Formal specification and proofs for the topology and classification of combinatorial surfaces

Dehlinger, Christophe; Dufourd, Jean-François

doi:10.1016/j.comgeo.2014.04.007

Cited by 3 publications

(2 citation statements)

References 15 publications

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“…The complexity of TGW for computing the unknown X d = A(S d−1 ) is obviously higher (see Section 3.3.3), and equates the standard in Solid Modeling. The main difference with our approach is that Alayrangues and colleagues start from a given Gmap cellular model, whose construction is quite complex and requires interactive operations with a graphical user interface or a symbolic logic systems (such as IN-RIA's Coq [85]) with a formal specification language. If simplicity metric matters, our linear algebraic representation of chains with sparse arrays compares well with chains of Gmaps.…”

Section: Relevant Literaturementioning

confidence: 99%

Topological computing of arrangements with (co)chains

Paoluzzi¹,

Shapiro²,

DiCarlo³

et al. 2019

Preprint

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In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences, adjacencies and ordering of cells, generally using disparate and often incompatible data structures and algorithms. This paper introduces computational topology algorithms to discover the 2D/3D space partition induced by a collection of geometric objects of dimension 1D/2D, respectively. Methods and language are those of basic geometric and algebraic topology. Only sparse vectors and matrices are used to compute both spaces and maps, i.e., the chain complex, from dimension zero to three.

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Section: Relevant Literaturementioning

confidence: 99%

Topological computing of arrangements with (co)chains

Paoluzzi¹,

Shapiro²,

DiCarlo³

et al. 2019

Preprint

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show abstract

“…This well-established model is highly regular, allowing for reasoning on the manipulation of subdivided objects. For instance, formal proofs with the Coq system have been studied in [18] with the help of generalized maps. Compared to generalized maps, oriented maps present the main advantage of a compact representation of objects to the detriment of regularity in dimension, making reasoning more difficult.…”

Section: Introductionmentioning

confidence: 99%

Topological consistency preservation with graph transformation schemes

Pascual

Gall

Arnould

et al. 2022

Science of Computer Programming

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Topological Computing of Arrangements with (Co)Chains

Paoluzzi

Shapiro

DiCarlo³

et al. 2020

ACM Trans. Spatial Algorithms Syst.

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In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences, adjacencies, and ordering of cells, generally using disparate and often incompatible data structures and algorithms. This article introduces computational topology algorithms to discover the two-dimensional (2D)/3D space partition induced by a collection of geometric objects of dimension 1D/2D, respectively. Methods and language are those of basic geometric and algebraic topology. Only sparse vectors and matrices are used to compute both spaces and maps, i.e., the chain complex, from dimension zero to three. The prototype software is written in Julia, the novel language for scientific computing. The applications may vary from 3D graphics to 3D printing, from images to scene understanding, and from games to building information modeling.

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Formal specification and proofs for the topology and classification of combinatorial surfaces

Cited by 3 publications

References 15 publications

Topological computing of arrangements with (co)chains

Topological computing of arrangements with (co)chains

Topological consistency preservation with graph transformation schemes

Topological Computing of Arrangements with (Co)Chains

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