{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T12:42:55Z","timestamp":1776429775983,"version":"3.51.2"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T00:00:00Z","timestamp":1673568000000},"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>Preston\u2019s equation is a general model describing the egg shape of birds. The parameters of Preston\u2019s equation are usually estimated after re-expressing it as the Todd-Smart equation and scaling the egg\u2019s actual length to two. This method assumes that the straight line through the two points on an egg\u2019s profile separated by the maximum distance (i.e., the longest axis of an egg\u2019s profile) is the mid-line. It hypothesizes that the photographed egg\u2019s profile is perfectly bilaterally symmetrical, which seldom holds true because of photographic errors and placement errors. The existing parameter estimation method for Preston\u2019s equation considers an angle of deviation for the longest axis of an egg\u2019s profile from the mid-line, which decreases prediction errors to a certain degree. Nevertheless, this method cannot provide an accurate estimate of the coordinates of the egg\u2019s center, and it leads to sub-optimal parameter estimation. Thus, it is better to account for the possible asymmetry between the two sides of an egg\u2019s profile along its mid-line when fitting egg-shape data. In this paper, we propose a method based on the optimization algorithm (optimPE) to fit egg-shape data and better estimate the parameters of Preston\u2019s equation by automatically searching for the optimal mid-line of an egg\u2019s profile and testing its validity using profiles of 59 bird eggs spanning a wide range of existing egg shapes. We further compared this method with the existing one based on multiple linear regression (lmPE). This study demonstrated the ability of the optimPE method to estimate numerical values of the parameters of Preston\u2019s equation and provide the theoretical egg length (i.e., the distance between two ends of the mid-line of an egg\u2019s profile) and the egg\u2019s maximum breadth. This provides a valuable approach for comparing egg shapes among conspecifics or across different species, or even different classes (e.g., birds and reptiles), in future investigations.<\/jats:p>","DOI":"10.3390\/sym15010231","type":"journal-article","created":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T05:09:32Z","timestamp":1673586572000},"page":"231","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A New Program to Estimate the Parameters of Preston\u2019s Equation, a General Formula for Describing the Egg Shape of Birds"],"prefix":"10.3390","volume":"15","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 Science, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lin","family":"Wang","sequence":"additional","affiliation":[{"name":"Bamboo Research Institute, College of Science, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Brady K.","family":"Quinn","sequence":"additional","affiliation":[{"name":"Biological Effects Section, St. Andrews Biological Station, Fisheries and Oceans Canada, St. Andrews, NB E5B 0E4, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"160","DOI":"10.2307\/4081145","article-title":"The shapes of birds\u2019 eggs","volume":"70","author":"Preston","year":"1953","journal-title":"Auk"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1111\/nyas.14680","article-title":"Egg and math: Introducing a universal formula for egg shape","volume":"1505","author":"Narushin","year":"2021","journal-title":"Ann. 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