Interpolation with radially symmetric thin plate splines

We present an interpolation method specially adapted to radially symmetric problems: given some tridimensional data (xi, yi, zi)iϵI, we determine the function σ which minimizes |ƒ|2=∫R21⩽x2+y2⩽R22∂2ƒ∂x2(x,y)2 − ∂2ƒ∂x∂y(x,y)2 + ∂2ƒ∂y2(x,y)2dxdy among all functions ƒ which are radially symmetric and satisfy ƒ (xi, Yi) = zi for all i in I. We determine σ by using the general theory of variational splines (or “abstract splines”). We analyse the difference between this function and the one obtained by rotating a univariate cubic spline.