{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:15:47Z","timestamp":1760242547480,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2017,10,1]],"date-time":"2017-10-01T00:00:00Z","timestamp":1506816000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/sym9100210","type":"journal-article","created":{"date-parts":[[2017,10,2]],"date-time":"2017-10-02T13:10:05Z","timestamp":1506949805000},"page":"210","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9923-0011","authenticated-orcid":false,"given":"Xiaowu","family":"Li","sequence":"first","affiliation":[{"name":"College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lin","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhinan","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Yichun University, Yichun 336000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Linke","family":"Hou","sequence":"additional","affiliation":[{"name":"Center for Economic Research, Shandong University, Jinan 250100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juan","family":"Liang","sequence":"additional","affiliation":[{"name":"Department of Science, Taiyuan Institute of Technology, Taiyuan 030008, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiaoyang","family":"Li","sequence":"additional","affiliation":[{"name":"Center for the World Ethnic Studies, Guizhou Minzu University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,10,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/0097-8493(95)00009-2","article-title":"Geometric algorithms for the intersection of curves and surfaces","volume":"19","author":"Limaiem","year":"1995","journal-title":"Comput. 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