Constructive Approximation

, Volume 49, Issue 1, pp 175–190 | Cite as

Ambient Approximation on Hypersurfaces

  • N. Lehmann
  • L.-B. MaierEmail author
  • S. Odathuparambil
  • U. Reif


We present a novel general approach for solving approximation problems on hypersurfaces drawing on standard approximation techniques in the ambient space. The method is capable of transferring approximation orders from the ambient space to the hypersurface under mild conditions. When based on tensor product B-splines of order n and knot spacing h, the resulting approximant is \(C^{n-2}\), while the error decays at the optimal rate \(h^n\). Further, the method is easy to implement and, when applying quasi interpolation techniques, optimally efficient as the computational expense is linear in the number of spline coefficients. Applications include the representation of smooth surfaces of arbitrary genus.


Spline approximation Manifold Closest point representation Multivariate spline Quasi-interpolation Implicit surface Data compression Local approximation order 

Mathematics Subject Classification

41A15 41A63 65D07 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • N. Lehmann
    • 1
  • L.-B. Maier
    • 1
    Email author
  • S. Odathuparambil
    • 1
  • U. Reif
    • 1
  1. 1.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

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