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Scaffolding a Skeleton

  • Athina Panotopoulou
  • Elissa Ross
  • Kathrin Welker
  • Evelyne Hubert
  • Géraldine Morin
Chapter
Part of the Association for Women in Mathematics Series book series (AWMS, volume 12)

Abstract

The goal of this paper is to construct a quadrilateral mesh around a one-dimensional skeleton that is as coarse as possible, the “scaffold.” A skeleton allows one to quickly describe a shape, in particular a complex shape of high genus. The constructed scaffold is then a potential support for the surface representation: it provides a topology for the mesh, a domain for parametric representation (a quad-mesh is ideal for tensor product splines), or, together with the skeleton, a grid support on which to project an implicit surface that is naturally defined by the skeleton through convolution. We provide a constructive algorithm to derive a quad-mesh scaffold with topologically regular cross-sections (which are also quads) and no T-junctions. We show that this construction is optimal in the sense that no coarser quad-mesh with topologically regular cross-sections may be constructed. Finally, we apply an existing rotation minimization algorithm along the skeleton branches, which produces a mesh with a natural edge flow along the shape.

Notes

Acknowledgements

This work has been partly supported by the NSF DMS-1619759 grant and the French Research National Agency (ANR) program CIMI, ANR-11-LABX-0040-CIMI and the German Research Foundation (DFG) within the priority program SPP 1962 under contract number Schu804/15-1.

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Copyright information

© The Author(s) and the Association for Women in Mathematics 2018

Authors and Affiliations

  • Athina Panotopoulou
    • 1
  • Elissa Ross
    • 2
  • Kathrin Welker
    • 3
  • Evelyne Hubert
    • 4
  • Géraldine Morin
    • 5
  1. 1.Dartmouth CollegeHanoverUSA
  2. 2.MESH Consultants Inc.Fields Institute for Research in the Mathematical SciencesTorontoCanada
  3. 3.Trier UniversityTrierGermany
  4. 4.INRIA MéditerranéeSophia AntipolisFrance
  5. 5.Toulouse Institute of Computer Science RToulouse, Garonne (Haute)France

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