Geometric Integration Theory

Whitney, Hassler

doi:10.1515/9781400877577

Cited by 970 publications

(556 citation statements)

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“…In fact, deRham's original work on the topology of differential forms effectively defined cochains as a way of discretizing differential forms [38]. Whitney showed how co-chains could be interpreted as continuous forms [48]. We will use the term one-form to emphasize this connection, as has been done before by others [5,32,21,18].…”

Section: Related Workmentioning

confidence: 98%

Discrete one-forms on meshes and applications to 3D mesh parameterization

Gortler

Gotsman

Thurston

2006

Computer Aided Geometric Design

109

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We describe how some simple properties of discrete one-forms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "spring-embedding" theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be non-convex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being non-convex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.

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Section: Related Workmentioning

confidence: 98%

Discrete one-forms on meshes and applications to 3D mesh parameterization

Gortler

Gotsman

Thurston

2006

Computer Aided Geometric Design

109

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“…The Whitney map [24,60] is an example of a conforming reconstruction operator whose range is in the Sobolev space Λ 1 (d, Ω). In contrast to the previous two reconstruction operators, the Whitney map builds a polynomial 1-form on c 2 from the cochain c 1 using a basis of polynomial 1-forms associated with the edges c i 1 .…”

Section: Operationmentioning

confidence: 99%

Principles of Mimetic Discretizations of Differential Operators

Bochev

Hyman

The IMA Volumes in Mathematics and Its Applications

158

206

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Abstract. Compatible discretizations transform partial differential equations to discrete algebraic problems that mimic fundamental properties of the continuum equations. We provide a common framework for mimetic discretizations using algebraic topology to guide our analysis. The framework and all attendant discrete structures are put together by using two basic mappings between differential forms and cochains. The key concept of the framework is a natural inner product on cochains which induces a combinatorial Hodge theory on the cochain complex. The framework supports mutually consistent operations of differentiation and integration, has a combinatorial Stokes theorem, and preserves the invariants of the De Rham cohomology groups. This allows, among other things, for an elementary calculation of the kernel of the discrete Laplacian. Our framework provides an abstraction that includes examples of compatible finite element, finite volume, and finite difference methods. We describe how these methods result from a choice of the reconstruction operator and explain when they are equivalent. We demonstrate how to apply the framework for compatible discretization for two scalar versions of the Hodge Laplacian.

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“…Then using a variant of Whitney's triangulation technique from [13], one can approximate the image of X by n-dimensional Euclidean polyhedra. But it is not clear how to construct the short maps between the various spaces, or even if the metrics of the polyhedra converge to the metric of X in the limit.…”

Section: Corollary 4 a Proper Metric Space X Admits An Intrinsic Isomentioning

confidence: 99%

Isometric Embeddings of Pro-Euclidean Spaces

Minemyer

2015

Analysis and Geometry in Metric Spaces

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Abstract:In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into E n if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a "nice" inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E n+ . Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C version of) the famous Nash isometric embedding theorem from [10].

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Geometric Integration Theory

Cited by 970 publications

References 0 publications

Discrete one-forms on meshes and applications to 3D mesh parameterization

Discrete one-forms on meshes and applications to 3D mesh parameterization

Principles of Mimetic Discretizations of Differential Operators

Isometric Embeddings of Pro-Euclidean Spaces

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