摘要
In this paper, we investigate how to compute the minimum distance between a point and a parametric surface, and then to return the nearest point (foot point) on the surface as well as its corresponding parameter, which is also called the point projection problem of a parametric surface. The geometric strategy algorithm (hereafter GSA) presented consists of two parts as follows. The normal curvature to a given parametric surface is used to find the corresponding foot point firstly, and then the Taylor's expansion of the parametric surface is employed to compute parameter increments and to get the iteration formula to calculate the orthogonal projection point of test point to the parametric surface. Our geometric strategy algorithm is essentially dependent on the geometric property of the normal curvature, and performs better than existing methods in two ways. Firstly, GSA converges faster than existing methods, such as the method to turn the problem into a root-finding of nonlinear system, subdividing methods, clipping methods, geometric methods (tangent vector and geometric curvature) and hybrid second-order method, etc. Specially, it converges faster than the classical Newton's iterative method. Secondly, GSA is independent of the initial iterative value, which we prove in Theorem 1. Many numerical examples confirm GSA's robustness and efficiency.
基金
This work is supported by the National Natural Science Foundation of China under Grant No.61263034
the Feature Key Laboratory for Regular Institutions of Higher Education of Guizhou Province of China under Grant No.[2016]003
the Key Laboratory of Advanced Manufacturing Technology of Ministry of Education of China with Guizhou University under Grant No.KY[2018]479
the Training Center for Network Security and Big Data Application of Guizhou Minzu University under Grant No.20161113006
the Shandong Provincial Natural Science Foundation of China under Grant No.ZR2016GM24
the Progress Project for Young Science and Technology Scholars of Guizhou Provincial Department of Education under Grant No.KY[2016]164.