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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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References

  1. Ma Y L, Hewitt W T. Point inversion and projection for NURBS curve and surface: Control polygon approach. Computer Aided Geometric Design, 2003, 20(2): 79-99.

    Article  MathSciNet  Google Scholar 

  2. Yang H P, Wang W P, Sun J G. Control point adjustment for B-spline curve approximation. Computer-Aided Design, 2004, 36(7): 639-652.

    Article  Google Scholar 

  3. Johnson D E, Cohen E. A framework for efficient minimum distance computations. In Proc. the 1998 IEEE International Conference on Robotics & Automation, May 1998, pp.3678-3684.

  4. Piegl L, Tiller W. Parametrization for surface fitting in reverse engineering. Computer-Aided Design, 2001, 33(8): 593-603.

    Article  Google Scholar 

  5. Pegna J, Wolter F E. Surface curve design by orthogonal projection of space curves onto free-form surfaces. Journal of Mechanical Design, 1996, 118: 45-52.

    Article  Google Scholar 

  6. Besl P J, McKay N D. A method for registration of 3-D shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1992, 14(2): 239-256.

    Article  Google Scholar 

  7. Zhou J M, Sherbrooke E C, Patrikalakis N. Computation of stationary points of distance functions. Engineering with Computers, 1993, 9(4): 231-246.

    Article  Google Scholar 

  8. Limaien A, Trochu F. Geometric algorithms for the intersection of curves and surfaces. Computers & Graphics, 1995, 19(3): 391-403.

    Article  Google Scholar 

  9. Polak E, Royset J O. Algorithms with adaptive smoothing for finite minimax problems. Journal of Optimization: Theory and Applications, 2003, 119(3): 459-484.

    MathSciNet  MATH  Google Scholar 

  10. Patrikalakis N, Maekawa T. Shape Interrogation for Computer Aided Design and Manufacturing (1st edition). Springer, 2002.

  11. Johnson D E, Cohen E. Distance extrema for spline models using tangent cones. In Proc. the 2005 Conference on Graphics Interface, May 2005, pp.169-175.

  12. Selimovic I. Improved algorithms for the projection of points on NURBS curves and surfaces. Computer Aided Geometric Design, 2006, 23(5): 439-445.

    Article  MathSciNet  Google Scholar 

  13. Cohen E, Lyche T, Riesebfeld R. Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics. Computer Graphics and Image Processing, 1980, 14(2): 87-111.

    Article  Google Scholar 

  14. Piegl L, Tiller W. The NURBS Book. Springer, 1995.

  15. Elber G, Kim M S. Geometric constraint solver using multivariate rational spline functions. In Proc. the 6th ACM Symposiumon Solid Modeling and Applications, June 2001, pp.1-10.

  16. Park C H, Elber G, Kim K J, Kim G Y, Seong J K. A hybrid parallel solver for systems of multivariate polynomials using CPUs and GPUs. Computer-Aided Design, 2011, 43(11): 1360-1369.

    Article  Google Scholar 

  17. Bartoň M. Solving polynomial systems using no-root elimination blending schemes. Computer-Aided Design, 2011, 43(12): 1870-1878.

    Article  Google Scholar 

  18. van Sosin B, Elber G. Solving piecewise polynomial constraint systems with decomposition and a subdivision-based solver. Computer-Aided Design, 2017, 90: 37-47.

    Article  MathSciNet  Google Scholar 

  19. Bartoň M, Elber G, Hanniel I. Topologically guaranteed univariate solutions of underconstrained polynomial systems via no-loop and single-component tests. Computer-Aided Design, 2011, 43(8): 1035-1044.

    Article  Google Scholar 

  20. Chen X D, Yong J H, Wang G Z, Paul J C, Xu G. Computing the minimum distance between a point and a NURBS curve. Computer-Aided Design, 2008, 40(10/11): 1051-1054.

    Article  Google Scholar 

  21. Chen X D, Xu G, Yong J H, Wang G Z, Paul J C. Computing the minimum distance between a point and a clamped B-spline surface. Graphical Models, 2009, 71(3): 107-112.

    Article  Google Scholar 

  22. Oh Y T, Kim Y J, Lee J, Kim M S, Elber G. Efficient point-projection to freeform curves and surfaces. Computer Aided Geometric Design, 2012, 29(5): 242-254.

    Article  MathSciNet  Google Scholar 

  23. Liu X M, Yang L, Yong J H, Gu H J, Sun J G. A torus patch approximation approach for point projection on surfaces. Computer Aided Geometric Design, 2009, 26(5): 593-598.

    Article  MathSciNet  Google Scholar 

  24. Hu S M, Wallner J. A second-order algorithm for orthogonal projection onto curves and surfaces. Computer Aided Geometric Design, 2005, 22: 251-260.

    Article  MathSciNet  Google Scholar 

  25. Li X, Wu Z, Hou L, Wang L, Yue C, Xin Q. A geometric orthogonal projection strategy for computing the minimum distance between a point and a spatial parametric curve. Algorithms, 2016, 9(1): Article No. 15.

    Article  MathSciNet  Google Scholar 

  26. Li X, Wang L, Wu Z, Hou L, Liang J, Li Q. Convergence analysis on a second-order algorithm for orthogonal projection onto curves. Symmetry, 2017, 9(10): Article No. 210.

    Article  Google Scholar 

  27. Hartmann E. On the curvature of curves and surfaces de- fined by normalforms. Computer Aided Geometric Design, 1999, 16(5): 355-376.

    Article  MathSciNet  Google Scholar 

  28. Hoschek J, Lasser D. Fundamentals of Computer Aided Geometric Design (1st edition). A K Peters/CRC Press, 1996.

  29. Hu S M, Sun J G, Jin T G, Wang G Z. Computing the parameter of points on Nurbs curves and surfaces via moving affine frame method. J. Software, 2000, 11(1): 49-53. (in Chinese).

    Google Scholar 

  30. Liang J, Hou L, Li X, Pan F, Cheng T, Wang L. Hybrid second-order method for orthogonal projection onto parametric curve in n-dimensional Euclidean space. Mathematics, 2018, 6(12): Article No. 306.

    Article  Google Scholar 

  31. Li X, Wang L, Wu Z, Hou L, Liang J, Li Q. Hybrid second- order iterative algorithm for orthogonal projection onto a parametric surface. Symmetry, 2017, 9(8): Article No. 146.

    Article  MathSciNet  Google Scholar 

  32. Li X, Pan F, Cheng T, Wu Z, Liang J, Hou L. Integrated hybrid second order algorithm for orthogonal projection onto a planar implicit curve. Symmetry, 2018, 10(5): Article No. 164.

    Article  Google Scholar 

  33. Ko K H, Sakkalis T. Orthogonal projection of points in CAD/CAM applications: An overview. Journal of Computational Design and Engineering, 2014, 1(2): 116-127.

    Article  Google Scholar 

  34. Wang X P, Zhang W Z, Huang X. Computation of point in-version and ray-surface intersection through tracing along the base surface. The Visual Computer, 2015, 31(11): 1487-1500.

    Article  Google Scholar 

  35. Piegl L A. Ten challenges in computer-aided design. Computer-Aided Design, 2005, 37(4): 461-470.

    Article  Google Scholar 

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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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