{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T03:27:13Z","timestamp":1752550033449,"version":"3.37.3"},"reference-count":22,"publisher":"Wiley","license":[{"start":{"date-parts":[[2023,7,25]],"date-time":"2023-07-25T00:00:00Z","timestamp":1690243200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009392","name":"Prince Sattam bin Abdulaziz University","doi-asserted-by":"publisher","award":["PSAU\/2023\/R\/1444"],"award-info":[{"award-number":["PSAU\/2023\/R\/1444"]}],"id":[{"id":"10.13039\/100009392","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Computational Intelligence and Soft Computing"],"published-print":{"date-parts":[[2023,7,25]]},"abstract":"<jats:p>There are substantial methods of degree reduction in the literature. Existing methods share some common limitations, such as lack of geometric continuity, complex computations, and one-degree reduction at a time. In this paper, an approximate geometric multidegree reduction algorithm of Wang\u2013Ball curves is proposed. <jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                        <a:msup>\n                           <a:mrow>\n                              <a:mi>G<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mn>0<\/a:mn>\n                           <\/a:mrow>\n                        <\/a:msup>\n                     <\/a:math>\n                  <\/jats:inline-formula>-, <jats:inline-formula>\n                     <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                        <c:msup>\n                           <c:mrow>\n                              <c:mi>G<\/c:mi>\n                           <\/c:mrow>\n                           <c:mrow>\n                              <c:mn>1<\/c:mn>\n                           <\/c:mrow>\n                        <\/c:msup>\n                     <\/c:math>\n                  <\/jats:inline-formula>-, and <jats:inline-formula>\n                     <e:math xmlns:e=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                        <e:msup>\n                           <e:mrow>\n                              <e:mi>G<\/e:mi>\n                           <\/e:mrow>\n                           <e:mrow>\n                              <e:mn>2<\/e:mn>\n                           <\/e:mrow>\n                        <\/e:msup>\n                     <\/e:math>\n                  <\/jats:inline-formula>-continuity conditions are applied in the degree reduction process to preserve the boundary control points. The general equation for high-order (G2 and above) multidegree reduction algorithms is nonlinear, and the solutions of these nonlinear systems are quite expensive. In this paper, <jats:inline-formula>\n                     <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <g:msup>\n                           <g:mrow>\n                              <g:mi>C<\/g:mi>\n                           <\/g:mrow>\n                           <g:mrow>\n                              <g:mn>1<\/g:mn>\n                           <\/g:mrow>\n                        <\/g:msup>\n                     <\/g:math>\n                  <\/jats:inline-formula>-continuity conditions are imposed besides the <jats:inline-formula>\n                     <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <i:msup>\n                           <i:mrow>\n                              <i:mi>G<\/i:mi>\n                           <\/i:mrow>\n                           <i:mrow>\n                              <i:mn>2<\/i:mn>\n                           <\/i:mrow>\n                        <\/i:msup>\n                     <\/i:math>\n                  <\/jats:inline-formula>-continuity conditions. While some existing methods only achieve the multidegree reduction by repeating the one-degree reduction method recursively, our proposed method achieves multidegree reduction at once. The distance between the original curve and the degree-reduced curve is measured with the <jats:inline-formula>\n                     <k:math xmlns:k=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                        <k:msub>\n                           <k:mrow>\n                              <k:mi>L<\/k:mi>\n                           <\/k:mrow>\n                           <k:mrow>\n                              <k:mn>2<\/k:mn>\n                           <\/k:mrow>\n                        <\/k:msub>\n                     <\/k:math>\n                  <\/jats:inline-formula>-norm. Numerical example and figures are presented to state the adequacy of the algorithm. The proposed method not only outperforms the existing method of degree reduction of Wang\u2013Ball curves but also guarantees geometric continuity conditions at the boundary points, which is very important in CAD and geometric modeling.<\/jats:p>","DOI":"10.1155\/2023\/5483111","type":"journal-article","created":{"date-parts":[[2023,7,25]],"date-time":"2023-07-25T18:35:33Z","timestamp":1690310133000},"page":"1-10","source":"Crossref","is-referenced-by-count":2,"title":["Geometric Degree Reduction of Wang\u2013Ball Curves"],"prefix":"10.1155","volume":"2023","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9319-8989","authenticated-orcid":true,"given":"Yusuf Fatihu","family":"Hamza","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sa\u2019adatu Rimi College of Education Kumbotso, Kano, Nigeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7111-5767","authenticated-orcid":true,"given":"Mukhtar Fatihu","family":"Hamza","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, College of Engineering in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2896-2290","authenticated-orcid":true,"given":"Abedallah","family":"Rababah","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, United Arab Emirates University, Al Ain, UAE"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1467-5426","authenticated-orcid":true,"given":"Salisu","family":"Ibrahim","sequence":"additional","affiliation":[{"name":"Department of Mathematics Education, Tishk International University, Erbil, Iraq"}]}],"member":"311","reference":[{"issue":"1","key":"1","first-page":"126","article-title":"Ball curve of high degree and its geometric properties","volume":"2","author":"G. Wang","year":"1987","journal-title":"Applied Mathematics: A Journal of Chinese Universities"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4485(74)90009-8"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4485(75)90068-8"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4485(77)90056-2"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1145\/77269.77275"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.3390\/sym11101242"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.3390\/sym11010015"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.3390\/math9182212"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-021-01499-0"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2022.12.001"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2022.01.018"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1007\/s10462-023-10416-4"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-8396(02)00093-6"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2009.01.006"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1155\/2016\/8140427"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinam.2022.100280"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2003.10.004"},{"issue":"1","key":"18","first-page":"12","article-title":"Weighted G0- and G1-degree reduction of disk B\u00e9zier curves","volume":"8","author":"A. Rababah","year":"2016","journal-title":"Journal of informatics and Mathematical Sciences"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4485(95)00047-X"},{"issue":"5","key":"20","first-page":"1045","article-title":"Explicit multi-degree reduction of said-b\u00e9zier generalized ball curves with endpoints constraints","volume":"4","author":"Q. Hu","year":"2007","journal-title":"Journal of Information and Computational Science"},{"key":"21","doi-asserted-by":"crossref","first-page":"959","DOI":"10.12733\/jics20101098","article-title":"Multiple degree reduction of wang-ball curves by using dual basis polynomials","volume":"10","author":"Z. Dong","year":"2013","journal-title":"Journal of Information and Computational Science"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1186\/s13660-015-0833-y"}],"container-title":["Applied Computational Intelligence and Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/acisc\/2023\/5483111.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/acisc\/2023\/5483111.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/acisc\/2023\/5483111.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,25]],"date-time":"2023-07-25T18:35:39Z","timestamp":1690310139000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/acisc\/2023\/5483111\/"}},"subtitle":[],"editor":[{"given":"Dimitrios A.","family":"Karras","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2023,7,25]]},"references-count":22,"alternative-id":["5483111","5483111"],"URL":"https:\/\/doi.org\/10.1155\/2023\/5483111","relation":{},"ISSN":["1687-9732","1687-9724"],"issn-type":[{"type":"electronic","value":"1687-9732"},{"type":"print","value":"1687-9724"}],"subject":[],"published":{"date-parts":[[2023,7,25]]}}}