{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:30:43Z","timestamp":1760243443761,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T00:00:00Z","timestamp":1665446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61263034","KY[2016]003","XXPY20050(700212)","GZMUZK[2021]YB20"],"award-info":[{"award-number":["61263034","KY[2016]003","XXPY20050(700212)","GZMUZK[2021]YB20"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Feature Key Laboratory for Regular Institutions of Higher Education of Guizhou Province","award":["61263034","KY[2016]003","XXPY20050(700212)","GZMUZK[2021]YB20"],"award-info":[{"award-number":["61263034","KY[2016]003","XXPY20050(700212)","GZMUZK[2021]YB20"]}]},{"name":"Shandong Youth University of Political Science Doctor Starting Project","award":["61263034","KY[2016]003","XXPY20050(700212)","GZMUZK[2021]YB20"],"award-info":[{"award-number":["61263034","KY[2016]003","XXPY20050(700212)","GZMUZK[2021]YB20"]}]},{"name":"Natural Science Research Project of Guizhou Minzu University","award":["61263034","KY[2016]003","XXPY20050(700212)","GZMUZK[2021]YB20"],"award-info":[{"award-number":["61263034","KY[2016]003","XXPY20050(700212)","GZMUZK[2021]YB20"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Point orthogonal projection onto an algebraic surface is a very important topic in computer-aided geometric design and other fields. However, implementing this method is currently extremely challenging and difficult because it is difficult to achieve to desired degree of robustness. Therefore, we construct an orthogonal polynomial, which is the ninth formula, after the inner product of the eighth formula itself. Additionally, we use the Newton iterative method for the iteration. In order to ensure maximum convergence, two techniques are used before the Newton iteration: (1) Newton\u2019s gradient descent method, which is used to make the initial iteration point fall on the algebraic surface, and (2) computation of the foot-point and moving the iterative point to the close position of the orthogonal projection point of the algebraic surface. Theoretical analysis and experimental results show that the proposed algorithm can accurately, efficiently, and robustly converge to the orthogonal projection point for test points in different spatial positions.<\/jats:p>","DOI":"10.3390\/axioms11100544","type":"journal-article","created":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T22:18:13Z","timestamp":1665526693000},"page":"544","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Application of Orthogonal Polynomial in Orthogonal Projection of Algebraic Surface"],"prefix":"10.3390","volume":"11","author":[{"given":"Xudong","family":"Wang","sequence":"first","affiliation":[{"name":"School of Economics, Capital University of Economics and Business, Beijing 100070, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9923-0011","authenticated-orcid":false,"given":"Xiaowu","family":"Li","sequence":"additional","affiliation":[{"name":"College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China"}]},{"given":"Yuxia","family":"Lyu","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Shandong Youth University of Political Science, Jinan 250100, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1115\/1.2826855","article-title":"Surface curve design by orthogonal projection of space curves onto free-form surfaces","volume":"118","author":"Pegna","year":"1996","journal-title":"J. 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