Advances in Computational Mathematics

, Volume 41, Issue 3, pp 489-505

First online:

Convergence of Wachspress coordinates: from polygons to curved domains

  • Jiří KosinkaAffiliated withComputer Laboratory, University of Cambridge Email author 
  • , Michael BartoňAffiliated withNumerical Porous Media Center, King Abdullah University of Science and Technology

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Given a smooth, strictly convex planar domain, we investigate point-wise convergence of the sequence of Wachspress coordinates defined over finer and finer inscribed polygonal approximations of the domain. Based on a relation between the discrete Wachspress case and the limit smooth case given by the Wachspress kernel defined by Warren et al., we show that the corresponding sequences of Wachspress interpolants and mappings converge as 𝓞(h 2) for a sampling step size h of the boundary curve of the domain as h → 0. Several examples are shown to numerically validate the results and to visualise the behaviour of discrete interpolants and mappings as they converge to their smooth counterparts. Empirically, the same convergence order is observed also for mean value coordinates. Moreover, our numerical tests suggest that the convergence of interpolants and mappings is uniform both in the Wachspress and mean value cases.


Barycentric coordinates Barycentric mapping Interpolation Convergence

Mathematics Subject Classification (2010)

65D17 26B10