Creating A Multidimensional Spline
Given the constraint from the definition formula we can conclude that the algorithm doesn't work for any sets of data. If we plan to use this algorithm for random points in a multidimensional space we need to find a solution to give as input to the algorithm sets of data where these constraints are met. A solution for this is to introduce a parameter so that the input data would be represented as single-valued functions depending on that parameter; after this the smoothing will be performed for each function. In a bidimensional space a solution would be to parametrize and so that they would become and where . A convenient solution for is the cumulating distance where .
A more detailed analysis on parametrization is done by E.T.Y Lee.
Read more about this topic: Smoothing Spline
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