{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,3]],"date-time":"2026-08-03T21:31:42Z","timestamp":1785792702412,"version":"3.56.0"},"reference-count":19,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,17]],"date-time":"2020-04-17T00:00:00Z","timestamp":1587081600000},"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>Many natural shapes exhibit surprising symmetry and can be described by the Gielis equation, which has several classical geometric equations (for example, the circle, ellipse and superellipse) as special cases. However, the original Gielis equation cannot reflect some diverse shapes due to limitations of its power-law hypothesis. In the present study, we propose a generalized version by introducing a link function. Thus, the original Gielis equation can be deemed to be a special case of the generalized Gielis equation (GGE) with a power-law link function. The link function can be based on the morphological features of different objects so that the GGE is more flexible in fitting the data of the shape than its original version. The GGE is shown to be valid in depicting the shapes of some starfish and plant leaves.<\/jats:p>","DOI":"10.3390\/sym12040645","type":"journal-article","created":{"date-parts":[[2020,4,21]],"date-time":"2020-04-21T05:48:52Z","timestamp":1587448132000},"page":"645","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["The Generalized Gielis Geometric Equation and Its Application"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4696-0130","authenticated-orcid":false,"given":"Peijian","family":"Shi","sequence":"first","affiliation":[{"name":"Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7514-3570","authenticated-orcid":false,"given":"David A.","family":"Ratkowsky","sequence":"additional","affiliation":[{"name":"Tasmanian Institute of Agriculture, University of Tasmania, Private Bag 98, Hobart 7001, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4536-3839","authenticated-orcid":false,"given":"Johan","family":"Gielis","sequence":"additional","affiliation":[{"name":"Department of Biosciences Engineering, University of Antwerp, B-2020 Antwerp, Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"333","DOI":"10.3732\/ajb.90.3.333","article-title":"A general geometric transformation that unifies a wide range of natural and abstract shapes","volume":"90","author":"Gielis","year":"2003","journal-title":"Am. J. Bot."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Gielis, J. (2017). The Geometrical Beauty of Plants, Atlantis Press.","DOI":"10.2991\/978-94-6239-151-2"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"856","DOI":"10.3389\/fpls.2015.00856","article-title":"Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape","volume":"6","author":"Shi","year":"2015","journal-title":"Front. 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Forest Res.","DOI":"10.1007\/s11676-018-0770-y"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Shi, P.J., Liu, M.D., Yu, X.J., Gielis, J., and Ratkowsky, D.A. (2019). Proportional relationship between leaf area and the product of leaf length and width of four types of special leaf shapes. Forests, 10.","DOI":"10.3390\/f10020178"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1186\/1687-2770-2013-253","article-title":"Fourier-Hankel solution of the Robin problem for the Helmholtz equation in supershaped annular domains","volume":"2013","author":"Caratelli","year":"2013","journal-title":"Bound. Value Probl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s00022-015-0269-z","article-title":"Gielis\u2019 superformula and regular polygons","volume":"106","author":"Matsuura","year":"2015","journal-title":"J. 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