# Changes between Version 127 and Version 128 of UvmatHelp

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Timestamp:
Dec 8, 2014, 12:47:37 AM (9 years ago)
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 v127 -'''Thin plate shell (tps) interpolation:''' This is a multi-dimensional generalisation of the spline interpolation/smoothing, an optimum way to interpolate data with minimal  curvature of the interpolating function. The result at an interpolation position vector ${\bf r}$ is expressed in the form, (see http://coriolis.legi.grenoble-inp.fr/spip.php?article73) This is a multi-dimensional generalisation of the spline interpolation/smoothing, an optimum way to interpolate data with minimal  curvature of the interpolating function. The result at an interpolation position vector ${\bf r}$ is expressed in the form, (see ThinPlateShell) $$\label{sol_gene} f({\bf r})=\sum S_i \phi({\bf|r-r_i}|)+a_0+a_1x+a_2y\;$$ where ${\bf r_i}$ are the positions of the measurement points (the ''centres''). Each ''centre'' can be viewed as the source of an axisymmetric field  $\phi$ of the form $\phi(r)=r^2\log (r)$. The weights $S_i$ and the linar coefficients $a_0,a_1,a_2$ are the thin plate shell (tps) coefficients which determine the interpolated value at any point. The spatial derivatives are similarly obtained at any point by analytical differentiation of the source functions $\phi(r)$. These tps weights, with the corresponding centre coordinates, therefore contain all the information needed for interpolation at any point. We call that a ''tps field representation''.^