Journal of Computer Science and Technology ›› 2019, Vol. 34 ›› Issue (6): 1279-1293.doi: 10.1007/s11390-019-1967-z

Special Issue: Artificial Intelligence and Pattern Recognition; Computer Graphics and Multimedia

• Computer Graphics and Multimedia • Previous Articles     Next Articles

A Geometric Strategy Algorithm for Orthogonal Projection onto a Parametric Surface

Xiaowu Li1, Zhinan Wu2, Feng Pan3, Senior Member, CCF, Juan Liang4, Jiafeng Zhang1, Linke Hou5,*   

  1. 1 College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China;
    2 School of Mathematics and Computer Science, Yichun University, Yichun 336000, China;
    3 School of Software Engineering, South China University of Technology, Guangzhou 510006, China;
    4 Department of Science, Taiyuan Institute of Technology, Taiyuan 030008, China;
    5 Center for Economic Research, Shandong University, Jinan 250100, China
  • Received:2019-01-13 Revised:2019-09-04 Online:2019-11-16 Published:2019-11-16
  • Contact: Linke Hou E-mail:abram75@163.com
  • About author:Xiaowu Li received his M.S. degree in computer science from Chongqing University, Chongqing, in 2006, and Ph.D. degree in mathematics science from Chongqing University, Chongqing, in 2011. He is currently a professor of College of Data Science and Information Engineering in Guizhou Minzu University, Guiyang. His research interests include numerical solution of partial differential equation, pattern recognition, computation geometry and computer aided geometric design.
  • Supported by:
    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.

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.

Key words: point projection problem; point inversion problem; normal curvature; normal curvature sphere; convergence analysis;

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