Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n-Dimensional Euclidean Space

Abstract

Orthogonal projection a point onto a parametric curve, three classic first order algorithms have been presented by Hartmann (1999), Hoschek, et al. (1993) and Hu, et al. (2000) (hereafter, H-H-H method). In this research, we give a proof of the approach’s first order convergence and its non-dependence on the initial value. For some special cases of divergence for the H-H-H method, we combine it with Newton’s second order method (hereafter, Newton’s method) to create the hybrid second order method for orthogonal projection onto parametric curve in an n-dimensional Euclidean space (hereafter, our method). Our method essentially utilizes hybrid iteration, so it converges faster than current methods with a second order convergence and remains independent from the initial value. We provide some numerical examples to confirm robustness and high efficiency of the method.

Point Orthogonal Projection onto a Spatial Algebraic Curve

“…Hartmann [2] proposed a first-order tangent line method to calculate point orthogonal projection onto parametric curve and surface. For a few cases of non-convergence and as supplement and perfection of the first-order tangent line method [2], Liang et al [3] and Li et al [4] presented hybrid second-order method for orthogonal projection onto parametric curve and surface, respectively. Hu et al [5] proposed a second-order geometric iterative algorithm with curvature information to approximate the orthogonal projection point of the given point on parametric curves and surfaces.…”