Triangular mesh parameterization with trimmed surfaces

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dc.citation.epage316spa
dc.citation.issue4spa
dc.citation.journalAbbreviatedTitleIJIDeMspa
dc.citation.journalTitleInternational Journal on Interactive Design and Manufacturing (IJIDeM)eng
dc.citation.journalTitleInternational Journal on Interactive Design and Manufacturingspa
dc.citation.spage303spa
dc.citation.volume9spa
dc.contributor.authorRuíz, Óscar E.
dc.contributor.authorMejía, Daniel
dc.contributor.authorCadavid, Carlos A.
dc.contributor.departmentUniversidad EAFIT. Departamento de Ingeniería Mecánicaspa
dc.contributor.researchgroupLaboratorio CAD/CAM/CAEspa
dc.date.accessioned2016-10-24T23:07:02Z
dc.date.available2016-10-24T23:07:02Z
dc.date.issued2015
dc.description.abstractGiven a 2manifold triangular mesh \(M \subset {\mathbb {R}}^3\), with border, a parameterization of \(M\) is a FACE or trimmed surface \(F=\{S,L_0,\ldots, L_m\}\) -- \(F\) is a connected subset or region of a parametric surface \(S\), bounded by a set of LOOPs \(L_0,\ldots ,L_m\) such that each \(L_i \subset S\) is a closed 1manifold having no intersection with the other \(L_j\) LOOPs -- The parametric surface \(S\) is a statistical fit of the mesh \(M\) -- \(L_0\) is the outermost LOOP bounding \(F\) and \(L_i\) is the LOOP of the ith hole in \(F\) (if any) -- The problem of parameterizing triangular meshes is relevant for reverse engineering, tool path planning, feature detection, redesign, etc -- Stateofart mesh procedures parameterize a rectangular mesh \(M\) -- To improve such procedures, we report here the implementation of an algorithm which parameterizes meshes \(M\) presenting holes and concavities -- We synthesize a parametric surface \(S \subset {\mathbb {R}}^3\) which approximates a superset of the mesh \(M\) -- Then, we compute a set of LOOPs trimming \(S\), and therefore completing the FACE \(F=\ {S,L_0,\ldots ,L_m\}\) -- Our algorithm gives satisfactory results for \(M\) having low Gaussian curvature (i.e., \(M\) being quasi-developable or developable) -- This assumption is a reasonable one, since \(M\) is the product of manifold segmentation preprocessing -- Our algorithm computes: (1) a manifold learning mapping \(\phi : M \rightarrow U \subset {\mathbb {R}}^2\), (2) an inverse mapping \(S: W \subset {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^3\), with \ (W\) being a rectangular grid containing and surpassing \(U\) -- To compute \(\phi\) we test IsoMap, Laplacian Eigenmaps and Hessian local linear embedding (best results with HLLE) -- For the back mapping (NURBS) \(S\) the crucial step is to find a control polyhedron \(P\), which is an extrapolation of \(M\) -- We calculate \(P\) by extrapolating radial basis functions that interpolate points inside \(\phi (M)\) -- We successfully test our implementation with several datasets presenting concavities, holes, and are extremely nondevelopable -- Ongoing work is being devoted to manifold segmentation which facilitates mesh parameterizationeng
dc.formatapplication/pdfeng
dc.identifier.doi10.1007/s12008-015-0276-1
dc.identifier.issn1955-2513
dc.identifier.urihttp://hdl.handle.net/10784/9544
dc.language.isoengeng
dc.publisherSpringer Verlagspa
dc.relation.ispartofInternational Journal on Interactive Design and Manufacturing (IJIDeM), Volume 9, Issue 4, pp 303-316spa
dc.relation.urihttp://link.springer.com/article/10.1007/s12008-015-0276-1
dc.rightsinfo:eu-repo/semantics/closedAccess
dc.rights.localAcceso cerradospa
dc.subject.keywordManifolds (Mathematics)spa
dc.subject.keywordGaussian quadrature formulasspa
dc.subject.keywordAlgorithmsspa
dc.subject.keywordNumerical grid generation (Numerical analysis)spa
dc.subject.keywordManifolds (Mathematics)eng
dc.subject.keywordGaussian quadrature formulaseng
dc.subject.keywordAlgorithmseng
dc.subject.keywordNumerical grid generation (Numerical analysis)eng
dc.subject.keywordSuperficies NURBS.keywor
dc.subject.keywordSuperficies B-Splines Racionales no Uniformes (NURBS).keywor
dc.subject.keywordSistemas CAD/CAM.keywor
dc.subject.keywordIngeniería inversa.keywor
dc.subject.lembVARIEDADES (MATEMÁTICAS)spa
dc.subject.lembCUADRATURA DE GAUSSspa
dc.subject.lembALGORITMOSspa
dc.subject.lembGENERACIÓN NUMÉRICA DE MALLAS (ANÁLISIS NUMÉRICO)spa
dc.titleTriangular mesh parameterization with trimmed surfaceseng
dc.typeinfo:eu-repo/semantics/articleeng
dc.typearticleeng
dc.typeinfo:eu-repo/semantics/publishedVersioneng
dc.typepublishedVersioneng
dc.type.localArtículospa

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