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47
Stellar Subdivision Grammars
, 2003
"... In this paper we develop a new description for subdivision surfaces based on a graph grammar formalism. Subdivision schemes are specified by a context sensitive grammar in which production rules represent topological and geometrical transformations to the surface’s control mesh. This methodology can ..."
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Cited by 12 (2 self)
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In this paper we develop a new description for subdivision surfaces based on a graph grammar formalism. Subdivision schemes are specified by a context sensitive grammar in which production rules represent topological and geometrical transformations to the surface’s control mesh. This methodology can be used for all known subdivision surface schemes. Moreover, it gives an effective representation that allows simple implementation and is suitable for adaptive computations.
Simplex and Diamond Hierarchies: Models and Applications
, 2010
"... Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through simplex bisection. Such decompositions, originally developed for finite elements, are extensively used ..."
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Cited by 11 (4 self)
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Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through simplex bisection. Such decompositions, originally developed for finite elements, are extensively used as the basis for multiresolution models of scalar fields, such as terrains, and static or timevarying volume data. They have also been used as an alternative to quadtrees and octrees as spatial access structures and in other applications. In this state of the art report, we distinguish between approaches that focus on a specific dimension and those that apply to all dimensions. The primary distinction among all such approaches is whether they treat the simplex or clusters of simplexes, called diamonds, as the modeling primitive. This leads to two classes of data structures and to different query approaches. We present the hierarchical models in a dimension–independent manner, and organize the description of the various applications, primarily interactive terrain rendering and isosurface extraction, according to the dimension of the domain.
Flipping Cubical Meshes
 ACM Computer Science Archive June
, 2001
"... We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation. ..."
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Cited by 10 (0 self)
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We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.
Simplicial Isosurface Compression
 Proc. Vision, Modeling, and Visualization Conf
, 2004
"... In this work, we introduce a new algorithm for direct and progressive encoding of isosurfaces extracted from volumetric data. A binary multi– triangulation is used to represent and adapt the 3D scalar grid. This simplicial scheme provides geometrical and topological control on the decoded isosurface ..."
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Cited by 8 (0 self)
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In this work, we introduce a new algorithm for direct and progressive encoding of isosurfaces extracted from volumetric data. A binary multi– triangulation is used to represent and adapt the 3D scalar grid. This simplicial scheme provides geometrical and topological control on the decoded isosurface. The resulting algorithm is an efficient and flexible isosurface compression scheme. 1
ON THE INFINITESIMAL RIGIDITY OF POLYHEDRA WITH VERTICES IN CONVEX POSITION
"... Abstract. Let P ⊂ R 3 be a polyhedron. It was conjectured that if P is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. P can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a wea ..."
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Cited by 7 (2 self)
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Abstract. Let P ⊂ R 3 be a polyhedron. It was conjectured that if P is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. P can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additionnal assumption of codecomposability. The proof relies on a result of independent interest concerning the HilbertEinstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In particular, if there are no interior vertices, then the Hessian is negative definite. 1.
Relations in Grassmann Algebra Corresponding to Three and FourDimensional Pachner Moves
"... Abstract. New algebraic relations are presented, involving anticommuting Grassmann variables and Berezin integral, and corresponding naturally to Pachner moves in three and four dimensions. These relations have been found experimentally – using symbolic computer calculations; their essential new fea ..."
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Cited by 6 (5 self)
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Abstract. New algebraic relations are presented, involving anticommuting Grassmann variables and Berezin integral, and corresponding naturally to Pachner moves in three and four dimensions. These relations have been found experimentally – using symbolic computer calculations; their essential new feature is that, although they can be treated as deformations of relations corresponding to torsions of acyclic complexes, they can no longer be explained in such terms. In the simpler case of three dimensions, we define an invariant, based on our relations, of a piecewiselinear manifold with triangulated boundary, and present example calculations confirming its nontriviality.
Invariants of threedimensional manifolds from fourdimensional Euclidean geometry
"... Abstract. This is the first in a series of papers where we will derive invariants of threemanifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a fourdimensional Euclidean space. Thus, the elements of the pseudotriangulation acquire Euclidean geom ..."
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Abstract. This is the first in a series of papers where we will derive invariants of threemanifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a fourdimensional Euclidean space. Thus, the elements of the pseudotriangulation acquire Euclidean geometric values such as volumes of different dimensions and various kinds of angles. Then we construct an acyclic complex made of differentials of these geometric values, and its torsion will lead, depending on the specific kind of this complex, to some manifold or knot invariants. In this paper, we limit ourselves to constructing a simplest kind of acyclic complex, from which a threemanifold invariant can be obtained. 1.
Minimal Triangulations of Manifolds
, 2007
"... Finding vertexminimal triangulations of closed manifolds is a very difficult problem. Except for spheres and two series of manifolds, vertexminimal triangulations are known for only few manifolds of dimension more than 2 (see the table given at the end of Section 5). In this article, we present ..."
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Finding vertexminimal triangulations of closed manifolds is a very difficult problem. Except for spheres and two series of manifolds, vertexminimal triangulations are known for only few manifolds of dimension more than 2 (see the table given at the end of Section 5). In this article, we present a brief survey on the works done in last 30 years on the following: (i) Finding the minimal number of vertices required to triangulate a given pl manifold. (ii) Given positive integers n and d, construction of nvertex triangulations of different ddimensional pl manifolds. (iii) Classifications of all the triangulations of a given pl manifold with same number of vertices. In Section 1, we have given all the definitions which are required for the remaining part of this article. A reader can start from Section 2 and come back to Section 1 as and when required. In Section 2, we have presented a very brief history of triangulations of manifolds. In Section 3, we have presented examples of several vertexminimal triangulations. In Section 4, we have presented some interesting results on triangulations of manifolds. In particular, we have stated the Lower Bound Theorem and the Upper Bound Theorem. In Section 5, we have stated several results on minimal triangulations without proofs. Proofs are available in the references mentioned there. We have also presented some open problems/conjectures in Sections 3 and 5.
Relational specification of surface subdivision algorithms
 In Proceedings of AGTIVE 2003
, 2003
"... Many polygon mesh algorithms operate in a local manner, yet are formally specified using global indexing schemes. We address this discrepancy by defining a set of local operations on polygon meshes in relational, indexfree terms. We also introduce the vv programming language to express these operat ..."
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Cited by 4 (3 self)
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Many polygon mesh algorithms operate in a local manner, yet are formally specified using global indexing schemes. We address this discrepancy by defining a set of local operations on polygon meshes in relational, indexfree terms. We also introduce the vv programming language to express these operations in a machinereadable form. We then apply vv to specify several surface subdivision algorithms. These specifications can be directly executed by the corresponding modeling software. 1
Euclidean geometric invariant of framed knots in manifolds. arXiv:math.GT/0605164
"... Abstract. We present an invariant of a three–dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. An important feature of ou ..."
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Abstract. We present an invariant of a three–dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. An important feature of our work is that we are not using any nontrivial representation of the manifold fundamental group or knot group. 1.