{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T18:11:59Z","timestamp":1777918319818,"version":"3.51.4"},"reference-count":29,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,10]],"date-time":"2022-05-10T00:00:00Z","timestamp":1652140800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12001079"],"award-info":[{"award-number":["12001079"]}]},{"name":"National Natural Science Foundation of China","award":["11801053"],"award-info":[{"award-number":["11801053"]}]},{"name":"National Natural Science Foundation of China","award":["3132022205"],"award-info":[{"award-number":["3132022205"]}]},{"name":"National Natural Science Foundation of China","award":["LJKZ0053"],"award-info":[{"award-number":["LJKZ0053"]}]},{"name":"National Natural Science Foundation of China","award":["GH19164"],"award-info":[{"award-number":["GH19164"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["12001079"],"award-info":[{"award-number":["12001079"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["11801053"],"award-info":[{"award-number":["11801053"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["3132022205"],"award-info":[{"award-number":["3132022205"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["LJKZ0053"],"award-info":[{"award-number":["LJKZ0053"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["GH19164"],"award-info":[{"award-number":["GH19164"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Scientific Research Foundation of Liaoning Education Department","award":["12001079"],"award-info":[{"award-number":["12001079"]}]},{"name":"Scientific Research Foundation of Liaoning Education Department","award":["11801053"],"award-info":[{"award-number":["11801053"]}]},{"name":"Scientific Research Foundation of Liaoning Education Department","award":["3132022205"],"award-info":[{"award-number":["3132022205"]}]},{"name":"Scientific Research Foundation of Liaoning Education Department","award":["LJKZ0053"],"award-info":[{"award-number":["LJKZ0053"]}]},{"name":"Scientific Research Foundation of Liaoning Education Department","award":["GH19164"],"award-info":[{"award-number":["GH19164"]}]},{"name":"Jilin Province education science \u201c13th five-year\u201d planning project","award":["12001079"],"award-info":[{"award-number":["12001079"]}]},{"name":"Jilin Province education science \u201c13th five-year\u201d planning project","award":["11801053"],"award-info":[{"award-number":["11801053"]}]},{"name":"Jilin Province education science \u201c13th five-year\u201d planning project","award":["3132022205"],"award-info":[{"award-number":["3132022205"]}]},{"name":"Jilin Province education science \u201c13th five-year\u201d planning project","award":["LJKZ0053"],"award-info":[{"award-number":["LJKZ0053"]}]},{"name":"Jilin Province education science \u201c13th five-year\u201d planning project","award":["GH19164"],"award-info":[{"award-number":["GH19164"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we consider Frenet type framed base curves that may have singular points and define one-parameter developable surfaces associated with such curves. By using the singularity theory, we classify the generic singularities of the developable surfaces, which are cuspidal edges and swallowtails. In order to characterize these singularities, two geometric invariants are discovered. At last, an example is given to demonstrate the main results.<\/jats:p>","DOI":"10.3390\/sym14050975","type":"journal-article","created":{"date-parts":[[2022,5,10]],"date-time":"2022-05-10T21:52:11Z","timestamp":1652219531000},"page":"975","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Geometry of Developable Surfaces of Frenet Type Framed Base Curves from the Singularity Theory Viewpoint"],"prefix":"10.3390","volume":"14","author":[{"given":"Qiming","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Mathematics, Jilin University of Finance and Economics, Changchun 130117, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lin","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Science, Dalian Maritime University, Dalian 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yongqiao","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Dalian Maritime University, Dalian 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cad.2019.05.001","article-title":"Multi-strip smooth developable surfaces from sparse design curves","volume":"114","author":"Bo","year":"2019","journal-title":"Comput. 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