Topology of Surfaces

Kinsey, L. Christine

doi:10.1007/978-1-4612-0899-0

Cited by 68 publications

(11 citation statements)

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“…Several equivalent definitions of topology on a set exist [Munkres 1975;Jänich 1983], and one such definition that is of relevance to solid modeling is based on the notion of nearness or neighborhood [Requicha 1980;Kinsey 1993]. In this definition, a topological space is a set X with a collection B of subsets N ⊆ X, called neighborhoods, such that:…”

Section: Finite Intersection Of the Sets Is An Open Setmentioning

confidence: 99%

Constructive topological representations

Raghothama

2006

Proceedings of the 2006 ACM Symposium on Solid and Physical Modeling

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Constructive representations, such as Constructive Solid Geometry (CSG) and its various feature-based extensions are inherently parametric in nature and are well suited for defining parametric family of solids. On the other hand, cell complex representations contain explicit shape elements (cells) and also their topology. However they are non-constructive, difficult to parameterize, and it is extremely difficult to enforce continuity in the usual cell complex topology (considered as a sub-space of Euclidean space). When cell complexes are used in conjunction with constructive representations, even if we can enforce continuity in limited cases, it is impossible to relate cellular operations with global semantics of constructive operations (such as Boolean and feature operations).Using the framework developed in our earlier work for defining part families on any solid representation, we propose constructive topological representations by identifying every constructive representation with its corresponding unique spatial decomposition. By applying the proposed definitions to spatial CSG representations we will systematically develop algorithms for topologizing a CSG and also enforcing continuity between two given topologized CSG representations. These algorithms have been implemented in a prototype system about which we will briefly discuss. Finally, we will illustrate some interesting applications of the proposed constructive topological representations: specification of constructive families and the enforcement of global semantics of feature operations.

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Section: Finite Intersection Of the Sets Is An Open Setmentioning

confidence: 99%

Constructive topological representations

Raghothama

2006

Proceedings of the 2006 ACM Symposium on Solid and Physical Modeling

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“…We start by defining some topological notions. For more background on topology refer to [RF90,Kin93,LW69,Moi77,Jan84].…”

Section: Barycentric Complexes and Mapsmentioning

confidence: 99%

Representing Topological Structures Using Cell-Chains

Cardoze

Phillips

2006

Geometric Modeling and Processing - GMP 2006

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Abstract.A new topological representation of surfaces in higher dimensions, "cell-chains" is developed. The representation is a generalization of Brisson's cell-tuple data structure. Cell-chains are identical to celltuples when there are no degeneracies: cells or simplices with identified vertices. The proof of correctness is based on axioms true for maps, such as those in Brisson's cell-tuple representation. A critical new condition (axiom) is added to those of Lienhardt's n-G-maps to give "cell-maps". We show that cell-maps and cell-chains characterize the same topological representations.

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“…This is exactly the case in Figure 1(c) where face fl and all of its bounding edges collapsed into the edge es and its bounding-vertices. This cell-by-cell continuous deformation is captured by the notion of a continuous cell map (modifkl from [12]). …”

Section: Cell Maps and Cell Homomorphismsmentioning

confidence: 99%

“…gp(alal +... +anan) = algp(m)+.. .+angp(an).Then it is known[1, 3,12] that the family of such chain maps satisfy the following commutative diagram.…”

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confidence: 93%

Necessary conditions for boundary representation variance

Raghothama

Shapiro

1997

Proceedings of the Thirteenth Annual Symposium on Computational Geometry - SCG '97

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One of the key unsolved problems in parametric solid modeling is a robust update (regeneration) of the solid's boundary representation, given a specified change in solid's parameter values. The fundamental difficulty lies in determining the mapping between boundary representations for solids in the same parametric family, also known as "persistent naming." We formulate the notion of Boundary Representation (BR-)variance for solids in the same parametric family based on the assumption of continuity: small changes in solid's parameter values should result in small changes in solid's boundary rep reeentation, which may include local collapses of cells in boundary representations. We show that our formulation gives the necessary conditions that must be satisfied by any BR-variant mappings between boundary representations. These conditions are powerful enough to identify invalid updates in many practical situations and provide a formal criteria for the recently proposed heuristic approaches to the "persistent naming" problem.1 Motivation Ambiguity of parametric solid modelingFrom its very inception solid modeling has been synonymous with unambiguous (informationally complete) representations of homogeneously n-dimensional subsets of Euclidean space PO,9,8]. On the other hand, the recent rise of solid modeling as a principal information medium first in engineering and now in con-"Complete address: 1513 University Avenue, Madison, sumer applications probably has to do more with development and successful marketing of new parametric ("feature-based" and "constraint-based" ) user interfaces than with the mathematical soundness of solid modeling systems. Th@e parametric interfaces allow user to define and modify solid models in terms of highlevel parametric definitions that are constructed to have intuitive and appealing meaning to the user and/or ap plication (see [11] for examples and references). The success of the parametric solid modeling came at a high price: the new solid modeling systems no longer guarantee that the parametric models are valid or unambiguous, and the results of modeling operations are not always predictable.

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Topology of Surfaces

Cited by 68 publications

References 0 publications

Constructive topological representations

Constructive topological representations

Representing Topological Structures Using Cell-Chains

Necessary conditions for boundary representation variance

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