An Algorithm for C 1 Surface Approximation with Large Variations

Curve and surface fitting using spline functions from rapidly varying data is a difficult problem. In the bivariate case and without information about the location of the large variations, the usual approximation methods lead to instability phenomenae or undesirable oscillations that can locally and even globally hinder the approximation. So, we propose a new method which uses scale transformations. The originality of the method consists in a pre-processing and a post-processing of the data. Instead of trying to find directly an approximant, we first apply a scale transformation to the z-values of the function. In the particular case of the approximation of surfaces, the originality of the method consists in removing the variations of the unknown function using a scale transformation in pre-processing. And so, the pre-processed data do not have great variations. So, we could use a usual approximant which will not create oscillations. We apply another scale transformation to map the approximant values back to the initial data. Numerical results are given.

Keywords:

• Approximation
• Scale Transformations
• Spline
• Surface Fitting