{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:13:46Z","timestamp":1760242426433,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2017,8,5]],"date-time":"2017-08-05T00:00:00Z","timestamp":1501891200000},"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>To compute the minimum distance between a point and a parametric surface, three well-known first-order algorithms have been proposed by Hartmann (1999), Hoschek, et al. (1993) and Hu, et al. (2000) (hereafter, the First-Order method). In this paper, we prove the method\u2019s first-order convergence and its independence of the initial value. We also give some numerical examples to illustrate its faster convergence than the existing methods. For some special cases where the First-Order method does not converge, we combine it with Newton\u2019s second-order iterative method to present the hybrid second-order algorithm. Our method essentially exploits hybrid iteration, thus it performs very well with a second-order convergence, it is faster than the existing methods and it is independent of the initial value. Some numerical examples confirm our conclusion.<\/jats:p>","DOI":"10.3390\/sym9080146","type":"journal-article","created":{"date-parts":[[2017,8,9]],"date-time":"2017-08-09T06:32:14Z","timestamp":1502260334000},"page":"146","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Hybrid Second-Order Iterative Algorithm for Orthogonal Projection onto a Parametric Surface"],"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":[[2017,8,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/S0167-8396(03)00021-9","article-title":"Point inversion and projection for NURBS curve and surface: Control polygon approach","volume":"20","author":"Ma","year":"2003","journal-title":"Comput. 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