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Triangular mesh parameterization with trimmed surfaces

  • Oscar E. RuizEmail author
  • Daniel Mejia
  • Carlos A. Cadavid
Original Paper

Abstract

Given a 2-manifold 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 1-manifold 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 i-th hole in \(F\) (if any). The problem of parameterizing triangular meshes is relevant for reverse engineering, tool path planning, feature detection, re-design, etc. State-of-art 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 pre-processing. 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 non-developable. Ongoing work is being devoted to manifold segmentation which facilitates mesh parameterization.

Keywords

Triangular mesh parameterization Trimmed surface Manifold learning NURBS RBFs 

Abbreviations

LOOP

Closed (piecewise linear or smooth) curve lying on a surface, and bounding a connected region on the surface. In this manuscript, LOOPs are denoted with \(\Gamma \) or \(\gamma \)

B-REP

Boundary representation

HLLE

Hessian locally linear embedding

NURBS

Non-uniform rational B-spline

RBF

Radial basis function

\(M\)

Triangular mesh (with boundary), composed by the set of triangles \(T=\{t_1,t_2,\ldots ,t_q\}\) with vertex set \(X=\{x_1,x_2,\ldots ,x_n\}\) (\(X \subset {\mathbb {R}}^3\))

\(\partial M\)

Boundary of \(M\), whose connected components are LOOPs (\(\partial M = \{\Gamma _0, \Gamma _1,\ldots ,\Gamma _k \}\))

\(\phi \)

An homeomorphic map \(\phi :M\rightarrow {\mathbb {R}}^2\), implemented here for dimensional reduction or manifold learning. \(\phi _{Isom}()\), \(\phi _{Lapl}()\), \(\phi _{\textit{HLLE}}()\), are the Isomap, Laplacian Eigenmap and Hessian locally linear embedding implementations, respectively. \(\phi ()\) is called forward map in this manuscript

\(U\)

\(U=\{u_1,u_2,\ldots ,u_n\}\) is the parametric image of vertices of \(M\) (\(U=\phi (X)\), \(U \subset {\mathbb {R}}^2\))

\(\partial (\phi (M))\)

Boundary of the parametric image of \(M\). For the sake of simplicity, we assume that \(\partial (\phi (M)) = \phi (\partial (M))\))

\(\gamma _i\)

i-th LOOPs of \(\partial (\phi (M))\)

\(\lambda _i\)

Re-sampling of a LOOP \(\gamma _i\)

\(W\)

Rectangular grid in \(R^2\) such that \(U\) lies in the convex hull of \(W\)

\(H(W)\)

Rectangular point set in \(R^2\) being the convex hull of \(W\)

\(P\)

Rectangular grid in \({\mathbb {R}}^3\) being the control polyhedron for the parametric surface \(f\)

\(f\)

Function \(f: W \rightarrow {\mathbb {R}}^3\) produces \(P\) the control polyhedron of \(S\) (\(P=f(W)\)) by calculating an extrapolation of \(M\) in \({\mathbb {R}}^3\)

\(S\)

\(S:{\mathbb {R}}^2 \rightarrow {\mathbb {R}}^3\) is a parametric surface which approximates and extends \(M\) in \({\mathbb {R}}^3\). \(S()\) is called backward map in this manuscript. To simplify notation, \(S\) refers here to both: (1) the parametric mapping (i.e., \(S()\)) and (2) the set of points product of the mapping \(S( )\) (i.e., \(S(H(W))= \{ S(w_1,w_2) | (w_1,w_2) \in H(W)\}\))

\(L_i\)

Trimming curve in \(M \subset {\mathbb {R}}^3\) defined as \(L_i = S(\lambda _i)\)

\(F\)

Trimmed surface (FACE) such that \(F \!=\! (S,\{L_0,L_1,\ldots \})\)

\(\partial F\)

Boundary of \(F\) approximated by the union of all \(L_i\)

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Copyright information

© Springer-Verlag France 2015

Authors and Affiliations

  • Oscar E. Ruiz
    • 1
    Email author
  • Daniel Mejia
    • 1
  • Carlos A. Cadavid
    • 1
  1. 1.Laboratorio de CAD CAM CAEUniversidad EAFITMedellínColombia

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