Engineering with Computers

, Volume 30, Issue 3, pp 289–300 | Cite as

Fitting polynomial surfaces to triangular meshes with Voronoi squared distance minimization

  • Vincent NivoliersEmail author
  • Dong-Ming Yan
  • Bruno Lévy
Original Article


This paper introduces Voronoi squared distance minimization (VSDM), an algorithm that fits a surface to an input mesh. VSDM minimizes an objective function that corresponds to a Voronoi-based approximation of the overall squared distance function between the surface and the input mesh (SDM). This objective function is a generalization of the one minimized by centroidal Voronoi tessellation, and can be minimized by a quasi-Newton solver. VSDM naturally adapts the orientation of the mesh elements to best approximate the input, without estimating any differential quantities. Therefore, it can be applied to triangle soups or surfaces with degenerate triangles, topological noise and sharp features. Applications of fitting quad meshes and polynomial surfaces to input triangular meshes are demonstrated.


Squared distance minimization Centroidal Voronoi tessellation Subdivision surface fitting 



The authors wish to thank Sylvain Lefebvre for a discussion (about an unrelated topic) that inspired this work, Rhaleb Zayer, Xavier Goaoc, Tamy Boubekeur, Yang Liu and Wenping Wang for many discussions, Loic Marechal, Marc Loriot and the AimAtShape repository for data. This project is partly supported by the European Research Council grant GOODSHAPE ERC-StG-205693 and ANR/NSFC (60625202,60911130368) Program (SHAN Project).


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Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  • Vincent Nivoliers
    • 1
    • 2
    Email author
  • Dong-Ming Yan
    • 1
    • 3
  • Bruno Lévy
    • 1
  1. 1.Project ALICE/Institut National de Recherche en Informatique et en Automatique (INRIA) Nancy Grand-Est, LORIANancyFrance
  2. 2.Institut National Polytechnique de Lorraine (INPL)NancyFrance
  3. 3.Geometric Modeling and Scientific Visualization CenterKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

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