Hybrid second-order iterative algorithm for orthogonal projection onto a parametric surface

Abstract

To compute the minimum distance between a point and a parametric surface, three well-known first-order algorithms have been proposed by Hartmann (1999), Hoschek, et al. (1993) and Hu, et al. (2000) (hereafter, the First-Order method). In this paper, we prove the method's first-order convergence and its independence of the initial value. We also give some numerical examples to illustrate its faster convergence than the existing methods. For some special cases where the First-Order method does not converge, we combine it with Newton's second-order iterative method to present the hybrid second-order algorithm. Our method essentially exploits hybrid iteration, thus it performs very well with a second-order convergence, it is faster than the existing methods and it is independent of the initial value. Some numerical examples confirm our conclusion.