Spline surface fitting using normal data and norm-like functions

This paper is motivated by the geometry reconstruction process for aircraft engines. In order to improve the overall smoothness of the resulting spline surface, we consider the simultaneous approximation of point and normal data. If the normal data to be approximated by one patch is taken from the boundary of its neighbors, this controls the behavior of the resulting spline patch along the boundary and ensures approximate G1-smoothness of the composite surface. We show that for every mesh size there exists a solution to the resulting optimization problem. Optimal convergence is achieved based on an appropriate choice of the weight controlling the relative influence of points and normals, taking the distinct approximation order of splines for points and derivatives into account. In addition we investigate the effect of using norm-like functions for measuring the errors.