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IMA Journal of Numerical Analysis Advance Access originally published online on October 26, 2007
IMA Journal of Numerical Analysis 2008 28(4):785-805; doi:10.1093/imanum/drm033
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Interpolation and scattered data fitting on manifolds using projected Powell–Sabin splines

Oleg Davydov{dagger}

Department of Mathematics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK

Larry L. Schumaker

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

{dagger} Email: oleg.davydov{at}strath.ac.uk

Received on 4 April 2007. Accepted for publication 31 August 2007.


  Abstract

We present methods either for interpolating data or for fitting scattered data on a 2D smooth manifold {Omega}. The methods are based on a local bivariate Powell–Sabin interpolation scheme, and make use of a family of charts {(U{xi}, {phi}{xi})}{xi} isin {Omega} satisfying certain conditions of smooth dependence on {xi}. If {Omega} is a C2-manifold embedded into R3, then projections into tangent planes can be employed. The data-fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function.

Key Words: interpolation; scattered data fitting; data on surfaces and manifolds; Powell-Sabin spline

Dedicated to Prof. M. J. D. Powell on the occasion of his 70th birthday.


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