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Article

Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves

by
Xiaowu Li
1,†,
Lin Wang
1,†,
Zhinan Wu
2,†,
Linke Hou
3,*,†,
Juan Liang
4,*,† and
Qiaoyang Li
5,*,†
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
Center for Economic Research, Shandong University, Jinan 250100, China
4
Department of Science, Taiyuan Institute of Technology, Taiyuan 030008, China
5
Center for the World Ethnic Studies, Guizhou Minzu University, Guiyang 550025, China
*
Authors to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Symmetry 2017, 9(10), 210; https://doi.org/10.3390/sym9100210
Submission received: 11 August 2017 / Revised: 6 September 2017 / Accepted: 27 September 2017 / Published: 1 October 2017
(This article belongs to the Special Issue Advances in Computer Graphics, Geometric Modeling, and Virtual and Augmented Reality)
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Abstract

Regarding the point projection and inversion problem, a classical algorithm for orthogonal projection onto curves and surfaces has been presented by Hu and Wallner (2005). The objective of this paper is to give a convergence analysis of the projection algorithm. On the point projection problem, we give a formal proof that it is second order convergent and independent of the initial value to project a point onto a planar parameter curve. Meantime, for the point inversion problem, we then give a formal proof that it is third order convergent and independent of the initial value.
Keywords: point projection; planar parametric curve; second order convergence; third order convergence point projection; planar parametric curve; second order convergence; third order convergence

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MDPI and ACS Style

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, 210. https://doi.org/10.3390/sym9100210

AMA Style

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):210. https://doi.org/10.3390/sym9100210

Chicago/Turabian Style

Li, Xiaowu, Lin Wang, Zhinan Wu, Linke Hou, Juan Liang, and Qiaoyang Li. 2017. "Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves" Symmetry 9, no. 10: 210. https://doi.org/10.3390/sym9100210

APA Style

Li, X., Wang, L., Wu, Z., Hou, L., Liang, J., & Li, Q. (2017). Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves. Symmetry, 9(10), 210. https://doi.org/10.3390/sym9100210

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