Dimension-independent modeling with simplicial complexes

Paoluzzi, Alberto; Bernardini, Fausto; Cattani, Carlo; Ferrucci, Vincenzo

doi:10.1145/169728.169719

Cited by 91 publications

(46 citation statements)

References 25 publications

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“…Let us mention here just some of them. Multidimensional simplicial complexes are used in [23] for dimension-independent geometric modeling for various applications. A representation of s-sets, i.e.…”

Section: Other Workmentioning

confidence: 99%

Cellular-functional modeling of heterogeneous objects

Adzhiev

Kartasheva

Kunii

et al. 2002

Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications

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The paper presents an approach to modeling heterogeneous objects as multidimensional point sets with multiple attributes (hypervolumes). A theoretical framework is based on a hybrid model of hypervolumes combining a cellular representation and a constructive representation using real-valued functions. This model allows for independent but unifying representation of geometry and attributes, and makes it possible to represent dimensionally non-homogeneous entities and their cellular decompositions. Hypervolume model components such as objects, operations and relations are introduced and outlined. The framework's inherent multidimensionality allowing, in particular, to deal naturally with time dependence promises to model complex dynamic objects composed of different materials with constructive building of their geometry and attributes. Attributes given at each point can represent properties of arbitrary nature (material, photometric, physical, statistical, etc.). To demonstrate a particular application of the proposed framework, we present an example of multimaterial modeling -a multilayer geological structure with cavities and wells. Another example illustrating the treatment of attributes other than material distributions is concerned with time-dependent adaptive mesh generation where function representation is used to describe object geometry and density of elements in the cellular model of the mesh. The examples have been implemented by using a specialized modeling language and software tools being developed by the authors.

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Section: Other Workmentioning

confidence: 99%

Cellular-functional modeling of heterogeneous objects

Adzhiev

Kartasheva

Kunii

et al. 2002

Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications

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“…One natural approach to the representation of objects in four or higher dimensions is based directly on classical mathematical methods in algebraic topology: this is the method of Regular Simplicial Complexes 4,11,20,21]. The basic building block used here is the simplex, which is the convex combination of n + 1 a nely independent points; combinations of such simplices satisfying certain wellformedness conditions are called complexes 11].…”

Section: Object Representationmentioning

confidence: 99%

“…For example, objects in E 4 can be represented by Regular Simplicial Complexes 4,11,20,21,22,23], with data structures that can be viewed as extensions of the standard winged-edge representation; alternatively, very general classes of objects in E 4 can be represented by using Selective Geometric Complexes 29].…”

Section: Introductionmentioning

confidence: 99%

Moving coordinate frames for representation and visualization in four dimensions

Egli

Petit

Stewart

1996

Computers & Graphics

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In this paper we generalize the method of sweeping, which may be used for both object representation and object visualization, to objects in E 4 . The generalized sweeping method involves a trajectory in E 4 , and a three-dimensional cross-section orthogonal to the trajectory. Three alternatives are considered for specifying the relationship between the trajectory and the cross-section, namely the Frenet frame, modi ed Frenet frames, and the standard viewing-reference coordinate (VRC) frame, where each of these is generalized to the case of four dimensions. We have implemented the method using the four-dimensional VRC frame, and the method is illustrated here by means of two examples involving visualization of objects embedded in E 4 .

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“…Although most work in the geometric modeling literature has been aimed at representing just three-dimensional manifold objects, several authors have pointed out the need of developing more general data structures, which can represent also higher dimensional and/or non-manifold and non-uniformly dimensional objects [8,15,10]. Non-manifold singularities in modeled objects occurs as a side-effect of feature extraction from images, 3D reconstruction or as a byproduct of severe discretization.…”

Section: Introductionmentioning

confidence: 99%

A Representation for Abstract Simplicial Complexes: An Analysis and a Comparison

Floriani

Morando

Puppo

2003

Discrete Geometry for Computer Imagery

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Abstract. Abstract simplicial complexes are used in many application contexts to represent multi-dimensional, possibly non-manifold and nonuniformly dimensional, geometric objects. In this paper we introduce a new general yet compact data structure for representing simplicial complexes, which is based on a decomposition approach that we have presented in our previous work [3]. We compare our data structure with the existing ones and we discuss in which respect it performs better than others.

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Dimension-independent modeling with simplicial complexes

Cited by 91 publications

References 25 publications

Cellular-functional modeling of heterogeneous objects

Cellular-functional modeling of heterogeneous objects

Moving coordinate frames for representation and visualization in four dimensions

A Representation for Abstract Simplicial Complexes: An Analysis and a Comparison

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