed decomposition, and show experimental results. Moreover, we discuss three applications which can highly benefit from ..."
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Cited by 36 (5 self)
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This paper addresses the problem of decomposing a polyhedral surface into “meaningful ” patches. We describe two decomposition algorithms – flooding convex decomposition and watershed decomposition, and show experimental results. Moreover, we discuss three applications which can highly benefit from surface decomposition. These applications include contentbased retrieval of threedimensional models, metamorphosis of threedimensional models and simplification.
Quadricbased simplification in any dimension
 ACM TRANS GRAPH
, 2005
"... We present a new method for simplifying simplicial complexes of any type embedded in Euclidean spaces of any dimension. At the heart of this system is a novel generalization of the quadric error metric used in surface simplification. We demonstrate that our generalized simplification system can pr ..."
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Cited by 35 (2 self)
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We present a new method for simplifying simplicial complexes of any type embedded in Euclidean spaces of any dimension. At the heart of this system is a novel generalization of the quadric error metric used in surface simplification. We demonstrate that our generalized simplification system can produce high quality approximations of plane and space curves, triangulated surfaces, and tetrahedralized volume data. Our method is both efficient and easy to implement. It is capable of processing complexes of arbitrary topology, including nonmanifolds, and can preserve intricate boundaries.
On Triangulating ThreeDimensional Polygons
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS
, 1996
"... A threedimensional polygon is triangulable if it has a nonselfintersecting triangulation which defines a simplyconnected 2manifold. We show that the problem of deciding whether a 3dimensional polygon is triangulable is NPComplete. We then establish some necessary conditions and some sufficie ..."
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Cited by 29 (3 self)
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A threedimensional polygon is triangulable if it has a nonselfintersecting triangulation which defines a simplyconnected 2manifold. We show that the problem of deciding whether a 3dimensional polygon is triangulable is NPComplete. We then establish some necessary conditions and some sufficient conditions for a polygon to be triangulable, providing special cases when the decision problem may be answered in polynomial time.
Safe Constraint Queries
 SIAM J. Comput
, 1998
"... ing with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Publications Dept, ACM Inc., fax +1 (212) 8690481, or permissions@acm.org. Safe Constraint Queries Michael Be ..."
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Cited by 27 (7 self)
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ing with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Publications Dept, ACM Inc., fax +1 (212) 8690481, or permissions@acm.org. Safe Constraint Queries Michael Benedikt Bell Laboratories 1000 E Warrenville Rd Naperville, IL 60566 Email: benedikt@research.belllabs.com Leonid Libkin Bell Laboratories 600 Mountain Avenue Murray Hill, NJ 07974 Email: libkin@research.belllabs.com Abstract We extend some of the classical characterization theorems of the relational theory  particularly those related to query safety  to the context where database elements come with fixed interpreted structure, and where formulae over elements of that structure can be used in queries. We show that the addition of common interpreted functions such as real addition and multiplication to the relational calculus preserves important characterization theorems ...
Interactive visualization of unstructured grids using hierarchical 3d textures
 In IEEE/SIGGRAPH Symposium and Volume Visualization and Graphics 2002
, 2002
"... planes rendered. Right: Refined to 98 textures with extra planes. We present a system for interactively rendering large, unstructured grids. Our approach is to voxelize the grid into a 3D voxel octree, and then to render the data using hierarchical, 3D texture mapping. This approach leverages the cu ..."
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Cited by 23 (3 self)
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planes rendered. Right: Refined to 98 textures with extra planes. We present a system for interactively rendering large, unstructured grids. Our approach is to voxelize the grid into a 3D voxel octree, and then to render the data using hierarchical, 3D texture mapping. This approach leverages the current 3D texture mapping PC hardware for the problem of unstructured grid rendering. We specialize the 3D texture octree to the task of rendering unstructured grids through a novel pad and stencil algorithm, which distinguishes between data and nondata voxels. Both the voxelization and rendering processes efficiently manage large, outofcore datasets. The system manages cache usage in main memory and texture memory, as well as bandwidths among disk, main memory, and texture memory. It also manages rendering load to achieve interactivity at all times. It maximizes a quality metric for a desired level of interactivity. It has been applied to a number of large data and produces high quality images at interactive, userselectable frame rates using standard PC hardware. 1
A fast solver for the Stokes equations with distributed forces in complex geometries
 J. Comput. Phys
"... We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded ..."
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Cited by 20 (9 self)
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We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded Boundary Integral method, is based on Anita Mayo’s work for the Poisson’s equation: “The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions”, SIAM Journal on Numerical Analysis, 21 (1984), pp. 285–299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting equations are discretized by Nyström’s method. The rectangular domain problem is discretized by finite elements for a velocitypressure formulation with equal order interpolation bilinear elements (£¥ ¤£¥ ¤). Stabilization is used to circumvent the ¦¨§�©������� � condition for the pressure space. For the integral equations, fast matrix vector multiplications are achieved via an ���¨���� � algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsify lowrank blocks. The regular grid solver is a Krylov method (Conjugate Residuals) combined with an optimal twolevel Schwartzpreconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates. Key Words: Stokes equations, fast solvers, integral equations, doublelayer potential, fast multipole methods, embedded domain methods, immersed interface methods, fictitious