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A Geometric Strategy Algorithm for Orthogonal Projection onto a Parametric Surface

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Abstract

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.

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Authors and Affiliations

  1. College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, China

    Xiaowu Li & Jiafeng Zhang

  2. School of Mathematics and Computer Science, Yichun University, Yichun, 336000, China

    Zhinan Wu

  3. School of Software Engineering, South China University of Technology, Guangzhou, 510006, China

    Feng Pan

  4. Department of Science, Taiyuan Institute of Technology, Taiyuan, 030008, China

    Juan Liang

  5. Center for Economic Research, Shandong University, Jinan, 250100, China

    Linke Hou

Authors
  1. Xiaowu Li
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  2. Zhinan Wu
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  3. Feng Pan
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  4. Juan Liang
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  5. Jiafeng Zhang
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  6. Linke Hou
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Correspondence to Linke Hou.

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Li, X., Wu, Z., Pan, F. et al. A Geometric Strategy Algorithm for Orthogonal Projection onto a Parametric Surface. J. Comput. Sci. Technol. 34, 1279–1293 (2019). https://doi.org/10.1007/s11390-019-1967-z

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  • DOI: https://doi.org/10.1007/s11390-019-1967-z

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