Weight Selection of Geometric Iterative Method for Curve and Surface Approximation

The geometric iterative method approximates the target curve (surface) by minimizing the distance between given data points and corresponding points on the approximate curve (surface). To address the issue of selecting weights for error vectors that affect the convergence and convergence rate in geometric iterative methods, this study takes the local approximation geometric iterative method as an example and proposes three types of weight selection methods for the progressive iterative process. Firstly, the range of weight values is analyzed to satisfy the convergence conditions of the algorithm. Then, different weighting methods are proposed based on the theoretical fastest rate, norm inequality of matrix eigenvalues, and the FFD(free-form deformation) method. To improve computational efficiency, a method of fixing weights using the initialized configuration matrix is also proposed. Finally, the performance of the weighting methods is evaluated through curve and surface approximation examples with different numbers of data points and control points. The experimental results indicate that, under the same number of iterations, the theoretical fastest rate and  norm or  norm weighting methods achieve smaller average errors in curve approximation, while the FFD-based method demonstrates smaller average errors in surface approximation, both exhibiting faster convergence rates. Under the same error threshold, the fixed weighting method takes less time, significantly improving the computational efficiency.